Unfolding Experiments

by Mohammad Ghomi

Initialization

Namiki and Fukuda's "UnfoldPolytope" Package with some modification

(*

        Unfolding  Convex Polytopes

             Version 1.0 Beta
              February, 1992

           Copyright (c) 1992 by
      Makoto Namiki and Komei Fukuda
      
This package contains Mathematica implementations
of unfolding 3-dimsional polytope.

This package is copyright 1992 by Makoto Namiki.
This package may be copied in its entirety for 
nonprofit purposes only. Sale, other than for
the direct cost of the media, is prohibited.  
This copyright notice must accompany all copies.

The authors make no representations, express or 
implied, with respond to this documentation, of 
the software it describes and contains, including
without limitations, any implied warranties of 
mechantability or fitness for a particular purpose,
all of which are expressly disclaimed.  The authors
shall in no event be liable for any indirect,
incidental, or consequential damages.

This beta release is designed to run under 
Version 1.2 & 2.0 of Mathematica. Any comments, 
bug reports, or requests to get on the 
UnfoldPolytope mailing list should be 
forwarded to:
  
   Makoto Namiki
   Department of Social Science,
   College of Arts and Sciences, University of Tokyo,
   3-8-1, Komaba, Meguro-ku, Tokyo, 153, Japan
   +81-3-3467-1171 ext. 245 or 244,
   namiki@is.titech.ac.jp
   
   Komei Fukuda
   Graduate School of Systems Management,
   The University of Tsukuba,
   3-29-1, Otsuka, Bunkyo-ku, Tokyo, 112, Japan
   +81-3-3942-6876
   fukuda@gssm.otsuka.tsukuba.ac.jp   	 

*)

BeginPackage["UnfoldPolytope`"]

UnfoldPolytope::usage="UnfoldPolytope[facets] gives graphics of unfoldings of a polytope which is given as a list of facets.";

OrderFacets::usage="OrderFacets[facets] returns a correctly order lists of facets.";

ProperSubsetQ::usage="ProperSubsetQ[t,s] returns True if t is a proper subset of s.";

MakeEdgesFromFacets::usage="MakeEdgesFromFacets[faces] returns a cordinats of lower faces and a adjacency of faces.";

MakeFacetsFromZerosets::usage="MakeFacetsFromZerosets[v,z] returns a list of facets consisting of cordinates of vertices, where v denotes a list of vertices and z denotes a list of zerosets.";
 
MakeAdjTable::usage="MakeAdjTable[n,l] or MakeAdjTable[n,l,costs] return an adjacency table where l denotes a list of unordered pairs consist of {1,2,...,n} and a costs denotes an adjacency cost.";

BreadthFirstSearch::usage="BreadthFirstSearch[adj_list,n] gives a incidence matrix representing a tree of a graph obtained by breadth first search from a vertex n, where this graph is given as a incidence matrix of adj_list."; 

DepthFirstSearch::usage="DepththFirstSearch[adj_list,n] gives a incidence matrix representing a tree of a graph obtained by depth first search from a vertex n, where this graph is given as a incidence matrix of adj_list."; 

SelectLeaf::usage="SelectLeaf[adj_Mat] returns {i,j} such that node i is a child of node j for adjacent matrix adj_Mat.";

ProjectVertex::usage="ProjectVetex[v1,v2,v3,v4] returns coordinates which is a projection of v4 onto affine space {v1,v2,v3} around the line {v1,v2}.";

ProjectFaces::usage="ProjectFaces[child,parent,faces,vervect] argument child is a list of faces, parent is a face, and faces is a list of all faces. This procedure project faces of child onto a face of parent by using ProjectVertices.";

Unfold::usage="Unfold[sptree,tree,maxf] open one of a leaf of Tree of Facet.";

Zonotope::usage="Zonotope[vlist] returns a list extreme points of zonotope which is defined by vlist, a list of vectors."; 

VerticalVector::usage="VerticalVector[maxf_List] returns a list of Vertical vector of each face.";

CenterOfTree::usage="Test dayon.";

Options[UnfoldPolytope] = { InternalGraphics -> False, ExactComputation -> False, OutputFile  -> "/dev/null", Tree -> "BFST", Embedding->False}
 
Begin["`private`"]

UnfoldPolytope[facets_List,opts___Rule]:=
	Block[{odfacets,edg,faAdj,vertices,veAdj,cent,xmin,xmax,ymin,ymax,
		t,tree,cotree,sptree,i,tr,vervec,optig,optec,OutputFile,Tree,emb},
		optig = InternalGraphics /. {opts} /. Options[UnfoldPolytope];
		optec = ExactComputation /. {opts} /. Options[UnfoldPolytope];
    	OutputFile = OutputFile /. {opts} /. Options[UnfoldPolytope];
    	Tree = Tree /. {opts} /. Options[UnfoldPolytope];
    	emb = Embedding /. {opts} /. Options[UnfoldPolytope];
    	odfacets = facets;
		{edg,faAdj} = MakeEdgesFromFacets[odfacets];
		vertices = Union[Flatten[facets,1]];
		veAdj = Flatten[{Position[vertices,#[[1]]],Position[vertices,#[[2]]]}]& /@ edg;
		If[Tree == "DFST",t = DepthFirstSearch[MakeAdjTable[Length[vertices],veAdj],1],
			t = BreadthFirstSearch[MakeAdjTable[Length[vertices],veAdj],1];
		];
		tree = Flatten[Position[veAdj,#]& /@ Position[t,1]];
		cotree = Complement[Table[i,{i,1,Length[edg]}],tree];
		sptree = MakeAdjTable[Length[odfacets],faAdj[[cotree]]];
		tr = Table[{i},{i,Length[facets]}];
		vervec = VerticalVector[odfacets];
		cent = CenterOfTree[sptree];
		If[optig,Show[Graphics3D[Polygon /@ odfacets],Boxed->False];];
		Do[{sptree,tr,odfacets} = Unfold[sptree,tr,odfacets,vervec,cent];
			If[optig,Show[Graphics3D[(Polygon /@ odfacets)],Boxed->False];];
			If[!optec,odfacets = N[odfacets];];
		,{i,1,Length[odfacets]-1}];
		If[emb,
			odfacets = Embed[odfacets,vervec,cent];
			xmin = Min[#[[1]]& /@ Flatten[odfacets,1]];
			xmax = Max[#[[1]]& /@ Flatten[odfacets,1]];
			ymin = Min[#[[2]]& /@ Flatten[odfacets,1]];
			ymax = Max[#[[2]]& /@ Flatten[odfacets,1]];		
			Show[Graphics[Line /@ odfacets],
				PlotRange->{{xmin,xmax},{ymin,ymax}},
				AspectRatio->(ymax-ymin)/(xmax-xmin)];
		];
		Write[OutputFile,odfacets];
			Show[Graphics3D[(Polygon /@ odfacets)],Boxed->False,DisplayFunction->Identity,ViewPoint->vervec[[cent]][[1]]]
	];


OrderFacets[facets_List]:= 
	Block[{i,edges,facetsAdj,f,out,cyc,cand,cand2cyc,ff},
		{edges,facetsAdj} = MakeEdgesFromFacets[facets];
		out = {};
		For[i=1,i<=Length[facets],i++,
			f = edges[[Transpose[Position[facetsAdj,i]][[1]]]];
			cand = Drop[f,1];
			cyc = {First[f]};
			ff = {};
			While[{} != (cand2cyc = Select[cand,(Length[Intersection[Last[cyc],#]]==1)&]),
				AppendTo[ff,Flatten[Intersection[Last[cyc],cand2cyc[[1]]],1]];
				AppendTo[cyc,cand2cyc[[1]]];
				cand = Complement[cand,{cand2cyc[[1]]}];
			];
			AppendTo[ff,Flatten[Intersection[Last[cyc],First[cyc]][[1]],1]];
			AppendTo[out,ff];
		];
		Return[out];
	];

MakeEdgesFromFacets[l_List] := 
	Block[{i,j,lf={},adj={}},
		For[i=1,i<Length[l],i++,
			For[j=i+1,j<=Length[l],j++,
				If[Length[Intersection[l[[i]],l[[j]]]] == 2,
					AppendTo[lf,Intersection[l[[i]],l[[j]]]];
					AppendTo[adj,{i,j}];
				];
			];
		];
		Return[{lf,adj}];
	];
	
MakeFacetsFromZerosets[v_List,z_List]:=
	Block[{i,j,flist,maxf,flag,out},
		flist = Transpose[Position[z,#]][[1]]& /@ Union[Flatten[z]];
		maxf = {};
		For[i=1,i<=Length[flist],i++,
			flag = 0;
			For[j=1,j<=Length[flist],j++,
				If[ProperSubsetQ[flist[[i]],flist[[j]]],
					flag = 1,
					If[SubsetQ[flist[[i]],flist[[j]]] && i<j,
						flag = 1;
					];
				];
			];
			If[flag == 0,
				AppendTo[maxf,flist[[i]]];
			];
		];
		(* Return[maxf]; *)
		out = v[[#]]& /@ (MakeOrder[#,z]& /@ maxf);
		Return[out];
	];
	
MakeOrder[ver_List,zerova_List]:=
	Block[{candidates, cycle ,candidate2cycle},
		candidates = Drop[ver,1];
		cycle = { First[ver] };
		(* when one is list, comparing needs other is list *)
		While[{} != (candidate2cycle = Select[candidates,AdjQ[Last[cycle],#,zerova] &]),
			AppendTo[cycle,candidate2cycle[[1]]];
			candidates = Complement[candidates,candidate2cycle[[{1}]]];
		];
		While[{} != (candidate2cycle = Select[candidates,AdjQ[First[cycle],#,zerova] &]),
			PrependTo[cycle,candidate2cycle[[1]]];
			candidates = Complement[candidates,candidate2cycle[[{1}]]];
		];
		cycle
	];


AdjQ[i_Integer,j_Integer,l_List]:=
    Block[{subface},
	subface=Intersection[l[[i]],l[[j]]];
	Length[Select[l,SubsetQ[subface,#]&]]==2];
		
ProperSubsetQ[s_List,t_List]:=
	Length[s]==Length[Intersection[s,t]] && Length[s]<Length[t];

SubsetQ[s_List,t_List]:=
	Length[s]==Length[Intersection[s,t]];
	
MakeAdjTable[n_Integer,l_List]:=
	Block[{t=Table[Table[0,{n}],{n}],i},
		For[i=1,i<=Length[l],i++,
			t[[l[[i,1]],l[[i,2]]]] = 1;
			t[[l[[i,2]],l[[i,1]]]] = 1;
		];
		Return[t];
	];
	
MakeAdjTable[n_Integer,l_List,c_List]:=
	Block[{t=Table[Table[0,{n}],{n}],i},
		For[i=1,i<=Length[l],i++,
			t[[l[[i,1]],l[[i,2]]]] = c[[i]];
			t[[l[[i,2]],l[[i,1]]]] = c[[i]];
		];
		Return[t];
	];

BreadthFirstSearch[adj_?MatrixQ,n_Integer]:=
	Block[{out={},search,find={},output,
			i,obj,cd,l=Length[adj]},
		search = {n};
		While[search != {},
			obj = search[[1]];
			search = Drop[search,1];
			cd = {};
			For[i=1, i<=l, i++,
				If[adj[[obj,i]] != 0, AppendTo[cd,i]];
			];
			cd = Complement[cd,search];
			cd = Complement[cd,find];
			For[i=1, i<=Length[cd],i++,
				AppendTo[search,cd[[i]]];
				AppendTo[out,{obj,cd[[i]]}];
				AppendTo[out,{cd[[i]],obj}];
			];
			AppendTo[find,obj];
		];
		Return[MakeAdjTable[l,out]];
	];
	
DepthFirstSearch[adj_?MatrixQ,n_Integer]:=
	Block[{out={},search,find={},output,
			i,obj,cd,l=Length[adj]},
		search = {n};
		While[search != {},
			obj = search[[1]];
			search = Drop[search,1];
			cd = {};
			For[i=1, i<=l, i++,
				If[adj[[obj,i]] != 0, AppendTo[cd,i]];
			];
			cd = Complement[cd,find];
			search = Complement[search,cd];
			If[cd != {},
				AppendTo[out,{obj,cd[[1]]}];
				AppendTo[out,{cd[[1]],obj}];
			];
			search = Join[cd,search];
			AppendTo[find,obj];
		];
		Return[MakeAdjTable[l,out]];
	];
	    
SelectLeaf[g_List,cent_Integer]:=
	Block[{size,i,j,deg,child,par,cp},
		size = Length[g];
		For[i=1,i<=size,i++,
			deg = 0;
			child = i;
			For[j=1,j<=size,j++,
				If[g[[i]][[j]] > 0, 
					deg = deg + 1;
				 	par = j;
				];
			];
			If[deg == 1 && child != cent,
				Return[{child,par}];
			];
		];
	]

ProjectFaces[child_List,par_Integer,face_List,vervec_List]:=
	Block[{v1,v2,nch,i,fl},
		{v1,v2} = Intersection[face[[child[[1]]]],face[[par]]];
		v3 = Complement[face[[par]],face[[child[[1]]]]][[1]];
		fl = {};
		For[i=1, i<=Length[child], i++, 
			AppendTo[fl,ProjectVertex[v1,v2,v3,#,vervec]& /@ face[[child[[i]]]]];
		];
		Return[fl];
	];

ProjectVertex[v1_List,v2_List,v3_List,v4_List,a_List]:=
	Block[{x,y},
		If [N[a[[1]]].N[v4] <= N[a[[2]]],
			x = (v2-v1).(v2-v4)/(v2-v1).(v2-v1)*v1 + (v1-v2).(v1-v4)/(v1-v2).(v1-v2)*v2;
			y = (v2-v1).(v2-v3)/(v2-v1).(v2-v1)*v1 + (v1-v2).(v1-v3)/(v1-v2).(v1-v2)*v2;
			Return[x + Sqrt[(v4-x).(v4-x)/(y-v3).(y-v3)]*(y-v3)],
			x = (v2-v1).(v2-v4)/(v2-v1).(v2-v1)*v1 + (v1-v2).(v1-v4)/(v1-v2).(v1-v2)*v2;
			y = (v2-v1).(v2-v3)/(v2-v1).(v2-v1)*v1 + (v1-v2).(v1-v3)/(v1-v2).(v1-v2)*v2;
			Return[x + Sqrt[(v4-x).(v4-x)/(y-v3).(y-v3)]*(v3-y)];
		];
	];

Unfold[sptree1_List,tree1_List,maxf1_List,vervec_List,cen_Integer]:=
	Block[{j,child,par,posch,ch,of,pospa,sptree=sptree1,tree=tree1,maxf=maxf1,vv=vervec},
		{child, par} = SelectLeaf[sptree,cen];
		sptree[[child,par]] = -1;
		sptree[[par,child]] = -1;
		
		{posch} = Transpose[Position[tree,child]][[1]]; 
		ch = tree[[posch]];
		of = ProjectFaces[ch,par,maxf,vv[[par]]];
		For[j=1, j<=Length[ch],j++,
			maxf[[ch[[j]]]] = of[[j]];
		];
		{pospa} = Transpose[Position[tree,par]][[1]]; 
		tree[[pospa]] = (AppendTo[tree[[pospa]],#]& /@ ch)[[Length[ch]]];
		tree = Drop[tree,{posch}];
		Return[{sptree,tree,maxf}];
	]
	
CenterOfTree[t_List]:=
	Block[{leafs,cand,count,i},
		cand = Table[i,{i,1,Length[t]}];
		While[True,
			leafs = {};
			For[i=1,i<=Length[cand],i++,
				If[Count[t[[cand[[i]]]][[cand]],1]==1,AppendTo[leafs,cand[[i]]]];
			];
			cand = Complement[cand,leafs];
			If[Length[cand]<=2,Return[cand[[1]]];];
		];
	]
			
Embed[faces_List,verv_List,c_Integer]:=
	Block[{i,j,k,f,ff,cen,v1,v2,v3,vv,cand,a1,a2,p1,p2,fa},
		cen = Apply[Plus,faces[[c]]]/Length[faces[[c]]];
		f = {};
		For[i=1,i<=Length[faces],i++,
			AppendTo[f,(#-cen)& /@ faces[[i]]];
		];
		cand = Table[k,{k,1,Length[verv[[c]][[1]]]}];
		cand = Select[cand
				,(Abs[N[verv[[c]][[1]][[#]]]] > 10.^(-15))&];
		If[Length[cand] == 1,cand = cand[[1]]; ff = f;,
			cand = 1;
			a1 = {1,0,0};
			a2 = verv[[c]][[1]];
			p1 = 1;
			a1 = a1/a1[[p1]];
			a2 = a2-a2[[p1]] a1;
			p2 = 2;
			While[Abs[N[a2[[p2]]]] <= 10.^(-15), p2++];
			a2 = a2/a2[[p2]];
			a1 = a1-a1[[p2]] a2;
			j = Complement[{1,2,3},{p1,p2}][[1]];
			v1 = {0,0,0};
			v1[[j]] = 1;
			v1[[p1]] = -a1[[j]];
			v1[[p2]] = -a2[[j]];
			v2 = -v1;
			v3 = {0,v1[[3]],-v1[[2]]};
			ff = {};
			For[i=1,i<=Length[f],i++,
				AppendTo[ff,ProjectVertex[v1,v2,v3,#,{-a1,0}]& /@ f[[i]]];
			];
		];
		fa = {};
		For[i=1,i<=Length[ff],i++,
			AppendTo[fa, Drop[#,{cand}]& /@ ff[[i]]];
		];
		ff = Append[#,First[#]]& /@ fa;
		Return[ff];
	]

Zonotope[vectors_List]:= 
	Block[{vlist},
		vlist = Sum[(#)[[i]],{i,Length[#]}]& /@ ((vectors * (#)&) /@ Pow[Length[vectors]]);
		Return[vlist];
	]

Pow[n_Integer]:=
	Block[{p,a,b},
		If[n == 1, 
			Return[{{1},{-1}}]
		];
		p = Pow[n-1];
		a = Append[#,1]& /@ p;
		b = Append[#,-1]& /@ p;
		Return[(AppendTo[a,#]& /@ b)[[Length[b]]]];
	]

NonEmptyFaces[zerova_List]:=
    Map[Function[Transpose[Position[zerova,#]][[1]]],Union[Flatten[zerova]]]

VerticalVector[maxf_List]:=
	Block[{i,j,vervec,a1,a2,b,v,vv={},p1,p2},
		For[i=1,i<=Length[maxf],i++,
			a1 = maxf[[i]][[3]]-maxf[[i]][[2]];
			a2 = maxf[[i]][[1]]-maxf[[i]][[3]];
			p1 = 1;
			While[Abs[N[a1[[p1]]]]<=10.^(-15),p1++];
			a1 = a1/a1[[p1]];
			a2 = a2-a2[[p1]] a1;
			p2 = 1;
			While[Abs[N[a2[[p2]]]]<=10.^(-15) || p1 == p2, p2++];
			a2 = a2/a2[[p2]];
			a1 = a1-a1[[p2]] a2;
			j = Complement[{1,2,3},{p1,p2}][[1]];
			vervec = {0,0,0};
			vervec[[j]] = 1;
			vervec[[p1]] = -a1[[j]];
			vervec[[p2]] = -a2[[j]];
			b = vervec.maxf[[i]][[1]];
			v = Complement[Union[Flatten[maxf,1]],maxf[[i]]][[1]];
			If[N[v.vervec] > N[b], vervec = -vervec; b = -b];
			AppendTo[vv,{vervec,b}];
		];
		Return[vv];
	]

End[]
EndPackage[]

Options for function 'UnfoldPolytope"

OutputFile->"filename"

One can obtain the resulting cordinates of faces after unfolding. Default is "/dev/null".

ExactComputation-> True or False

In order to save time for computation, we employ the approximate computation. If one need to compute exactly, this option is needed. Default is False.

Embedding-> True or False

This option gives 2D-Graphics primitive also for unfoldings and 2D cordinates if one employ "OutputFile" option. Default is False.

InternalGraphics-> True or False

This option gives internal graphics during unfolding.

Tree-> "BFST" or "DFST"

If one employs "BFST", one can unfold polytopes along with the breadth first search tree. "DFST" means depth first search tree. Unfoldings with "DFST" tends to be long shaped. The default is "BFST".

See the following example.

vertices = {{0,0,0}, {1,0,0},{0,1,0},{0,0,1}};

faces = vertices[[#]]& /@ {{1, 2, 3}, {1, 3, 4}, 
                           {1, 4, 2}, {2, 4, 3}};

UnfoldPolytope[faces,OutputFile->"stdout",
					 ExactComputation->True,
					 Embedding->True,
					 Tree->"DFST"];

Fukuda and Mizukoshi's "VertexEnum" Package

(*

  Vertex Enumeration for Convex Polytopes
             and Arrangements

             Version 0.41 Beta (mod 1.02)
              November 14, 1992

           Copyright (c) 1991,1992 by
    Komei Fukuda and Ichiro Mizukoshi

***    
Beta1.01 eww: PolyhedronFromInequalities added
Beta1.02 eww: MathWorld context added
***

This package contains Mathematica implementations
of Avis-Fukuda algorithms for enumerating all 
vertices of a convex polyhedra given by a system
of linear inequalities, and all vertices (points)
of an arrangement given similarly.

This package is copyright 1991 by Komei Fukuda and
Ichiro Mizukoshi.  This package may be copied in 
its entirety for nonprofit purposes only.  
Sale, other than for the direct cost of the media,
is prohibited.  This copyright notice must
accompany all copies.

The authors make no representations, express or 
implied, with respond to this documentation, of 
the software it describes and contains, including
without limitations, any implied warranties of 
mechantability or fitness for a particular purpose,
all of which are expressly disclaimed.  The authors
shall in no event be liable for any indirect,
incidental, or consequential damages.

This beta release is designed to run under 
Version 1.2 & 2.0 of Mathematica. Any comments, 
bug reports, or requests to get on the 
VertexEnumeration mailing list should be 
forwarded to:

  Komei Fukuda  or Ichiro Mizukoshi
  Graduate School of Systems Management
  University of Tsukuba, Tokyo
  3-29-1 Otsuka, Bunkyo-ku
  Tokyo 112, Japan

  +81-3-3942-6876
  fukuda@gssm.otsuka.tsukuba.ac.jp
  or mizukosi@gssm.otsuka.tsukuba.ac.jp
 
*)


BeginPackage["MathWorld`VertexEnum`"]

Unprotect[VE]
Unprotect[PE]
Unprotect[VertexEnumeration]
Unprotect[PointEnumeration]
Unprotect[CrissCrossSolve]
Unprotect[BlandSolve]
Unprotect[Faces]
Unprotect[Polyhedron3D]
Unprotect[SubsetQ]
Unprotect[AdjacentQ]
Unprotect[EdgesOfPolyhedron]
Unprotect[MaximalQ]
Unprotect[MaximalSets]
Unprotect[LinesOfArrangement]

VE::usage="VE[m,b] is a short form of VertexEnumeration[m,b].";

PE::usage="PE[m,b] is a short form of PointEnumeration[m,b].";

VertexEnumeration::usage="VertexEnumeration[m,b] gives the list of all vertices of the polyhedron represented as the solution set of the linear inequality system  m.x <= b, x>=0. If you want to get results on progress MonitoringFile -> \"stdout\" ";

PointEnumeration::usage="PointEnumeration[m,b]  gives the list of all vertices of the arrangement of hyperplanes associated with the linear inequality system m.x <= b, x>=0.";

CrissCrossSolve::usage="CrissCrossSolve[c,m,b] solves the linear program  maximize c.x subject to m.x<=b and x>=0 by the Criss-Cross method. The output is {x*, v*, y*} where x* is an optimal solution, v* is an optimal value, and y* is a dual optimal solution. It warns if it is infeasible or dualinfeasible";

BlandSolve::usage="BlandSolve[c,m,b] solves the linear program  maximize c.x subject to m.x<=b and x>=0 by the Simplex method with Bland's rule, provided b is nonnegative.  The output is {x*, v*, y*} where x* is an optimal solution, v* is an optimal value, and y* is a dual optimal solution. It warns if it is unbounded.";

Faces::usage="Faces[list] returns a list of faces for the zero-variable list.  The kth face corresponds to kth inequality in x>=0, m.x<=b, for VertexEnumeration[m,b].";

Polyhedron3D::usage="Polyhedron3D[ver_List,zerova_List] returns 3D graphic primitives for drawing a 3D polytope.  The two arguments correspond to the first two list given by VertexEnumeration[m,b].";

SubsetQ::usage="SubsetQ[s,t] returns true if s is a subset of t."

AdjacentQ::usage="AdjacentQ[i_Integer,j_Integer,zerova_List] returns True if and only if the zeroset zerova[[i]] and the zeroset zerova[[j]] correspond to adjacent vertices (the intersection of zerova[[i]] and zerova[[j]] is contained in no other zerosets)."

EdgesOfPolyhedron::usage="EdgesOfPolyhedron[zerova_List] returns the set of adjacent pairs of vertices.  See AdjacentQ."

MaximalQ::usage="MaximalQ[i_Integer,l_List] returns True if and only if l[[i]] is a maximal set."

MaximalSets::usage="MaximalSets[l_List] returns the sublist of maximal sets.  See, MaximalQ."

LinesOfArrangement::usage="LinesOfArrangement[zerova_List] returns the list of lines as [1] sets of colinear vertices, and as [2] zerosets."


(* added by eww *)

PolyhedronFromInequalities::usage = 
"PolyhedronFromInequalities[ineqs,{x,y,z},<offset>] returns the polygons of a polyhedron \
obtained by vertex enumeration of the given list of inequalities in variables x, y, and z \
for x,y,z>0.  An offset may be specified either as {dx,dy,dz} or dx (assumed then to be \
multiplied by {1,1,1} in order to make sure the inequalities have solutions in only the \
positive octant, as needed by VertexEnumeration."

Options[VertexEnumeration] = {SearchTree -> False, ZeroVariables -> True, MonitoringFile -> "/dev/null"}

Options[PointEnumeration] = {SearchTree -> False, ZeroVariables -> True, MonitoringFile -> "/dev/null"}

Begin["`Private`"]

(* added by eww *)

MatrixFromInequalities[ineqs_,{x_,y_,z_}]:=Module[
	{i=First/@ineqs-Last/@ineqs,m,b},
	m=Coefficient[#,{x,y,z}]&/@i;
	b=-#/.Thread[{x,y,z}:>0]&/@i;
	{m,b}
]


(* VertexEnumeration does not work well with approximate numbers *)

(*
PolyhedronFromInequalities[ineqs_,{x_,y_,z_}]:=Module[
	{i=First/@ineqs-Last/@ineqs,m,b,offset},
    {m,b}=MatrixFromInequalities[ineqs,{x,y,z}];
    offset=Tr[LinearProgramming[-#,m,-b]&/@IdentityMatrix[3],List];
    Print[offset];
    Polyhedron3D@@VertexEnumeration[m,Chop[b+m.Ceiling[offset]]]
]
*)

PolyhedronFromInequalities[ineqs_,{x_,y_,z_},offset_List]:=
  Module[{i=First/@ineqs-Last/@ineqs,m,b},
    {m,b}=MatrixFromInequalities[ineqs,{x,y,z}];
    (Polyhedron3D@@VertexEnumeration[m,Chop[b+m.offset]])/.
    	Polygon[l_]:>Polygon[#-offset&/@l]
]

PolyhedronFromInequalities[ineqs_,{x_,y_,z_},offset_:10]:=
	PolyhedronFromInequalities[ineqs,{x,y,z},offset{1,1,1}]
    
(* end *)

VertexEnumeration::LpInfeasible = "linear program is infeasible."
VertexEnumeration::LpDualInfeasible = "linear program is dual infeasible."
VertexEnumeration::LpUnbounded = "linear program is unbounded."
VertexEnumeration::Infeasible = "linear inequality system is infeasible."

(* comments for private routine *)
Pivot::usage="Pivot[mat,r,s,b,n] performs a pivot operation on mat with row:r and 
column:s and updates basis b and nonbasis n. Last 2 arguments are optional.";

GetPivotSequence::usage="GetPivotSequence[startbv,startnbv,goalbv,goalnbv] gives a set of pivots (variable pairs) which transfers the start to the goal.";

SortDict::usage="SortDict[mat,b,n] returns the new {mat,b,n} sorted by b and n reordered in ascending order.";

Dic2Vertex::usage="Dic2Vertex[value of bv, name of bv, name of nbv, ver] returns the vertex associated with ver.";

MakeSequence::usage="MakeSequence[dic_?MatrixQ,bv_?VectorQ,nbv_?VectorQ,bases_?MatrixQ] returns pivot sequence.";

GetChangeBaseVariableNames::usage="GetChangeBaseVariableNames[startbv,startnbv,goalbv,goalnbv] gives a set of pivots (variable pairs) which transfers the start to the goal.";

ExecPivotSequence::usage="ExecPivotSequence[dic,bv,nbv,sequence] performs a sequence of pivots to {dic,bv,nbv} according to the given sequence."

NonEmptyFaces::usage="NonEmptyFaces[list] returns a list of nonempty faces.";

OptimalBasesEnumeration::usage="OptimalBasesEnumeration[mat,b] gives the set of variable
pairs obtaining an optimal basis.";

LexMin::usage="LexMin[mat,bv,nbv,m,n] returns True if the dictionary {mat,bv,nbv} is lexicographically minimum for representing the current solution.";

CrissCrossSelect::usage="CrissCrossSelect[mat,bv,nbv,m,n] gives the pivot position
selected by the Criss-Cross rule.";

CrissCrossSelectQ::usage="CrissCrossSelectQ[mat,bv,nbv,i,j] checks whether {i,j} is the pivot position selected by the Criss-Cross rule.";

CrissCrossFilter::usage="CrissCrossFilter[mat,bv,nbv,i,j] is filter (necesarily True) for {i,j}==CrissCrossSelect[mat,bv,nbv].";

CrissCross::usage="CrissCross[mat,bv,nbv,m,n] solves the linear program given by a dictionary {mat,bv,nbv} and outputs a terminal dictionary.";

NegSmallest::usage="Ratio Function used by the Simplex Method";

FirstElement::usage="FirstElement[function,list1[,list2]] gives a position on list1 (ordered by list2),that is True at first time.";

NonNegativeVectorQ::usage="NonNegativeVectorQ[vec] returns True if and only if vec is a nonnegative vector.";

BlandSelect::usage="BlandSelect[mat,bv,nbv] gives a pivot position selected by Bland's rule.";

BlandSelectQ::usage="BlandSelectQ[mat,bv,nbv,i,j] returns True if and only if {i,j}==
BlandSelect[mat,bv,nbv].";

BlandFilter::usage="BlandFilter[mat,bv,nbv,i,j,m,n] is filter (necesarily True) for {i,j}==
BlandSelect[mat,bv,nbv].";

Bland::usage="Bland[mat,bv,nbv,m,n] solves the linear program given by a feasible dictionary {mat,bv,nbv} and outputs a terminal dictionary.";

DegenerateQ::usage="DegenerateQ[vec] gives True if vec contains 0. It is used to recognize degenerate dictionary.";

InfeasibleQ::usage="InfeasbleQ[mat] gives True if mat is infeasible.";

DualInfeasibleQ::usage="DualInfeasbleQ[mat] gives True if mat is dual infeasible.";

Zerovar::usage="Zerovar[Value of bv, Name of bv,Name of nbv] shows the variable names with value 0."

BSearch::usage="BSearch[mat,b,n] returns the set of all basic feasible solutions for the input mat, b, n, {value of bv, name of bv, name of nbv}.";

CSearch::usage="CSearch[mat,b,n] returns the set of all basic solutions for the input mat, b, n, {value of bv, name of bv, name of nbv}.";

Options[CrissCross] = { NegativeTest-> Function[Negative[Chop[N[Simplify[#]]]]]}

Options[Bland] = { NegativeTest-> Function[Negative[Chop[N[Simplify[#]]]]]}

Options[ZeroVar] = { NegativeTest-> Function[Negative[Chop[N[Simplify[#]]]]]}

Options[BSearch] = {SearchTree -> False, NegativeTest-> Function[Negative[Chop[N[Simplify[#]]]]], MonitoringFile->"/dev/null"}

Options[CSearch] = {SearchTree -> False, NegativeTest-> Function[Negative[Chop[N[Simplify[#]]]]], MonitoringFile->"/dev/null"}

VE[mat_?MatrixQ,vec_?VectorQ, opts___Rule]:=VertexEnumeration[mat,vec,opts]

PE[mat_?MatrixQ,vec_?VectorQ, opts___Rule]:=PointEnumeration[mat,vec,opts]

Bout2Dlist[bout_List,dlist0_List]:=
	Block[{i,j,dlist,boutlength,dlist0rule},
		dlist =  Map[ Function[Father2Edge[Dlist2Father[#[[2]]]]], bout];
		boutlength = Map[Function[Length[#[[1]]]],bout];
		boutlength = Prepend[Table[Sum[boutlength[[i]],{i,1,j}],{j,Length[boutlength]-1}],0];
		dlist += boutlength;
		dlist0rule = Table[ Rule[i, boutlength[[i]] + 1], {i,Length[boutlength]}];
		AppendTo[dlist,dlist0 /. dlist0rule];
		{Flatten[dlist,1],boutlength+1}
	]


VertexEnumeration[mat_?MatrixQ,vec_?VectorQ,opts___Rule]:=
	Block[{dic,i,j,m,n,bv,nbv,flattenbout,bout,obe,output,dlist,opt1,opt2},
		opt1 = SearchTree /. {opts} /. Options[VertexEnumeration];
		opt2 = ZeroVariables /. {opts} /. Options[VertexEnumeration];
		{m,n} = Dimensions[mat];
		dic = Transpose[Append[Transpose[-mat],vec]];
		AppendTo[dic,Append[Table[0,{n}],0]];
	(* get dic, bv, nbv for Bland. Warnig nbv has been chanded.*)
		{dic,bv,nbv} = CrissCross[dic,Range[n+1,m+n],Range[n],m+1,n+1,opts];
	(* feasiblity check *)
		If[InfeasibleQ[dic], Message[VertexEnumeration::Infeasible];Return[{}]];
	(* shape up dic for Bland *)
		dic = Drop[dic,-1];
		If[DegenerateQ[Last[Transpose[dic]]],
		(* There is degeneracy *)
			(* obe looks like {{{dic,bv,nbv},{dic,bv,nbv}}, dlist0} *)
			obe= OptimalBasesEnumeration[Table[-dic[[i,j]],{i,m},{j,n}],Table[dic[[i,n+1]],{i,m}],bv,nbv];
			(* bout looks like
			 {{{{Value of bv,bv,nbv},{Value of bv,bv,nbv},...},dlist},{{{Value of bv,bv,nbv},...},dlist},...} *)
			bout = Map[ Function[BSearch[#[[1]],#[[2]],#[[3]],opts]],obe[[1]]];
			(* flattenbout looks like {{Value of bv,bv,nbv},{Value of bv,bv,nbv},....} but no dupilicat startpoint*)
			flattenbout = Flatten[Map[Function[#[[1]]],bout],1];
			output = {Map[Function[Dic2Vertex[ #[[1]],#[[2]],#[[3]],Range[n]]],flattenbout]};
			If[opt2, AppendTo[output, Map[ Function[ZeroVar[#[[1]],#[[2]],#[[3]],opts]], flattenbout]]];
			If[opt1, 
				dlist = Bout2Dlist[bout,obe[[2]]];
				AppendTo[output,dlist[[1]]];
				AppendTo[output,dlist[[2]]]],
		(* There is not degeneracy *)
			(* dic and mat must be same shape *)
			AppendTo[dic,Append[Table[-1,{n}],0]];
			bout = BSearch[dic,bv,nbv,opts];
			output = {Map[ Function[Dic2Vertex[ #[[1]],#[[2]],#[[3]],Range[n]]],bout[[1]]]};
			If[opt2, AppendTo[output, Map[ Function[ZeroVar[#[[1]],#[[2]],#[[3]],opts]],
			                            bout[[1]]]]];
			If[opt1, AppendTo[output, Father2Edge[Dlist2Father[bout[[2]]]]]]
		];
		output
	]


PointEnumeration[mat_?MatrixQ,vec_?VectorQ,opts___Rule]:=
	Block[{dic,i,j,m,n,bv,nbv,tmp,flattenbout,bout,obe,output,opt1,opt2},
		opt1 = SearchTree /. {opts} /. Options[PointEnumeration];
		opt2 = ZeroVariables /. {opts} /. Options[PointEnumeration];
		{m,n} = Dimensions[mat];
(*
		dic = Table[
			Which[
				SimpleNegative[vec[[i]]], Join[mat[[i]],vec[[i]]],
				True, Join[-mat[[i]],-vec[[i]]]],
			{i,m}
		]
*)
		dic = Transpose[Append[Transpose[-mat],vec]];
		AppendTo[dic,Append[Table[-1,{n}],0]];
		bv = Range[n+1,m+n];
		nbv = Range[n];
		If[DegenerateQ[Last[Transpose[dic]]],
		(* There is degeneracy *)
			(* obe looks like {{{dic,bv,nbv},{dic,bv,nbv}}, dlist0} *)
			obe= OptimalBasesEnumeration[Table[-dic[[i,j]],{i,m},{j,n}],Table[dic[[i,n+1]],{i,m}],bv,nbv];
			(* bout looks like
			 {{{{Value of bv,bv,nbv},{Value of bv,bv,nbv},...},dlist},{{{Value of bv,bv,nbv},...},dlist},...} *)
			bout = Map[ Function[CSearch[#[[1]],#[[2]],#[[3]],opts]],obe[[1]]];
			(* flattenbout looks like {{Value of bv,bv,nbv},{Value of bv,bv,nbv},....} but no dupilicat startpoint*)
			flattenbout = Flatten[Map[Function[#[[1]]],bout],1];
			output = {Map[Function[Dic2Vertex[ #[[1]],#[[2]],#[[3]],Range[n]]],flattenbout]};
			If[opt2, AppendTo[output, Map[ Function[ZeroVar[#[[1]],#[[2]],#[[3]],opts]], flattenbout]]];
			If[opt1, 
				dlist = Bout2Dlist[bout,obe[[2]]];
				AppendTo[output,dlist[[1]]];
				AppendTo[output,dlist[[2]]]],
		(* There is not degeneracy *)
			(* dic and mat must be same shape *)
		tmp = CSearch[dic,bv,nbv,opts];
		output = {Map[ Function[Dic2Vertex[ #[[1]],#[[2]],#[[3]],Range[n]]],tmp[[1]]]};
		If[opt2, AppendTo[output, Map[ Function[ZeroVar[#[[1]],#[[2]],#[[3]],opts]],
		                            tmp[[1]]]]];
		If[opt1, AppendTo[output, Father2Edge[Dlist2Father[tmp[[2]]]]]]
		];
		output
	]

Faces[zerova_List]:=
	Map[Function[x,
	      If[Position[zerova,x]=={},{},
	         Transpose[Position[zerova,x]][[1]]
	      ]
	    ], 
	    Range[Max[zerova]]
	]

NonEmptyFaces[zerova_List]:=
	Map[Function[Transpose[Position[zerova,#]][[1]]],Union[Flatten[zerova]]]

MakeLink[ver_List,zerova_List]:=
		Block[{candidates, cycle ,candidate2cycle},
		candidates = Drop[ver,1];
		cycle = { First[ver] };
		(* when one is list, comparing needs other is list *)
		While[{} != (candidate2cycle = Select[candidates,AdjacentQ[Last[cycle],#,zerova] &]),
			AppendTo[cycle,candidate2cycle[[1]]];
			candidates = Complement[candidates,candidate2cycle[[{1}]]];
		];
		While[{} != (candidate2cycle = Select[candidates,AdjacentQ[First[cycle],#,zerova] &]),
			PrependTo[cycle,candidate2cycle[[1]]];
			candidates = Complement[candidates,candidate2cycle[[{1}]]];
		];
		cycle
	]

Polyhedron3D[ver_List,zerova_List]:=
	Polygon /@ (ver[[#]]&) /@ ( MakeLink[#,zerova] & ) /@ NonEmptyFaces[zerova]

ZeroVar[ValueOfBv_?VectorQ,bv_?VectorQ,nbv_?VectorQ,opts___Rule]:=
	Block[{m,out={}},
		m=Length[ValueOfBv];
		SimpleNegative := NegativeTest /. {opts} /. Options[ZeroVar];
		out=Map[Function[SimpleSameQ[#,0]],ValueOfBv];
		Join[nbv,Map[Function[bv[[#[[1]]]]],Position[out,True]]]
	]
		
ForAllQ[func_,list1_List]:= Apply[And, Map[func, list1]]
	

Father2Edge[fathers_?VectorQ]:=	Map[{#,fathers[[#]]}&,Range[2,Length[fathers]]]

Dlist2Father[dlist_?VectorQ]:=
	Block[{po,dist,father,fathers={1}},
	(* first element in dist must be 0 and Length[dlist] => 2*)
		For[po = 2, po <= Length[dlist], ++po,
			dist =  dlist[[po]];
			father = po - 1;
			--dist;
		(* At first step a element in dist must be 1 *)
			While[dist > 0,
				father = fathers[[father]];
				--dist
			];
			AppendTo[fathers,father];
		];
		fathers
	]

Dic2Vertex[valueofbv_?VectorQ,bv_?VectorQ,nbv_?VectorQ,ver_?VectorQ]:=
	Map[Function[If[ MemberQ[nbv,#],
	                   0,valueofbv[[Position[bv,#][[1,1]]]]]],
	    ver]


LexMin[dic_?MatrixQ,bv_?VectorQ,nbv_?VectorQ]:=
	Block[{m,n},
		{m,n}=Dimensions[dic];
		LexMin[dic,bv,nbv,m,n]
	]

LexMin[dic_?MatrixQ,bv_?VectorQ,nbv_?VectorQ,m_Integer,n_Integer]:=
	Block[{i,j},
		For[i = 1, i < m , ++i,
			If[SimpleSameQ[0,dic[[i,n]]],
				For[j = 1, j < n, ++j,
					If[ Not[SimpleSameQ[0,dic[[i,j]]]] && bv[[i]]  > nbv[[j]], 
					    Return[False]]
					]
				]
			];
		True
	]

CSearch[dict_?MatrixQ,opts___Rule]:=
	Block[{m,n,bv,nbv},
		{m,n}=Dimensions[dict];
		bv = Range[m-1];
		nbv = Range[m,m+n-2];
		CSearch[dict,bv,nbv,opts]
	]

CSearch[origdict_?MatrixQ,origbv_?VectorQ,orignbv_?VectorQ,opts___Rule]:=
	Block[{ bv=origbv,nbv=orignbv,dict=origdict,m,n,i=1,j=1,opt1,MonitoringFile,
	        result,dist=1,distlist={0}},
		opt1 = SearchTree /. {opts} /. Options[CSearch];
		MonitoringFile = MonitoringFile /. {opts} /. Options[CSearch];
		SimpleNegative := NegativeTest /. {opts} /. Options[CSearch];
		{m,n}=Dimensions[dict];
		If[opt1,
			result ={{Drop[Transpose[origdict][[n]],-1],origbv,orignbv}};
			Write[MonitoringFile,Last[result][[3]]],
			If[LexMin[origdict,origbv,orignbv,m,n], 
				result ={{Drop[Transpose[origdict][[n]],-1],origbv,orignbv}};
				Write[MonitoringFile,Last[result][[3]]],
				result = {}]];
		While[ ((i < m )|| (origbv != bv)),
			While[ ((i <  m ) && Not[CrissCrossSelectQ[dict,bv,nbv,i,j]]),
				++j;
				If[ j >= n, j = 1; ++i];
			];
			If[ i < m,
				{dict,bv,nbv} = Pivot[dict,bv,nbv,i,j,m,n];
				If[opt1,
					AppendTo[result,{Drop[Transpose[dict][[n]],-1],bv,nbv}];
					Write[MonitoringFile,Last[result][[3]]];
					AppendTo[distlist,dist];
					dist = 1,
					If[LexMin[dict,bv,nbv,m,n],
						AppendTo[result,{Drop[Transpose[dict][[n]],-1],bv,nbv}];
						Write[MonitoringFile,Last[result][[3]]];
						AppendTo[distlist,dist];
						dist =1;
					];
				];
				{i,j}={1,1},
				{i,j}=CrissCrossSelect[dict,bv,nbv,m,n];
				If[ (i < m && j < n),
					{dict,bv,nbv}=Pivot[dict,bv,nbv,i,j,m,n];
					++j;
					++dist;
					If[ j >= n, j = 1; ++i]
				];
			];
		];
	{result,distlist}
	]

BSearch[dict_?MatrixQ,opts___Rule]:=
	Block[{m,n,bv,nbv},
		{m,n}=Dimensions[dict];
		bv = Range[m-1];
		nbv = Range[m, m+n-2];
		BSearch[dict,bv,nbv,opts]
	]

BSearch[origdict_?MatrixQ,origbv_?VectorQ,orignbv_?VectorQ,opts___Rule]:=
	Block[{ bv=origbv,nbv=orignbv,dict=origdict,m,n,i=1,j=1,opt1,MonitoringFile,
	        result,dist=1,distlist = {0},bvposi,nbvposi},
		opt1 = SearchTree /. {opts} /. Options[BSearch];
		MonitoringFile = MonitoringFile /. {opts} /. Options[BSearch];
		SimpleNegative := NegativeTest /. {opts} /. Options[BSearch];
		{m,n}=Dimensions[dict];
		bvposi = Flatten[Map[Position[bv,#] & ,Sort[bv]]];
		nbvposi = Flatten[Map[Position[nbv,#] & ,Sort[nbv]]];
		If[opt1,
			result ={{Drop[Transpose[origdict][[n]],-1],origbv,orignbv}};
			Write[MonitoringFile,Last[result][[3]]],
			If[LexMin[origdict,origbv,orignbv,m,n], 
				result ={{Drop[Transpose[origdict][[n]],-1],origbv,orignbv}};
				Write[MonitoringFile,Last[result][[3]]],
				result = {}]];
		While[ ((i < m) || (origbv != bv)),
			While[ (i <  m && Not[BlandSelectQ[dict,bv,nbv,i,j,m,n,bvposi]]),
				++j;
				If[ j >= n, j = 1; ++i]
			];
			If[ i < m,
				{dict,bv,nbv} = Pivot[dict,bv,nbv,i,j,m,n];
				bvposi = Flatten[Map[Position[bv,#] & ,Sort[bv]]];
				nbvposi = Flatten[Map[Position[nbv,#] & ,Sort[nbv]]];
				
(*
				Print["."];
*)
				If[opt1,
					AppendTo[result,{Drop[Transpose[dict][[n]],-1],bv,nbv}];
					Write[MonitoringFile,Last[result][[3]]];
					AppendTo[distlist,dist];
					dist = 1,
					If[LexMin[dict,bv,nbv,m,n],
						AppendTo[result,{Drop[Transpose[dict][[n]],-1],bv,nbv}];
						Write[MonitoringFile,Last[result][[3]]];
						AppendTo[distlist,dist];
						dist = 1,
						Null
					];
				];
				{i,j}={1,1},
				{i,j}=BlandSelect[dict,bv,nbv,bvposi,nbvposi];
				If[ (i < m && j < n),
					{dict,bv,nbv}=Pivot[dict,bv,nbv,i,j,m,n];
					bvposi = Flatten[Map[Position[bv,#] & ,Sort[bv]]];
					nbvposi = Flatten[Map[Position[nbv,#] & ,Sort[nbv]]];
					++j;
					++dist;
					If[ j >= n, j = 1; ++i]
				];
			];
		];
	{result,distlist}
	]

BlandFilter[dict_?MatrixQ,bv_?VectorQ,nbv_?VectorQ,i_Integer,j_Integer]:=
	Block[{m,n},
		{m,n}=Dimensions[dict];
		BlandFilter[dict,bv,nbv,i,j,m,n,Flatten[Map[Position[bv,#] & ,Sort[bv]]]]
	]

BlandFilter[dict_?MatrixQ,bv_?VectorQ,nbv_?VectorQ,i_Integer,j_Integer,m_Integer,n_Integer]:=
		BlandFilter[dict,bv,nbv,i,j,m,n,Flatten[Map[Position[bv,#] & ,Sort[bv]]]]

BlandFilter[dict_?MatrixQ,bv_?VectorQ,nbv_?VectorQ,i_Integer,j_Integer,m_Integer,n_Integer,bvposi_?VectorQ]:=
	( SimpleNegative[dict[[i,j]]] && SimpleNegative[dict[[m,j]]] &&
	  SimpleSameQ[-dict[[i,n]]/dict[[i,j]],
	              NegSmallest[Transpose[dict][[j]],bvposi,Last[Transpose[dict]]][[2]]]
	)

BlandSelectQ[origdict_?MatrixQ,origbv_?VectorQ,orignbv_?VectorQ,i_Integer,j_Integer,m_Integer,n_Integer]:=
	BlandSelectQ[origdict,origbv,orignbv,i,j,m,n,Flatten[Map[Position[origbv,#] & ,Sort[origbv]]]]
	
BlandSelectQ[origdict_?MatrixQ,origbv_?VectorQ,orignbv_?VectorQ,i_Integer,j_Integer,m_Integer,n_Integer,bvposi_?VectorQ]:=
	Block[{dict,bv,nbv},
		BlandFilter[origdict,origbv,orignbv,i,j,m,n,bvposi] &&
		(
		 {dict,bv,nbv}=Pivot[origdict,origbv,orignbv,i,j,m,n];
		 SameQ[{i,j},BlandSelect[dict,bv,nbv]]
		)
	]

NonNegativeVectorQ[vec_?VectorQ]:=
	ForAllQ[Function[Not[SimpleNegative[#]]],vec]


BlandSolve[cost_?VectorQ,mat_?MatrixQ,b_?NonNegativeVectorQ,opts___Rule]:=
	Block[{dic,m,n,ccdic,ccbv,ccnbv},
		dic = Transpose[Append[Transpose[Append[-mat,cost]],Append[b,0]]];
		{m,n}=Dimensions[dic];
		{ccdic, ccbv, ccnbv}= Bland[dic,opts];
		If[DualInfeasibleQ[ccdic], Message[VertexEnumeration::LpUnbounded];Return[{}]];
		{Dic2Vertex[ Drop[Transpose[ccdic][[n]],-1], ccbv, ccnbv,Range[n-1]],ccdic[[m,n]], Dic2Vertex[ Drop[-ccdic[[m]],-1],ccnbv, ccbv, Range[n,n+m-2]]}
	]

Bland[dict_?MatrixQ,opts___Rule]:=
	Block[{m,n,bv,nbv},
		{m,n}=Dimensions[dict];
		bv = Range[n,n+m-2];
		nbv = Range[n-1];
		Bland[dict,bv,nbv,m,n,opts]
	]

Bland[origdict_?MatrixQ,origbv_?VectorQ,orignbv_?VectorQ,opts___Rule]:=
	Block[{m,n},
		{m,n}=Dimensions[origdict];
		Bland[origdict,origbv,orignbv,m,n,opts]
	]

Bland[origdict_?MatrixQ,origbv_?VectorQ,orignbv_?VectorQ,m_Integer,n_Integer,opts___Rule]:=
	Block[{bv=origbv,nbv=orignbv,dict=origdict,i,j},
		SimpleNegative := NegativeTest /. {opts} /. Options[Bland];
		For[{i,j}=BlandSelect[dict,bv,nbv], (i < m && j < n),
		    {i,j}=BlandSelect[dict,bv,nbv],
		    {dict,bv,nbv}=Pivot[dict,bv,nbv,i,j,m,n]
		];
		{dict,bv,nbv}
	]

BlandSelect[dict_?MatrixQ,bv_?VectorQ,nbv_?VectorQ]:=
	Block[{bvposi,nbvposi},
		bvposi = Flatten[Map[Position[bv,#] & , Sort[bv]]];
		nbvposi = Flatten[Map[Position[nbv,#] & , Sort[nbv]]];
		BlandSelect[dict,bv,nbv,bvposi,nbvposi]
	]

BlandSelect[dict_?MatrixQ,bv_?VectorQ,nbv_?VectorQ,bvposi_?VectorQ,nbvposi_?VectorQ]:=
	Block[{i,j},
		j = FirstElement[SimplePositive,Last[dict],nbvposi];
		i = NegSmallest[
		                Transpose[dict][[j]],bvposi,
		                Last[Transpose[dict]]
		               ][[1]];
		{i,j}
	]

CrissCrossFilter[dict_?MatrixQ,bv_?VectorQ,nbv_?VectorQ,i_Integer,j_Integer]:=
	Block[{m,n},
		{m,n}=Dimensions[dict];
		CrissCrossFilter[dict,bv,nbv,i,j,m,n]
	]

CrissCrossFilter[dict_?MatrixQ,bv_?VectorQ,nbv_?VectorQ,i_Integer,j_Integer,m_Integer,n_Integer]:=
	((
	 SimplePositive[dict[[i,n]]] && SimplePositive[dict[[i,j]]] && 
	 ForAllQ[Function[Not[SimpleNegative[dict[i,#]]] &&
	 nbv[#] < bv[i]],Range[n-1]]
	)
	||
	(
	 SimpleNegative[dict[[m,j]]] && SimpleNegative[dict[[i,j]]] &&
	 ForAllQ[Function[Not[SimplePositive[dict[#,j]]] &&
	 bv[#] < nbv[j]],Range[m-1]]
	))
	
CrissCrossSelectQ[origdict_?MatrixQ,origbv_?VectorQ,orignbv_?VectorQ,i_Integer,j_Integer]:=
	Block[{dict,bv,nbv,m,n},
		{m,n}=Dimensions[origdict];
		CrissCrossFilter[origdict,origbv,orignbv,i,j,m,n] &&
		(
		 {dict,bv,nbv}=Pivot[origdict,origbv,orignbv,i,j,m,n];
		 SameQ[{i,j},CrissCrossSelect[dict,bv,nbv,m,n]]
		)
	]

CrissCrossSolve[cost_?VectorQ,mat_?MatrixQ,b_?VectorQ,opts___Rule]:=
	Block[{dic,m,n,ccdic,ccbv,ccnbv},
		dic = Transpose[Append[Transpose[Append[-mat,cost]],Append[b,0]]];
		{m,n}=Dimensions[dic];
		{ccdic, ccbv, ccnbv}=CrissCross[dic,opts];
		If[InfeasibleQ[ccdic], Message[VertexEnumeration::LpInfeasible];Return[{}]];
		If[DualInfeasibleQ[ccdic], Message[VertexEnumeration::LpDualInfeasible];Return[{}]];
		{Dic2Vertex[ Drop[Transpose[ccdic][[n]],-1], ccbv, ccnbv,Range[n-1]],ccdic[[m,n]], Dic2Vertex[ Drop[-ccdic[[m]],-1],ccnbv, ccbv, Range[n,n+m-2]]}
	      ]

CrissCross[dict_?MatrixQ,opts___Rule]:=
	Block[{m,n,bv,nbv},
		{m,n}=Dimensions[dict];
		bv = Range[n,m+n-2];
		nbv = Range[n-1];
		CrissCross[dict,bv,nbv,m,n,opts]
	]

CrissCross[origdict_?MatrixQ,origbv_?VectorQ,orignbv_?VectorQ,opts___Rule]:=
	Block[{m,n},
		{m,n}=Dimensions[origdict];
		CrissCross[origdict,origbv,orignbv,m,n,opts]
	]

CrissCross[origdict_?MatrixQ,origbv_?VectorQ,orignbv_?VectorQ,m_Integer,n_Integer,opts___Rule]:=
	Block[{bv=origbv,nbv=orignbv,dict=origdict,i,j},
		SimpleNegative := NegativeTest /. {opts} /. Options[CrissCross];
		For[{i,j}=CrissCrossSelect[dict,bv,nbv,m,n],(i < m && j < n),
			{i,j}=CrissCrossSelect[dict,bv,nbv,m,n],
			{dict,bv,nbv}=Pivot[dict,bv,nbv,i,j,m,n]
		];
		{dict,bv,nbv}
	]

CrissCrossSelect[dict_?MatrixQ,bv_?VectorQ,nbv_?VectorQ]:=
	Block[{m,n},
		{m,n}=Dimensions[dict];
		CrissCrossSelect[dict,bv,nbv,m,n]
	]

CrissCrossSelect[dict_?MatrixQ,bv_?VectorQ,nbv_?VectorQ,m_Integer,n_Integer]:=
	Block[{i,j,bvposi,nbvposi},
		bvposi = Flatten[Map[ Position[bv,#] & ,Sort[bv]]];
		nbvposi = Flatten[Map[ Position[nbv,#] & ,Sort[nbv]]];
		i = FirstElement[SimpleNegative,Last[Transpose[dict]],bvposi];
		j = FirstElement[SimplePositive,Last[dict],nbvposi];
		Which[
			i == m, i = FirstElement[SimpleNegative,Transpose[dict][[j]],bvposi],
			j == n, j = FirstElement[SimplePositive,dict[[i]],nbvposi],
			nbv[[j]] > bv[[i]], j = FirstElement[SimplePositive,dict[[i]],nbvposi],
			True,   i = FirstElement[SimpleNegative,Transpose[dict][[j]],bvposi];
		];
		{i,j}
	]
	 
FirstElement[func_,list1_List,list2_List]:=
	Block[{i},
		For[i = 1,i <= Length[list2],++i,
			If[func[list1[[list2[[i]]]]],Return[list2[[i]]]]];
			i
	]

FirstElement[func_,list1_List]:=
	Block[{i},
		For[i = 1, i <= Length[list1], ++i,
		If[func[list1[[i]]],Return[i]]];
		i
	]

NegSmallest[list1_List,list2_List,list3_List]:=
	Block[{i,posi,tmpval,val=Infinity},
		For[i = 1,i <= Length[list2],++i,
			If[
				(
				 SimpleNegative[list1[[list2[[i]]]]] &&
				 SimpleGreater[val,
				               tmpval = -(list3[[list2[[i]]]]/list1[[list2[[i]]]])]
				),
				posi = i;
				val = tmpval
			];
		];
		(* val = Infinity -> nothing happens. Never happens *)
		If[SameQ[Infinity,val], {i,val}, {list2[[posi]],val}]
	]

Pivot[d_?MatrixQ,bv_?VectorQ,nbv_?VectorQ,i_Integer,j_Integer,m_Integer,n_Integer]:=
	Block[{tmpbv,tmpnbv},
		{tmpbv,tmpnbv} = {bv,nbv};
		{tmpbv[[i]],tmpnbv[[j]]}={tmpnbv[[j]],tmpbv[[i]]};
		{Pivot[d,i,j,m,n],tmpbv,tmpnbv}
	]

Pivot[d_?MatrixQ,bv_?VectorQ,nbv_?VectorQ,i_Integer,j_Integer]:=
	Block[{tmpbv,tmpnbv},
		{tmpbv,tmpnbv} = {bv,nbv};
		{tmpbv[[i]],tmpnbv[[j]]}={tmpnbv[[j]],tmpbv[[i]]};
		{Pivot[d,i,j],tmpbv,tmpnbv}
	]

Pivot[d_?MatrixQ,r_Integer,s_Integer]:=
	Block[{m,n},
		{m,n}=Dimensions[d];
		Pivot[d,r,s,m,n]
	]

Pivot[d_?MatrixQ,r_Integer,s_Integer,m_Integer,n_Integer]:=
  Block[{i,j},
	Simplify[
	  Table[
	    Which[ (i == r) && (j == s), 1 / d[[i,j]],
		   (i == r) && (j != s), -(d[[i,j]]/d[[r,s]]),
		   (i != r) && (j == s), d[[i,j]]/d[[r,s]],
		   True, d[[i,j]] - ((d[[r,j]] * d[[i,s]]) / d[[r,s]])
		 ],{i,m},{j,n}
	       ]
	     ]
	   ]


GetChangeBaseVariableNames[startbv_?VectorQ,startnbv_?VectorQ,goalbv_?VectorQ,goalnbv_?VectorQ]:=
	{Intersection[startbv,goalnbv],Intersection[startnbv,goalbv]}

MakeSequence[origdic_?MatrixQ,origbv_?VectorQ,orignbv_?VectorQ,bases_?MatrixQ]:=
	Block[{dic = origdic, bv = origbv, nbv = orignbv, bvposi, nbvposi, tmpsequence},
		(* name to position *)
		bvposi = Flatten[Map[Position[bv,#] & ,bases[[1]]]];
		nbvposi = Flatten[Map[Position[nbv,#] & ,bases[[2]]]];
		(* make all possible sequences *)
		tmpsequence = Map[Transpose[{bvposi,#}] &, Permutations[nbvposi]];
		(* choose one *)
		If[{} == tmpsequence,tmpsequence,tmpsequence[[FirstElement[# &,Map[Apply[And,Map[Not[SimpleSameQ[0,origdic[[#[[1]],#[[2]]]]]] &,#]] &,tmpsequence]]]]]
	]

GetPivotSequence[startbv_?VectorQ,startnbv_?VectorQ,goalbv_?VectorQ,goalnbv_?VectorQ]:=
	Transpose[{Intersection[startbv,goalnbv],Intersection[startnbv,goalbv]}]

ExecPivotSequence[origdic_?MatrixQ,origbv_?VectorQ,orignbv_?VectorQ,sequence_?MatrixQ]:=
	Block[{dic = origdic, bv = origbv, nbv = orignbv, i},
		Do[{dic,bv,nbv} = Pivot[dic,bv,nbv,sequence[[i,1]],sequence[[i,2]]],{i,Length[sequence]}];
		{dic,bv,nbv}
	]

OptimalBasesEnumeration[mat_?MatrixQ,vec_?VectorQ,bv_?VectorQ,nbv_?VectorQ]:=
	Block[{dic,degenerateBasicVariables,i,m,n,tmp,output},
		{m,n} = Dimensions[mat];
		degenerateBasicVariables = Flatten[Position[vec,0]];
		dic = mat[[degenerateBasicVariables]];
		AppendTo[dic,Table[1,{n}]];
		dic = Append[Transpose[dic],Append[Table[-1,{Length[degenerateBasicVariables]}],0]];
(*
		bv = Table[i,{i,n+1,m+n}];
		nbv = Table[i,{i,n}];
*)
(* this is dual, so bv and nbv are exchanged *)
		tmp = BSearch[dic,nbv,bv[[degenerateBasicVariables]],SearchTree->True,MonitoringFile->"/dev/null"];
		dic = Transpose[Append[Transpose[-mat],vec]];
		AppendTo[dic,Append[Table[-1,{n}],0]];
		output = Map[ Function[MakeSequence[dic,bv,nbv,GetChangeBaseVariableNames[bv,nbv,#[[3]],#[[2]]]]], tmp[[1]]];
		output = {Map[Function[ExecPivotSequence[dic,bv,nbv,#]], output]};
		AppendTo[output,  Father2Edge[Dlist2Father[tmp[[2]]]]];
		output
	]

SubsetQ[s_List,t_List]:=
   Length[s]==Length[Intersection[s,t]];

AdjacentQ[i_Integer,j_Integer,l_List]:=
    Block[{subface}, 
      subface=Intersection[l[[i]],l[[j]]];
      Length[Select[l,SubsetQ[subface,#]&]]==2]


EdgesOfPolyhedron[l_List]:=
    Block[{i,j,edges={}},
      Do[
          Do[If[AdjacentQ[i,j,l],AppendTo[edges,{i,j}]],
             {j,i+1,Length[l]}
          ],{i,Length[l]}
      ];
    edges]

    
MaximalQ[i_Integer,l_List]:=
    Block[{candidate}, 
      candidate=l[[i]];
      Length[Union[Select[l,SubsetQ[candidate,#]&]]]==1]

MaximalSets[l_List]:=
    Block[{i,maximals={}},
        Do[If[MaximalQ[i,l],
            AppendTo[maximals,l[[i]]]
           ], {i,Length[l]}
        ];
    maximals]
LinesOfArrangement[zerova_List]:=
    Block[{i,j,p=Length[zerova],zeros={},lines={}},
      Do[Do[AppendTo[zeros,
              Intersection[zerova[[i]],zerova[[j]]]],{j,i+1,p}
         ],{i,p}
      ];zeros=MaximalSets[Union[zeros]];
      Do[AppendTo[lines,
           Select[Range[p],SubsetQ[zeros[[i]],zerova[[#]]]&]
         ],{i,Length[zeros]}
      ];
      {lines,zeros} 
    ]

SortDict[origdict_?MatrixQ,origbv_?VectorQ,orignbv_?VectorQ]:=
	Block[{bv=origbv,nbv=orignbv,dict=origdict},
		dict = Append[
		              Map[ 
		                  Function[dict[[Position[bv,#][[1,1]]]]],
		                  Sort[bv]],
		              Last[dict]];
		dict = Append[
		              Map[
		                  Function[Transpose[dict][[Position[nbv,#][[1,1]]]]],
		                  Sort[nbv]],
		              Last[Transpose[dict]]];
		{Transpose[dict],Sort[bv],Sort[nbv]}
	]

DegenerateQ[vec_?VectorQ]:=Count[vec,0]!=0

InfeasibleRowQ[vec_?VectorQ]:=
	SimpleNegative[Last[vec]] && ForAllQ[Function[Not[SimplePositive[#]]],Drop[vec,-1]]

InfeasibleQ[mat_?MatrixQ]:=
	Apply[Or,Map[InfeasibleRowQ,Drop[mat,-1]]]

DualInfeasibleQ[mat_?MatrixQ]:=
	InfeasibleQ[Transpose[-mat]]

SimplePositive:=Function[SimpleNegative[-#]]

SimpleSameQ:=
	Function[( Not[SimpleNegative[#1 - #2]] && Not[SimplePositive[#1 - #2]] )]

SimpleGreater:=Function[SimplePositive[#1 - #2]]

End[(* "`Private`" *)]

Protect[VE]
Protect[PE]
Protect[VertexEnumeration]
Protect[PointEnumeration]
Protect[CrissCrossSolve]
Protect[BlandSolve]
Protect[Faces]
Protect[Polyhedron3D]
Protect[SubsetQ]
Protect[AdjacentQ]
Protect[EdgesOfPolyhedron]
Protect[MaximalQ]
Protect[MaximalSets]
Protect[LinesOfArrangement]
EndPackage[]

Eric Weisstein's ``MathWorld" Packages

Polygons

(*:Mathematica Version: 5.2 *)

(*:Package Version: 1.00 *)

(*:Name: MathWorld`Polygons` *)

(*:Author: Eric W. Weisstein *)

(*:URL:
  http://mathworld.wolfram.com/packages/Polygons.m
*)

(*:Summary:
*)

(*:History:
  v1.00 (2005-02-15): Written
  
  (c) 2005 Eric W. Weisstein
*)

(*:Keywords:
  
*)

(*:Requirements: None. *)

(*:Discussion:
*)

(*:References: *)

(*:Limitations: None known. *)

BeginPackage["MathWorld`Polygons`"]


PolygonName::usage =
"PolygonName[n] gives the name of the n-gon."

PolygonalName::usage =
"PolygonalName[n] gives the name of the n-gonal."

Begin["`Private`"]

PolygonName[3]:="triangle"
PolygonName[4]:="square"
PolygonName[5]:="pentagon"
PolygonName[6]:="hexagon"
PolygonName[7]:="heptagon"
PolygonName[8]:="octagon"
PolygonName[9]:="nonagon"
PolygonName[10]:="decagon"
PolygonName[11]:="undecagon"
PolygonName[12]:="dodecagon"
PolygonName[14]:="tetradecagon"
PolygonName[20]:="icosagon"

PolygonalName[3]:="triangular"
PolygonalName[4]:="square"
PolygonalName[5]:="pentagonal"
PolygonalName[6]:="hexagonal"
PolygonalName[7]:="heptagonal"
PolygonalName[8]:="octagonal"
PolygonalName[9]:="nonagonal"
PolygonalName[10]:="decagonal"
PolygonalName[11]:="undecagonal"
PolygonalName[12]:="dodecagonal"
PolygonalName[14]:="tetradecagonal"
PolygonalName[20]:="icosagonal"

End[]

EndPackage[]

(* Protect[  ] *)

Polyhedron Operations

(*:Mathematica Version: 5.2 *)

(*:Package Version: 1.34 *)

(*:Name: MathWorld`PolyhedronOperations` *)

(*:Author:

Eric W. Weisstein (eww@wolfram.com)

*)

(*:Summary:

Adds functionality to construct and perform operations on polyhedra.

*)

(*:URL:
	Current version available from
	
	http://mathworld.wolfram.com/packages/PolyhedronOperations.m
*)


(*:History:
  v1.00 (2000-11-13): Broke off from Polyhedra.m and added partition
  v1.01 (2000-11-30): Documentation cleaned up
  v1.02 (2001-02-21): ConsphericQ.
  v1.03 (2002-01-25): Moved Bevel here from SolidGeometry.m.
                      Fixed up date
  v1.04 (2002-07-17): SphericalPolyhedron
  v1.05 (2002-08-01): PolyhedronInequalities
  v1.06 (2002-08-08): Wrapped And around PolyhedronInequalities.
                      SplitPolyhedron.  Fixed and improved VertexLabels.
  v1.07 (2002-10-03): Removed duplications with PolyhedronOperations`
  v1.08 (2002-10-15): Circumradius, Inradius, Midradius
  v1.09 (2002-10-26): VertexFigures now returns polygons, added Convexify.
                      Convexify now takes tolerance, fixed projection plane
                      determination.  VertexFigures now more efficient and takes
                      Convex->True option
  v1.10 (2002-11-07): PolyhedronFaces, Polyhedron, PolyhedronVertices
  v1.11 (2002-11-09): Net, NetFaces, NetVertices, PolyhedronDual, PolyhedronName.
                      Moved Net here from JohnsonSolids.m and combined Net and
                      ColoredNet.
  v1.12 (2002-11-20): ToExact rewritten for speed.  Polygon2D now returns a
                      normalized polygon, with first vertex {0,0} and last on the
                      x-axis
  v1.13 (2002-12-11): ToExact moved to Simplify.m
  v1.14 (2002-12-27): Fixed roty, fixed Polygon2D
  v1.15 (2002-12-31): Modified PolyhedronRotationCompound; added PolyhedronPolyhedra.
                      Polyhedron now evaluates only if PolyhedronFaces and PolyhedronVertices
                      are defined, and returns a list of lists if PolyhedronPolyhedra is
                      defined.  Moved RotationMatrixX/Y/Z here.  Colorize.
  v1.16 (2003-01-11): NetPieces.  Remove NonConvexHull in favor of Polyhedron[poly,Split->True].
                      NetDiagram, NetEdgeLengths, NetEdges.
  v1.17 (2003-01-25): Fixed RotationMatrixY.
  v1.18 (2003-02-20): Fixed up Orientation stuff and added OrientedQ
                      Fixed Convexify.  Added HalfIcosahedralGroup and IcosahedralGroup
                      courtesy of Ed Pegg, Jr.  Junked obsolete routines.
  v1.19 (2003-02-21): Moved Wireframe here and moved IcosahedralGroup routines
                      to Groups.m
  v1.20 (2003-04-04): SphericalPolyhedron now accepts a symbolic polyhedron (without
                      Polyhedron[] wrapped around it)
  v1.21 (2003-07-28): Added BoxRatios->{1,1,1} and option passing to HiddenWireframe
  v1.22 (2003-08-24): Centroid[Polygon[]]
  v1.23 (2003-09-14): NetPrintout.  Fixed Net.
  v1.24 (2003-09-17): NetPrintout now properly inhibits character substitution
  v1.25 (2003-10-05): Fixed Centroid
  v1.26 (2003-10-19): updated contexts
  v1.27 (2003-11-09): Cumulate now accepts any input containing Polygons at any level
                      Added PDFDirectory for NetPrintout
  v1.28 (2003-11-15): FaceIntersections, NetSurfaceArea
  v1.29 (2003-12-07): Replaced PolyhedronFaceLists with PolyhedronFaces[list]
  v1.30 (2003-12-08): Added Diameters
  v1.31 (2003-12-09): NonparallelVectors
  v1.32 (2003-12-20): Expansion, Snubify, SymbolicUnion, TruncateByPlanes.
                      Added OrientPolygon message for degenerate points.  Removed EdgeLengths.
  v1.33 (2004-09-14): Defined NetFaces[list]
  v1.34 (2004-11-10): Fixed Inradius, moved PointPlaneDistance to SolidGeometry`Distanced3D

  (c) 2000-2005 Eric W. Weisstein
*)

(*:Keywords:
*)

(*:Requirements:
*)

(*:Discussion:

*)

(*:References: *)

(*:Limitations:
	
	This package is still under construction and is not intended to
	be a finished product.
	
	Documentation not complete.
	
	SphericalPolyhedron requires a future version of
	Mathematica to render the concave polygons properly
*)

BeginPackage["MathWorld`PolyhedronOperations`",
	{
	"DiscreteMath`Combinatorica`",
	"DiscreteMath`ComputationalGeometry`",
	"Geometry`Rotations`",
	"Graphics`Colors`",
	"Utilities`FilterOptions`",
	"MathWorld`ConvexHull`",
	"MathWorld`SolidGeometry`",
	"MathWorld`VertexEnum`"
	}
]

Bevel::usage =
"Bevel[poly-list,w,<h>] takes a list of Polygons and returns a list of \
polygonal 'frames' on the same vertices of width w.  If h>0 is specified, \
then the edges of the frame are extended inwards with sides taken along \
radii towards (0,0,0).  Note that these inward-pointing planes are \
currently not internally capped and so are left open."

BeveledPolygon::usage =
"BeveledPolygon[{v1,...,vn},w,<h>] returns a \
polygonal 'frame' on the same vertices of width w.  If h>0 is specified, \
then the edges of the frame are extended inwards with sides taken along \
radii towards (0,0,0)."

Centroid::usage =
"Centroid[{v1,...,vn}] or Centroid[Polygon[{v1,...,vn}]] gives the centroid \
of the specified polygon."

Circumradius::usage =
"Circumradius[polyhedron] gives the distance from the origin (assumed \
to be the polyhedron's center) to the first vertex of its first face."

ClosedLine::usage =
"ClosedLine[l] gives a line that adds an edge between the first and last \
vertices."

Colored::usage =
"Colored->(True|False) is an option for Net."

Colorize::usage =
"Colorize[polyhedron-compound,colors] colors a Polyhedron object.  If specified, \
Colors->{color1,...} is a list of colors to use cyclically.  If not specified, \
Automatic is used which chooses Hue[i/n] for polyhedron number i=1, 2, ..., n \
in the compound.  A complete sample syntax is \
Show[Graphics3D[Colorize[Polyhedron[Cube3Compound],Colors->{Red,Yellow,Blue}]]."

Colors::usage =
"Colors->{3->color1, 4->color2,...} is an option to Net that \
specifies the colors to paint faces with specified number of vertices.  \
Colors->{color1,color2,...} is an option for Colorize[polyhedron-compound] \
that allows specification of a list of colors that will be used cyclically."

Convex::usage =
"Convex->(True|False) is an option to VertexFigures that fills in the polygonal \
holes centered on the original face centroids."

Convexify::usage =
"Convexify[polygon] uses Mathematica's 2-D ConvexHull routine \
to return a 3-dimensional polygon with the same vertices as the original, \
but specified in convex order.  Convexify[polygon,tol] uses tolerance tol \
for zero determination when finding an appropriate projection plane."

Cumulate::usage =
"Cumulate[polyhedron,h] replaces faces of a polyhedron with pyramids \
of height h on the same vertices. h may be negative, zero, or \
positive.  If h is a list of the form {{n1,h1},...{nk,hk}}, then \
ni-gonal faces are cumulated with height hi, and any faces not in the \
list are not cumulated."

CumulateFace::usage =
"CumulateFace[poly,h] gives a pyramid of height h with base on the \
vertices of the specified polygon."

CumulateWireframe::usage =
"CumulateWireframe[polyhedron,scale] replaces faces of a polyhedron with  \
pyramids of height scale on the same vertices, drawing the pyramid edges \
as lines on top of the original polyhedron."

Diameters::usage =
"Diameters[named-polyhedron] gives the longest vertex-vertex distances of \
Polyhedron[named-polyhedron]."

DualModel::usage =
"DualModel."

Expansion::usage =
"Expansion[poly,r] moves the specified polygona outward by a distance r \
and fills the gaps with new faces.  Expansion[poly,r,fn] uses the \
specified function for simplification before passing on to ConvexHull3D."

FaceIntersections::usage =
"FaceInterscetions[poly] gives the set of all lines of intersections \
of the faces in the specified polyhedron."

FromFaces::usage =
"FromFaces[l] gives the graph corresponding to the connectivtity \
table represented by the list of face indices."

HiddenWireframe::usage =
"HiddenWireframe[poly,<opts>] returns a Graphics object representing the list of \
Polygons poly such that hidden lines are distinguished from visible lines.  Options \
ViewPoint->{x,y,z} and PlotStyles->{frontstyle,backstyle} can be specified."

HullSpiderWeb::usage =
"HullSpiderWeb[poly,hull] generates a list of two Graphics3D objects, the first is a
spiderweb overlaid on poly and the second is the convex hull itself.  If name is Self,
then the solid itself is used as the convex hull."

Inradius::usage =
"Inradius[polyhedron] gives the distance from the origin (assumed \
to be the polyhedron's center) to the centroid of its first face."

Labeled::usage =
"Labeled->(True|False) is an option for Net."

Lines::usage =
"Lines->(True|False) is an option for Net."

ListEdges::usage =
"ListEdges."

Midpoint::usage =
"Midpoint[{v1,v2}] gives the midpoint of the given line segment."

MidpointCumulate::usage =
"MidpointCumulate[poly,scale] replaces the list of polygons with \
flat triangular pyramids on the face and a pyramid of height h with \
vertices on the edge midpoints.  scale may be negative, zero, or \
positive.  If scale is a list of the form {{n1,s1},...{nk,sk}}, then \
ni-gonal faces are cumulated with height si, and any faces not in the \
list are not cumulated."

Midradius::usage =
"Midradius[polyhedron] gives the distance from the origin (assumed \
to be the polyhedron's center) to the midpoint of the edge of its \
first face)."

Net::usage =
"Net[polyhedron] give a list of 2-D Polygons representing the net of the specified \
polyhedron.  For Johnson solids, the vertices of the net are \
specified exactly when known, and the scale is chosen so that all edges \
have length unity.  Options to Net include Colored, Labeled, Lines, and Colors."

NetConnectivity::usage =
"NetConnectivity[net] gives a list of the vertices adjacent to each \
vertex of a net specified as a series of lines."

NetDiagram::usage =
"NetDiagram[polyhedron] plots the net of the polyhedron with edge lengths labeled in \
increasing order."

NetEdgeLengths::usage =
"NetEdgeLengths[polyhedron] gives a list of the length of the edges of the \
specified polyhedron in the ordering of NetEdges[polyhedron]."

NetEdges::usage =
"NetEdges[polyhedron] gives a list of the indices of the edges of the net \
for the specified polyhedron, where the ordering is that of NetVertices[polyhedron].  \
Use in conjunction with Net[polyhedron]."

NetFaces::usage =
"NetFaces[polyhedron] gives a list of the indices of NetVertices[polyhedron] that specify \
the faces of a net.  Use in conjunction with Net[polyhedron]."

NetPieces::usage =
"NetPieces[polyhedron] is a list of the form {{n1,{f11,f12,..}},..} \
that indicates the indices fij of the NetFaces that form separate \
pieces of the net for a concave polyhedron that cannot be constructed from a single \
connected net.  The first element of each list is the count of number of pieces \
of that type needed."

NetPrintout::usage =
"NetPrintout[polyhedron,filename,opts] exports a copy of specified polyhedron to a PDF \
file.  NetPrintout accepts the same options as Net."

NetSurfaceArea::usage =
"NetSurfaceArea[polyhedron] gives the surface area for a polyhedron with a net \
defined."

NetVertices::usage =
"NetVertices[polyhedron] gives a list of vertices of the net of the specified solid.  \
Use in conjunction with Net[polyhedron].  NetVertices[line-list] gives the vertices \
of a net specified as a series of lines."

NonparallelVectors::usage =
"NoneparallelVectors[{v1,v2,...}] gives a subset of the given vectors with any parallel vectors \
removed."

NormalizedNet::usage =
"NormalizedNet[poly] gives a normalized form of the specified net."

Orient::usage =
"Orient[{polygon1,polygon2,...}] orients the faces of polygons with normal pointing outward, \
where outward is assumed to mean in a direction increasing the distance from {0,0,0}.  \
Orient[{polygon1,polygon2,...},{x,y,z}] assumes that {x,y,z} is at the center, so outward \
means the direction increasing the distance from {x,y,z}."

OrientedQ::usage =
"OrientedQ[{polygon1,polygon2,...}] returns a list of Booleans indicating if each polygon \
is positively oriented.  OrientedQ[{polygon1,polygon2,...},{x,y,z}] uses the point {x,y,z} \
as the center."

OrientPolygon::usage =
"OrientPolygon[list] takes a list of vertices and orients then so that the \
first and last edges form an outward-pointing cross product.  It returns a \
Polygon object."

PDFDirectory::usage =
"PDFDirectory is an option for NetPrintout."

PDFName::usage =
"PDFName is an option for NetPrintout."

PlotStyles::usage =
"PlotStyles->{style1,style2} is an option for HiddenWireframe."

Polygon2D::usage =
"Polygon2D[{v1,v2,...}] given a list of 3-D vertices, Polygon2D returns an \
equiavelent Polygon[] object with vertices in the plane that is congruent to the \
original 3-D polygon.  The resulting vertices will contain ArcTans of complicated \
expressions, but can sometimes be converted to algebraic numbers using ToExact.  \
The returned polygon has first vertex at {0,0}."

Polygon3D::usage =
"Polygon3D[Polygon[poly2d],z,vec] takes a 2-D polygon and transforms \
it back into 3-space."

PolygonArea::usage =
"PolygonArea[list] gives the area of an arbitrary convex planar polygon in \
3-D computed by triangulating it."

Polyhedron::usage =
"Polyhedron[polyhedron] gives a list of Polygons with exact vertices \
representing the given polyhedron, assuming that PolyhedronVertices[polyhedron] \
and PolyhedronFaces[polyhedron] are defined.  If PolyhedronPolyhedra[polyhedron] is \
also defined, the expression returned is a list of lists of Polygons, with each \
sublist representing a single polyhedron in a compound."

PolyhedronCompoundSurfaceArea::usage =
"PolyhedronCompoundSurfaceArea[poly] gives the volume of a compound \
of a polyhedron and its dual, making the assumption that poly \
is in the form {{rim1,pyr1},{rim2,pyr2}...}."

PolyhedronCompoundVolume::usage =
"PolyhedronCompoundVolume[poly] gives the volume of a compound \
of a polyhedron and its dual, making the assumption that poly \
is in the form {{rim1,pyr1},{rim2,pyr2}...}."

PolyhedronDual::usage =
"PolyhedronDual[polyhedron] gives the name of the polyhedron that is \
dual to the specified polyhedron."

PolyhedronEdges::usage =
"PolyhedronEdges[poly-list] gives the explicit edges of the given \
polyhedron specified as a list of Polygons.  PolyehdronEdges[polyhedron] \
gives a list of indices of PolyhedronVertices[polyhedron] that are the \
edges."

PolyhedronEdgeLengths::usage =
"PolyhedronEdgeLengths[vertex-list] gives a sorted list of the \
vertex-vertex distances among the specified vertices.  \
PolyhedronEdgeLengths[polyhedron] gives a sorted list of edge lengths \
using the analytic representation of polyhedron."

PolyhedronFaces::usage =
"PolyhedronFaces[polyhedron] is a list of the indices of \
PolyhedronVertices[polyhedron] that specify the faces of the given \
polyhedron.  Use in conjunction with Polyhedron[polyhedron]."

PolyhedronFaceCounts::usage =
"PolyhedronFaceCounts[polyhedron] gives a list {{f1,n1},...} where \
fi is the number of sides of face i and ni is the number of fi-gons \
faces in a given polyhedron."

PolyhedronFaceCenters::usage =
"PolyhedronFaceCenters gives a list of the centers of faces of a \
polyhedron."

PolyhedronFaceGraphics::usage =
"PolyhedronFaceGraphics[poly-list] gives an array of 2-D graphics objects \
corresponding to the faces of poly."

PolyhedronInequalities::usage =
"PolyhedronInequalities[{polygon1, polygon2, ...}, {x,y,z}] returns a list \
of inequalities in variables x, y, and z specifying the given set of \
polygons, assumed to \
bound a polyhedron.  The polygons are assumed to all be oriented \
with face normals pointing away from the polyhedron center."

PolyhedronName::usage =
"PolyhedronName[polyhedron] gives a string that is the name of the specified \
polyhedron."

PolyhedronPolyhedra::usage =
"PolyhedronPolyhedra[polyhedron] gives a list of lists of faces comprising \
the individual solids in a polyhedron compound."

PolyhedronRotationCompound::usage =
"PolyhedronRotationCompound[polyhedron,n] for a polyhedron of k faces \
(specifed as a list of k polygons) gives the compound of k polyhedra obtained \
by rotating one copy of the polyhedron by an angle Pi/n about each axes \
determined by the origin-face centroid line.  Note that depending on symmetry, \
some of the polyhedra in the compound may be conincident.  The specified \
polyhedron is _assumed_ to have centroid {0,0,0}."

PolyhedronSurfaceArea::usage =
"PolyhedronSurfaceArea[poly] gives the surface area of the polyhedron \
specified by a list of Polygons.  PolyhedronSurfaceArea[poly,{n1,n2,...}] \
gives the surface area of the polyhedron assuming that all n1-gons, \
n2-gons, etc. contribute equally.  If each polygon of a given number \
of sides contributes the same as all others having the same number of \
sides, then use PolyhedronSymmetricSurfaceArea."

PolyhedronSymmetricSurfaceArea::usage =
"PolyhedronSymmetricSurfaceArea[poly] gives the surface area of the \
polyhedron specified by a list of Polygons assuming that each polygon \
of a given number of sides contributes the same as all others having \
the same number of sides.  If this is not the case, use 
PolyhedronSurfaceArea."

PolyhedronSymmetricVolume::usage =
"PolyhedronSymmetricVolume[poly] gives the volume of the polyhedron \
specified by a list of Polygons assuming that each polygon of a given \
number of sides contributes the same as all others having the same \
number of sides.  If this is not the case, use PolyhedronVolume."

PolyhedronVertexDistances::usage =
"PolyhedronVertexDistances"

PolyhedronVertices::usage =
"PolyhedronVertices[polyhedron] is a list of the exact vertices of \
the specified polyhedron.  Use in conjunction with Faces[poly] and \
Polyhedron[poly].  \
PolyhedronVertices[polygon-list] extracts the vertices from the \
specified list of polygons.  Vertices can be equivalent but not obviously \
identical, in which case the user is responsible for simplifiying \
them and taking their union."

PolyhedronVolume::usage =
"PolyhedronVolume[poly] gives the volume of the polyhedron specified by \
a list of Polygons.  PolyhedronVolume[poly,{n1,n2,...}] gives the volume \
of the polyhedron assuming that all n1-gons, n2-gons, etc. contribute \
equally to the volume.  If each polygon of a given number of sides \
contributes the same as all others having the same number of sides, then \
use PolyhedronSymmetricVolume."

PyramidVolume::usage =
"PyramidVolume[{v1,...,vn}] gives the volume of the pyramid with apex at (0,0,0) \
and specified base."

RectifiedPair::usage =
"RectifiedPair[poly] gives poly and a rectified version of it."

Rectify::usage =
"Rectify[poly] gives the rectified polyhedron."

RotateAboutAxis::usage =
"RotateAboutAxis[v,n,ang] rotates a vector about the normal n by angle ang.  \
RotateAboutAxis[polygon,n,ang] rotates the polygon."

RotationFormula::usage =
"RotationFormula[r,n,angle] rotates a vector r about the unit \
vector n by angle."

RotationMatrix::usage =
"RotationMatrix[angle] gives the 2-D rotation matrix for an angle \
ang in the plane."

RotationMatrixX::usage =
"RotationMatrixX[angle] gives the 3-D rotation matrix for an angle \
ang about the x-axis."

RotationMatrixY::usage =
"RotationMatrixY[angle] gives the 3-D rotation matrix for an angle \
ang about the y-axis."

RotationMatrixZ::usage =
"RotationMatrixZ[angle] gives the 3-D rotation matrix for an angle \
ang about the z-axis."

Snubify::usage =
"Snubify[poly,d,t] constructs the snubification of the given polyhedron \
with radial displacement d and with twist angle t."

SolidCentroid::usage =
"SolidCentroid[poly-list] returns the centroid of the list of polygons."

SphericalPolyhedron::usage =
"SphericalPolyhedron[poly,n] gives the spherical polyhedron formed by projecting \
the edges of the specified list of 3-dimensional polygons onto a unit sphere \
with n equally spaced steps along each arc.  SphericalPolyhedron[poly,n,r] \
displays the arcs as radial \"walls\" extending from a radius r to radius 1."

SpiderWeb::usage =
"SpiderWeb[Polygon[...]] gives a web of lines around the rim
of the polygon and from the centroid to vertices and edge
midpoints."

SymbolicUnion::usage =
"SymbolicUnion[l] takes a list of analytic objects and unions them \
by treating objects that are numerically equal as symbolically equal \
and discarding the duplicated.  WorkingPrecision can be specified."

TruncateByPlanes::usage =
"TruncateByPlanes[poly,r] truncates the given polyhedron by converting \
its faces to inequalities, adding the inequalities obtained by adding \
cutting planes at distances r from the center along the vertices, \
and computing the new vertices."

VertexDistances::usage =
"VertexDistances"

VertexFigures::usage =
"VertexFigures[poly] gives the vertex figures of the specified polyhedron. \
VertexFigures[poly,Convex->True] fills in the polygonal holes centered on \
the original face centroids."

VertexLabels::usage =
"VertexLabels[poly] gives numbers at the vertices of the polygon."

WenningerModel::usage =
"WenningerModel."

Wireframe::usage =
"Wireframe[{polygon1,...}] gives a wireframe of the input.  Unlike \
Graphics`Shapes`WireFrame, the input does not already have to be a \
Graphics3D object."


(***********************************)
(* Set your favorite defaults here *)
(***********************************)

Options[HiddenWireframe]={
	BoxRatios->{1,1,1},
	ViewPoint->{1.3,-2.4,2.},
	PlotStyles->{Automatic,{GrayLevel[.5],Dashing[{.02}]}}
}

Options[Net]:={
	Lines->True,
	Colored->False,
	Colors->{3->Red,4->Yellow,5->Blue,6->Green,8->Violet,10->Orange},
	Labeled->False
};

Options[NetPrintout]:={
	PDFDirectory->"/troves/MathOzTeX/pdf/",
	PDFName->""
};

Options[VertexFigures]={
	Convex->False
};

Options[Colorize]={
	Colors->Automatic,
	Lighting->False
};

Options[SymbolicUnion]={
	WorkingPrecision->100
};


Begin["`Private`"]

Area[Polygon[l_List?MatrixQ]]:=
  Plus@@(Det/@Partition[l,2,1,1])/2/;Dimensions[l][[2]]==2

BeveledPolygon[l_List,w_,h_:0]:=Module[
	{pts,bev,bis,i,n=Length[l],edge=Partition[l,3,1,1]},
    pts=#2-(bis=Unit[Unit[#2-#1]+Unit[#2-#3]])w/Sqrt[1-(bis.Unit[#2-#1])^2]&@@@
		edge;
    bev=Table[{edge[[i,1]],edge[[i,2]],pts[[i]],pts[[Mod[i-2,n]+1]]},{i,n}];
    Polygon/@Join[bev,
		If[h<=0,{},{#3,#4,#4-h Unit[#4],#3-h Unit[#3]}&@@@bev]
	]
]

Bevel[l_List,w_,h_:0]:=l/.Polygon[x_]:>BeveledPolygon[x,w,h]

Centroid[v_List?MatrixQ]:=Plus@@v/Length[v]

Centroid[Polygon[p_List?MatrixQ]]:=Centroid[p]

Circumradius[poly_List]:=norm[poly[[1,1,1]]]

CircumsphereRadius[poly_]:=Sqrt[Max[#.#&/@PolyhedronVertices[poly]]]

ClosedLine[l_List]:=Line[Append[l,First[l]]]

(* Colorize *)

Colorize[p_List,opts___]:=Module[
	{
	color=Colors/.{opts}/.Options[Colorize],
	n=Length[p],
	gopts=FilterOptions[Graphics3D,opts,Options[Colorize]]
	},
	Sequence@@{
	If[color===Automatic,
		MapIndexed[{FaceForm[Sequence@@Table[Hue[#2[[1]]/n],{2}]],#1}&,p],
		MapIndexed[{FaceForm[Sequence@@Table[color[[Mod[#2[[1]]-1,Length[color]]+1]],{2}]],#1}&,p]
	],gopts
	}
]

(* Convexify *)

(* the triangle is always convex *)

Convexify[poly_Polygon]:=poly /; Length[poly[[1]]]==3

(* but higher n-gons are unfortunately not *)

Convexify::singular = "Normal vector is singular for vector `1`";
Convexify::fatal = "Fatal error while finding projection plain.";

Convexify[poly_Polygon,eps_:10^-6]:=Module[
	{v=poly[[1]],normal,indx,p,nv},
	nv=N[v,100];
	normal=Cross@@(#-nv[[1]]&/@{nv[[2]],nv[[-1]]});
	(* pick a nonsingular projection plane *)
	p=Flatten[Position[normal,_?(Abs[#]<eps&),{1},Heads->False]];
	If[Length[p]==3,
	  Message[Convexify::singular,normal];
	  $Failed,
	(* Else *)
		indx=Switch[Length[p],
		3,Message[Convexify::fatal],
		2,Complement[Range[3],p],
		1,{Complement[Range[3],p][[1]]},
		_,{3}];
		Polygon[v[[ConvexHull[Drop[#,indx]&/@nv]]]]
	]
]

(* CoplanarQ---from SolidGeometry.m *)

CoplanarQ[v_List?MatrixQ,eps_:10^-6]:=True /;Length[v]==3

CoplanarQ[v_List?MatrixQ,eps_:10^-6]:=
  Module[{n=Cross[v[[2]]-v[[1]],v[[3]]-v[[1]]]},
      And@@(Abs[n.(v[[1]]-#)]<eps&/@Drop[v,3])
] /; Length[v]>3
  
(* Cumulate *)

CumulateFace[Polygon[face_List],s_]:=Module[{ctr=Plus@@face/Length[face],tip},
      tip = ctr-Unit[Cross[face[[1]]-ctr,face[[-1]]-ctr]]s;
      Polygon[{tip,First[#],Last[#]}]&/@Partition[face,2,1,{1,1}]
]

CumulateFace[face_,s_]:=Module[{ctr=Plus@@face/Length[face],tip},
      tip = ctr-Unit[Cross[face[[1]]-ctr,face[[-1]]-ctr]]s;
      Polygon[{tip,First[#],Last[#]}]&/@Partition[face,2,1,{1,1}]
]

Cumulate[poly_List,scales_List]:=Module[
	{inpoly=poly/.Polygon:>Identity,n,s,i},
	{n,s}=Transpose[scales];
	Table[
		Map[CumulateFace[#,s[[i]]]&,Select[inpoly,Length[#]==n[[i]]&]],
		{i,Length[n]}
	]
]

(*
Cumulate[poly_List,scale_] :=Module[{inpoly=poly /. Polygon:>Identity},
	CumulateFace[#,scale]&/@inpoly
]
*)

Cumulate[g_,scale_]:=g/.Polygon[v_]:>CumulateFace[v,scale]

CumulateWireframe[poly_,s_]:=Module[{inpoly = Map[#[[1]]&,poly]},
	{
	Thickness[.015],Map[CumulateWires[#,s]&,inpoly],
	poly
	}
]

CumulateWires[face_,s_] :=Module[{ctr=Plus@@face/Length[face],tip},
      tip = ctr-Unit[Cross[face[[1]]-ctr,face[[-1]]-ctr]]s;
      Map[Line[{tip,#}]&,face]
]

Diameters[poly_,eps_:10^-10]:=Module[{e=KSubsets[PolyhedronVertices[poly],2],m},
	m=Max[Norm/@e];
	Select[e,(Abs[Norm[#]-m]<eps&)]
]

DualModel[n_]:=Module[{l=Select[WenningerDuals,#[[3]]==n&]},
	{#[[4]],"p. "<>ToString[#[[5]]],
		"Dual of "<>#[[2]]<>
		" ("<>udualstring[#[[2]]]<>", W"<>ToString[#[[1]]]<>")"}&/@l
]

Expansion[poly_,r_]:=
  ConvexHull3D[
    PolyhedronVertices[
      poly/.Polygon[l_]:>Polygon[#+r Unit[Centroid[Polygon[l]]]&/@l]
    ]
  ]

FaceIntersections[poly_]:=Module[{pairs=KSubsets[Take[#,3]&/@First/@poly,2]},
    DeleteCases[Intersection3D@@@Map[Plane,pairs,{2}],{}]
    ]

FromFaces[l_List]:=
	FromUnorderedPairs[Union[Flatten[Sort/@Partition[#,2,1,1]&/@l,1]]]

HiddenWireframe[poly_,opts___]:=Module[
	{front,all,back,frontstyle,backstyle,view,gopts},
	view=ViewPoint/.{opts}/.Options[HiddenWireframe];
	{frontstyle,backstyle}=PlotStyles/.{opts}/.Options[HiddenWireframe]
		/.Automatic->{};
	gopts=FilterOptions[Graphics3D,opts,Options[HiddenWireframe]];
	front=Select[
		Flatten[Show[Graphics[
		Graphics3D[{Wireframe[poly],EdgeForm[],poly},
		gopts,RenderAll->False,ViewPoint->view]],AspectRatio->Automatic,
		DisplayFunction->Identity][[1]]],
		Head[#]===Line&];
	all=Select[Flatten[Show[Graphics[
		Graphics3D[Wireframe[poly],gopts,ViewPoint->view,AspectRatio->Automatic]],
		DisplayFunction->Identity][[1]]],
		Head[#]===Line&];
	(* back lines are subset of all which are _not_ in front.
	   Sort to make sure order of vertices is the same during the
	   comparison
	*)
	back=Complement[Map[Sort,all,2],Map[Sort,front,2]];
	Graphics[{{Sequence@@frontstyle,front},{Sequence@@backstyle,back}},
		AspectRatio->Automatic][[1]]
]

HullSpiderWeb[poly_,Self]:=HullSpiderweb[poly,poly]

HullSpiderWeb[poly_,hullname_String]:=Module[
	{
		hull=ConvexHull3D[Union[Flatten[Map[#[[1]]&,poly],1]]]
	},
	{
		Graphics3D[{poly,Thickness[.01],SpiderWeb/@hull},Boxed->False],
		Graphics3D[hull,Boxed->False]
	}
]

HullSpiderWeb[poly_]:=HullSpiderweb[poly,poly]

HullSpiderWeb[poly_,hull_]:=Module[{},
	{
	Graphics3D[{#,Thickness[.01],SpiderWeb/@#}&@poly,Boxed->False],
	Graphics3D[hull,Boxed->False]
	}
]

Inradius[poly_List]:=Distance3D[Flatten[poly][[1]]]

LineLineIntersection[{l1_,l2_},{l3_,l4_}]:=Module[{a,b},
	l1+(l2-l1)a /. Solve[l1+(l2-l1)a==l3+(l4-l3)b,{a,b}][[1]]
]

(* must sort to make sure smallest vertex of pair is first *)

ListEdges[l_List]:=Sort/@(l[[#]]&/@Partition[Range[Length[l]],2,1,1])

lline[l_List]:=Module[{i,j},
	Table[ClosedLine[Table[RotationMatrixZ[j Pi/2].l[[i]],{i,Length[l]}]],{j,0,3}]
]

Midpoint[Line[l_List?MatrixQ]]:=Point[Midpoint[l]] /; Dimensions[l][[1]]==2

Midpoint[l_List /; Length[l]==2]:=Plus@@l/2

addn[n_,k_,m_]:=Mod[n+k-1,m]+1
MidpointCumulateFace[Polygon[poly_List],h_]:=Module[
	{mid,n,v},
	n=Length[poly];
	v=Centroid[poly];
	mid=Plus@@#/2&/@Partition[poly,2,1,1];
	{
	Polygon/@Table[{poly[[i]],mid[[i]],mid[[addn[i,-1,n]]]},{i,n}],
	Polygon/@Table[{mid[[i]],mid[[addn[i,1,n]]],v+h Unit[v]},{i,n}]
	}
]

MidpointCumulate[poly_List,scales_List]:=Module[
	{n,s,i,inpoly=poly/.Polygon:>Identity},
	{n,s}=Transpose[scales];
	Table[
		Map[MidpointCumulateFace[Polygon[#],s[[i]]]&,
			Select[inpoly,Length[#]==n[[i]]&]],
		{i,Length[n]}
	]
]

MidpointCumulate[poly_List,h_]:=MidpointCumulateFace[#,h]&/@poly

Midradius[poly_List]:=Distance3D[Line[Take[Flatten[poly][[1,1]],2]]]

(* Net *)

Net::counts="To create the solid, the following numbers of pieces of each type are \
needed: `1`.";

Net[poly__,opts___?OptionQ]:=Module[
	{
		colored=Colored/.{opts}/.Options[Net],
		lines=Lines/.{opts}/.Options[Net],
		colors=Colors/.{opts}/.Options[Net],
		labeled=Labeled/.{opts}/.Options[Net],
		v=NetVertices[poly],
		net,
		c,piece
	},
	(*
	If[Cases[DownValues[NetPieces],Verbatim[HoldPattern[NetPieces[poly]]],Infinity]!={},
		Message[Net::counts,First/@NetPieces[poly]];
		c=Last/@NetPieces[poly],
		(* Else *)
		c=Range[Length[NetFaces[poly]]]
	];
	*)
	net=Polygon[v[[#]]]&/@NetFaces[poly];
	{	
		If[colored,Transpose[{Length@@@net/.colors,net}],Sequence@@{}],
		If[lines,ClosedLine@@@net,Sequence@@{}],
		If[labeled,
			MapIndexed[Text[#2[[1]],#1,Background->White]&,v],
			Sequence@@{}
		]
	}
] 

NetConnectivity[polynet_List]:=
  Module[{NetVert = NetVertices[Flatten[polynet]]},
    Flatten[polynet] /. Line[x_List] :> Drop[x, -1] /. 
      Thread[NetVert -> Range[Length[NetVert]]]
]

NetDiagram[poly__,opts___?OptionQ]:=Module[
    {
      s=RootReduce/@NetEdgeLengths[poly],
      sides,
      lengths
      },
    sides=Sort[Union[s],Less];
    lengths=
      Sort[Transpose[{s/.Thread[sides->Range[Length[sides]]],
            NetEdges[poly]}]];
    {
      Net[poly,opts],
      Text[
            Subscript[
              StyleForm["s",FontSize->12,FontFamily->"Times",
                FontSlant->"Italic"],#1],Plus@@NetVertices[poly][[#2]]/2,
            Background->White]&@@@lengths
      }
    ]

NetEdgeLengths[poly__]:=
	SymbolicUnion[
	Sqrt[#.#]&/@Map[Subtract@@#&,NetVertices[poly][[#]]&/@NetEdges[poly],{1}]
	]

NetEdges[poly__]:=Union[Sort/@Flatten[Partition[#,2,1,1]&/@NetFaces[poly],1]]

size[poly_]:=Module[{v=Transpose[NetVertices[poly]]//N},
    Subtract@@{Max/@v,Min/@v}
]

NetFaces[l_List]:=Module[{v=NetVertices[l]},
    l/.{Line[x_List]:>Drop[x,-1], Polygon[x_List]:>x}/.Thread[v->Range[Length[v]]]
]

NetPrintout[poly_,file_,opts___]:=Module[{sz=size[poly],
	dir=PDFDirectory/.{opts}/.Options[NetPrintout],
	html=PDFName/.{opts}/.Options[NetPrintout]
	},
	If[html=="",html=StringDrop[file,-4]];
    Export[dir<>file,Graphics[
	{
		NormalizedNet[poly,opts],
		Text[StyleForm["Net for the "<>PolyhedronName[poly],
			PrivateFontOptions->{"OperatorSubstitution"->False}],
			{.15Min[sz],0},{0,1},{0,-1},
			TextStyle->{FontFamily->"Times",FontSlant->"Italic",FontSize->18}],
		Text[StyleForm["Downloaded from http://mathworld.wolfram.com/pdf/"<>html<>".pdf",
			PrivateFontOptions->{"OperatorSubstitution"->False}],
            {.1Min[sz],0},{0,1},{0,-1},
			TextStyle->{FontFamily->"Times",FontSlant->"Italic",FontSize->12}],
		Text[StyleForm["For more information, see the MathWorld entry http://mathworld.wolfram.com/"<>html<>".html",
			PrivateFontOptions->{"OperatorSubstitution"->False}],
            {.07Min[sz],0},{0,1},{0,-1},
			TextStyle->{FontFamily->"Times",FontSlant->"Italic",FontSize->12}]
	},
	AspectRatio->Automatic,
	PlotRange->All]
	]
]

NetSurfaceArea[poly__]:=Module[{v=NetVertices[poly],f=NetFaces[poly]},
	Plus@@@Abs/@Map[
		Area[Polygon[#]]&,
		Last/@NetPieces[poly]/.Thread[Range[Length[f]]->f]/.Thread[Range[Length[v]]->v],
		{2}
	].First/@NetPieces[poly]
]

NetVertices[l_List]:=
	Union[Flatten[(Flatten[l]) /. {Line[x_List]:>Drop[x,-1], Polygon[x_List]:>x},1],
	SameTest->(Abs[(#1-#2).(#1-#2)]<10^-6&)]

NonparallelVectors[v_List?MatrixQ,eps_:10^-10]:=Union[v,SameTest->(Norm[Cross[#1,#2]]<eps&)]

(* Norm *)

norm[a_List,b_List]:=norm[a-b]

norm[a_List]:=Sqrt[normsq[a]]

normsq[a_List,b_List]:=normsq[a-b]

normsq[a_List]:=a.a

NormalizedNet[poly_,opts___]:=Module[
	{
		v=Transpose[NetVertices[poly]]//N,
		dx,size,net
	},
    size=Subtract@@{Max/@v,Min/@v};
    net=If[Subtract@@size>0,
        dx={Max[v[[2]]],Min[v[[1]]]}; Net[poly,opts]/.{Line[x_]:>Line[Reverse/@x],
        	Polygon[x_]:>Polygon[Reverse/@x]},
        dx={Max[v[[1]]],Min[v[[2]]]}; Net[poly,opts]];
    net/.{Line[x_]:>Line[{#[[1]],-#[[2]]}&/@(#-dx&/@x)],
	    Polygon[x_]:>Polygon[{#[[1]],-#[[2]]}&/@(#-dx&/@x)]}
]

(* Orient *)

(* Returns a polyhedron recentered at a given internal point and all faces
pointing outward.  If the internal point is not specified, the centroid
is used.
*)

Orient[poly_,ctr_:{0,0,0}]:=poly/.Polygon[v_List]:>OrientPolygon[v,ctr]

OrientedQ[poly_,ctr_:{0,0,0}]:=poly/.Polygon[v_List]:>(Polygon[v]===OrientPolygon[v,ctr])

(* Check to see a point on the polygon offset by a small right-hand normal 
is closer or farther away to the origin.  The centroid of the polygon
can be used instead of face[[1]], but I'm not sure it really matters.
*)

OrientPolygon[face_List?MatrixQ]:=OrientPolygon[face,{0,0,0}]

OrientPolygon::degen = 
"Coincident vertices encountering in vertex list `1`."

OrientPolygon[face_List?MatrixQ,origin_List?VectorQ,eps_:0.001]:=Module[
	{
		c=Centroid[face],
		tip=Cross@@N[{face[[2]]-face[[1]],face[[-1]]-face[[1]]},100]
	},
	If[Norm[tip]==0.,
		Message[OrientPolygon::degen,face];
		face,
	(* Else *)
		tip=eps Unit[tip];

		(* Chop[N[#]] needed here to avoid approx real + exact real = 
		approx real + exact imaginary result possible in v4.2; see bug 42124
		*)
	
		Polygon[
			If[Chop[N[normsq[c+tip,origin]]] < Chop[N[normsq[c-tip,origin]]],
				Reverse[face],
			(* Else *)
				face
			]
		]
	]
]

rotmm::Singular="Singular value ArcTan[0,0] encounted: (`1`,`2`,`3`)"

rotmm[{x_, y_, z_}] := Module[{a, b,eps=.001},
	If[Sqrt[N[y^2+z^2]]<eps && Abs[N[x]]<eps,
		Message[rotmm::Singular,x,y,z],
		{a, b} = {If[Abs[N[y]]<eps && Abs[N[z]]<eps,0,ArcTan[z, y]], \
ArcTan[Sqrt[y^2 + z^2], x]};
		    {{Cos[b],0,-Sin[b]},{0,1,0},{Sin[b],0,Cos[b]}}.
    		{{1,0,0},{0,Cos[a],-Sin[a]},{0,Sin[a],Cos[a]}}
	]
]

Polygon2D::nonz="z-coordinates are not zero but should be: `1`";
Polygon2D::nonplanar="Polygon is not planar."
Polygon2D::dimension="Polygon vertices must be of the form {x,y,z}."

Polygon2D[p_List?MatrixQ,eps_:10^-16]:=Module[{v=#-p[[1]]&/@p,l},
	(* rotate last point above x-axis so y[n-1]=0 *)
	If[(l=Drop[v[[-1]],{3}])!={0,0},
		v=RotationMatrixZ[ArcTan@@l].#&/@v
	];
	
	(* rotate last point onto x-axis so y[n-1]=0, z[n-1]=0 *)
	If[(l=Drop[v[[-1]],{2}])!={0,0},
		v=RotationMatrixY[-ArcTan@@l].#&/@v
	];
	
	(* rotate 2nd point into xy-plane so z[2]=0 *)
	If[(l=Drop[v[[2]],{1}])!={0,0},
		v=RotationMatrixX[ArcTan@@l].#&/@v
	];

	(* check that all z-coordinates are actually 0 *)
	If[#.#&[Last/@v]>eps,
		Message[Polygon2D::nonz,Last/@v],
		Polygon[Drop[#,-1]&/@v]
	]
]/;Last[Dimensions[p]]==3&&CoplanarQ[p]

Polygon2D[Polygon[p_List?MatrixQ],eps_:10^-16]:=
  Message[Polygon2D::dimension]/;Last[Dimensions[p]]!=3

Polygon2D[Polygon[p_List?MatrixQ],eps_:10^-16]:=
  Message[Polygon2D::nonplanar]/;Not[CoplanarQ[p]]
  
Polygon3D[Polygon[poly2d_],z_,vec_]:=Module[
	{
	poly3d=Map[Append[#,z]&,poly2d,{2}]
	},
	Map[Transpose[rotmm[vec]].#&,poly3d,{2}]
]

PolygonArea[l_List]:=Module[
	{triangulate=Join[{1},#]&/@Partition[Range[2,Length[l]],2,1]},
    Plus@@(TriangleArea3D[l[[#]]]&/@triangulate)
]

(*

To convert an existing polyhedron, use

poly=Flatten[TriakisIcosahedronExact];
FullSimplify[v=PolyhedronVertices[poly]]//InputForm
First/@poly/.Thread[v->Range[Length[v]]]

*)

Polyhedron[p__]:=Module[{poly=VerticesToSolid[PolyhedronVertices[p],PolyhedronFaces[p]]},
	If[Cases[
			DownValues[PolyhedronPolyhedra],Verbatim[HoldPattern[PolyhedronPolyhedra[p]]],
			Infinity
		]=={},
		poly,
		poly[[#]]&/@PolyhedronPolyhedra[p]
	]
] 
(* otherwise, Prism[n] doesn't work *)
(*/;
Cases[DownValues[PolyhedronVertices],Verbatim[HoldPattern[PolyhedronVertices[p]]],Infinity]!={}&&
Cases[DownValues[PolyhedronFaces],Verbatim[HoldPattern[PolyhedronFaces[p]]],Infinity]!={}
*)

PolyhedronCompoundVolume[poly_]:=Module[
    {l=Length/@poly,l2,i,n=Length[poly]},
    l2=Length/@Table[poly[[i,1,1]],{i,n}];
    Plus@@Table[
        l[[i]]l2[[i]](PolyhedronVolume[{poly[[i,1,1,1]]}]+
              PolyhedronVolume[{poly[[i,1,2,1]]}]),{i,n}]
]

PolyhedronCompoundSurfaceArea[poly_]:=Module[
    {l=Length/@poly,l2=Length/@{poly[[1,1,1]],poly[[2,1,1]],i}},
    Plus@@Table[
        l[[i]]l2[[i]](PolyhedronSurfaceArea[{poly[[i,1,1,1]]}]+
              PolyhedronSurfaceArea[{poly[[i,1,2,1]]}]),{i,2}]
]

PolyhedronEdges[poly_List]:=
	SymbolicUnion[Flatten[Flatten[poly] /. Polygon:>ListEdges,1]]

PolyhedronEdges[poly_]:=Union[
	Sort/@Flatten[Partition[#,2,1,1]&/@PolyhedronFaces[poly],1]
]

PolyhedronEdgeLengths[poly_List]:=
	SymbolicUnion[norm@@@(PolyhedronEdges[Flatten[poly]])]

PolyhedronEdgeLengths[poly_]:=Module[{v=PolyhedronVertices[poly]},
	SymbolicUnion[Sqrt[#.#&/@Subtract@@@(v[[#]]&/@PolyhedronEdges[poly])]]
]

PolyhedronFaceCounts[poly_]:={Length[#],#[[1]]}&/@
    Split[Sort[(Flatten[poly]) /. Polygon:>Length]]

PolyhedronFaceCenters[poly_]:=
	Centroid/@(Flatten[poly]/.Polygon:>Identity)

PolyhedronFaceCenters[poly_,n_]:=
	Centroid/@(Select[Flatten[poly]/.Polygon:>Identity,Length[#]==n&])

PolyhedronFaceGraphics[poly_List]:=Module[{i,p=Flatten[poly]},
	Table[
		Graphics[p[[i]] /.Polygon:>Polygon2D,AspectRatio->Automatic],
		{i,Length[p]}
	]
]

PolyhedronFaces[l_List]:=Module[{v=PolyhedronVertices[l]},
    First/@Flatten[l]/.Thread[v->Range[Length[v]]]
]

FaceDet[Polygon[f_],x_List]:=Det[Append[#,1]&/@Prepend[Take[f,3],x]]<0

PolyhedronInequalities[poly_List,x_List]:=poly/.p_Polygon:>FaceDet[p,x]

PolyhedronRotationCompound[poly_,n_]:=Module[
	{
	j,
	dirs=PolyhedronFaceCenters[poly]
	},
	Table[RotateAboutAxis[#,dirs[[j]],Pi/n]&/@poly,{j,Length[dirs]}]
]

PolyhedronSurfaceArea::Faces="Using {facetypes,facecounts} = `1`"

PolyhedronSurfaceArea[poly_,f_List]:=Module[
	{faces,i},
	faces=Table[Select[poly,Length[#[[1]]]==f[[i]]&]//Flatten,{i,Length[f]}] /.
		Polygon:>Identity;
	Message[PolyhedronSurfaceArea::Faces,{Length[#[[1]]],Length[#]}&/@faces];
	Plus@@(PolygonArea[#[[1]]]Length[#]&/@faces)
]

PolyhedronSurfaceArea[poly_]:=
	Plus@@(PolygonArea[# /. Polygon:>Identity]&/@(Flatten[poly]))

PolyhedronSymmetricSurfaceArea[poly0_List]:=Module[
	{poly=Flatten[poly0]},
	PolyhedronSurfaceArea[poly,Union[Length/@(poly /. Polygon:>Identity)]]
]

PolyhedronSymmetricSurfaceArea[poly_]:=
	PolyhedronSymmetricSurfaceArea[Polyhedron[poly]]

PolyhedronSymmetricVolume[poly0_List]:=Module[
	{poly=Flatten[poly0]},
	PolyhedronVolume[Flatten[poly],Union[Length/@(poly /. Polygon:>Identity)]]
]

PolyhedronSymmetricVolume[poly_]:=
	PolyhedronSymmetricVolume[Polyhedron[poly]]

PolyhedronVertexDistances[poly_]:=
	VertexDistances[PolyhedronVertices[Flatten[poly]]]

(* Make have to simplify and re-union *)

PolyhedronVertices[poly_List]:=
	Union[Flatten[(Flatten[poly]) /. Polygon:>Identity,1],SameTest->SameQ]

PolyhedronVolume[poly_List]:=
	Plus@@(PyramidVolume[#/.Polygon:>Identity]&/@Flatten[poly])

PolyhedronVolume::Faces="Using {facetypes,facecounts} = `1`"

PolyhedronVolume[poly_List,f_List]:=Module[
	{faces,i},
	faces=Table[Select[Flatten[poly],Length[#[[1]]]==f[[i]]&]//Flatten,{i,Length[\
f]}] /. Polygon:>Identity;
	Message[PolyhedronVolume::Faces,{Length[#[[1]]],Length[#]}&/@faces];
	Plus@@(PyramidVolume[#[[1]]]Length[#]&/@faces)
]

PyramidVolume[v_List]:=Module[{n=Unit[Cross[v[[2]]-v[[1]],v[[-1]]-v[[1]]]]},
    1/3 PolygonArea[v] Abs[v[[1]].n]
]

RectifiedPair[poly_] := 
  Module[{v = Transpose[PolyhedronVertices[poly]], m, eps = .01},
    m = {Min[#] - eps, Max[#] + eps} & /@ v;
    {Graphics3D[{poly, VertexFigures[poly]}, Boxed -> False, PlotRange -> m],
      Graphics3D[Rectify[poly], Boxed -> False, PlotRange -> m]}
]

Rectify[poly_List]:=Module[
	{
      vf = VertexFigures[poly],
      v = PolyhedronVertices[poly],
      cent = Centroid /@ (poly /. Polygon :> Identity),
      mid = Midpoint /@ PolyhedronEdges[poly],
      dist2, i, eps = .01
	},
    dist2 = normsq[Midpoint[Take[poly[[1]] /. Polygon :> Identity, 2]], 
        Centroid[poly[[1]] /. Polygon :> Identity]];
    {
	vf /. Line[l_List] :> Polygon[Drop[l, -1]],
	Convexify/@Polygon /@ 
		Table[Select[mid, N[Abs[normsq[cent[[i]], #] - dist2]] < eps &],
		{i,Length[cent]}]
	}
]

(* Rotation Stuff *)

RotationMatrixX[x_]:={{1,0,0},{0,Cos[x],Sin[x]},{0,-Sin[x],Cos[x]}}
RotationMatrixY[x_]:={{Cos[x],0,-Sin[x]},{0,1,0},{Sin[x],0,Cos[x]}}
RotationMatrixZ[x_]:={{Cos[x],Sin[x],0},{-Sin[x],Cos[x],0},{0,0,1}}

RotationMatrix[x_]:={{Cos[x],Sin[x]},{-Sin[x],Cos[x]}}

RotationFormula[r_List?VectorQ,n0_List?VectorQ,ang_]:=Module[
	{n=Unit[n0]},
	r Cos[ang]+n(n.r)(1-Cos[ang])+Cross[r,n]Sin[ang]
]

RotateAboutAxis[Polygon[l_List],n_List,ang_]:=
	Polygon[RotationFormula[#,n,ang]&/@l]

RotateAboutAxis[l_List,n_List,ang_]:=l/.Polygon[x_]:>RotateAboutAxis[Polygon[x],n,ang]

RotateE[g_,n_,ang_]:=Module[{i},
	Polygon/@Table[RotateAboutAxis[g[[i]],n,ang],{i,Length[g]}]
]

Snubify[poly_,d_,rot_]:=
  ConvexHull3D[
    PolyhedronVertices[poly/.Polygon[v_]:>RotateAboutAxis[
            Polygon[d Unit[c=Centroid[v]]+#&/@v],c,rot]]]
    
SolidCentroid[poly_]:=Module[{facectrs=poly /. Polygon:>Centroid},
	Chop[Plus@@facectrs/Length[facectrs]]
]

SphericalPolyhedronArcs[p_,n_]:=Module[{i},
    Table[Unit[#[[1]]+(#[[2]]-#[[1]])i/n],{i,0,n}]&/@PolyhedronEdges[p]
]

SphericalPolyhedron[p_List,n_:10]:=Line/@SphericalPolyhedronArcs[p,n]

SphericalPolyhedron[p_List,n_,r_]:=
    Polygon[Join[#,r Reverse[#]]]&/@SphericalPolyhedronArcs[p,n]

SphericalPolyhedron[p_,n_:10]:=SphericalPolyhedron[Polyhedron[p],n]
SphericalPolyhedron[p_,n_,r_]:=SphericalPolyhedron[Polyhedron[p],n,r]

SpiderWeb[Polygon[l_List]]:=Module[{c=Centroid[l]},
	{
		ClosedLine[l],
		Line[{#,c}]&/@l,
		Line[{c,Plus@@l[[#]]/2}]&/@Partition[Range[Length[l]],2,1,1]
	}
]

SymbolicUnion[v_,opts___]:=Module[
	{
	nv,
	prec=WorkingPrecision/.{opts}/.Options[SymbolicUnion]
	},
	nv=Chop[N[v,prec]];
	First/@Split[Sort[nv]]/.Thread[nv->v]
]

(* defined in SolidGeometry.m *)
TriangleArea3D[{x1_,x2_,x3_}]:=Module[{i={1,1,1},x,y,z},
      {x,y,z}=Transpose[{x1,x2,x3}];
      Sqrt[Det[{y,z,i}]^2+Det[{z,x,i}]^2+Det[{x,y,i}]^2]/2
]

TruncateByPlanes[poly_,r_]:=
  Module[{i=PolyhedronInequalities[Polyhedron[poly],{x,y,z}],i2},
    i2=Numerator[Simplify[#.({x,y,z}-r Unit[#])]]<0&/@
        PolyhedronVertices[poly];
    PolyhedronFromInequalities[Join[i,i2],{x,y,z}]
]

Unit[v_List]:=v/Sqrt[v.v]

VertexDistances[v_]:=Module[{i,j},
	Sort[Union[
		FullSimplify[Together//@
			Flatten[
				Table[Sqrt[(v[[i]]-v[[j]]).(v[[i]]-v[[j]])],
					{i,Length[v]},{j,i+1,Length[v]}
				]
			]
		],SameTest->SameQ
	],Less]
]

VertexFigures[poly_List,opts___]:=Module[
	{
		v=PolyhedronVertices[poly],
		e=PolyhedronEdges[poly],
		i,n,l,m,
		figures,holes,
		convex=Convex/.{opts}/.Options[VertexFigures]
	},
	n=Length[v];
	l=e/.Thread[v->Range[n]];
	m=Midpoint/@e;
	figures=Table[First/@Position[l,i],{i,n}];
	Flatten[Function[f,Convexify/@Polygon/@Map[m[[#]]&,f,{2}]]/@
		If[convex,
			{figures,holes=Map[Sort,
    			Partition[#,2,1,1]&/@First/@poly/.Thread[v->Range[n]],
    			{2}]/.Thread[l->Range[Length[l]]]},
			{figures}
		]
	]
]

VertexLabels[Polygon[l_List]]:=
  MapIndexed[Text[ToString[Sequence@@#2],1.05#1]&,l]

VerticesToSolid[v_List?MatrixQ,f_List]:=Polygon[v[[#]]]&/@f

WenningerModel[n_]:=Module[{l=Select[WenningerDuals,#[[1]]==n&]},
	{#[[2]]<>" ("<>ustring[#[[2]]]<>")",
		"Wenninger dual "<>ToString[#[[3]]]<>" ("<>#[[4]]<>")"}&/@l
]

Wireframe[l_List]:=Union[frame[l]]
frame[p_]:=Switch[p,
	_Cuboid,Union[(p/.Cuboid->cuboid)/.Polygon->ClosedLine],
	_List,Union[(p/.Cuboid->cuboid)/.Polygon->ClosedLine],
	_Graphics3D,Union[(p[[1]]/.Cuboid->cuboid)/.Polygon->ClosedLine]
]

End[]

(* Protect[  ] *)

EndPackage[]

Platonics

(*:Mathematica Version: 5.2 *)

(*:Package Version: 1.45 *)

(*:Name: MathWorld`Platonic` *)

(*:Author: Eric W. Weisstein *)

(*:URL:
	http://mathworld.wolfram.com/packages/Platonic.m
*)

(*:Summary:
*)

(*:History:
  v1.00 (1999-08-26): Split off from Polyhedra
  v1.01 (1999-08-27): DodecahedronIcosahedronExact, CubeOctahedronExact,
                      StellaOctangulaExact
  v1.02 (1999-09-02): Moved Cube-n-Compounds here.
                      Cube5CompoundExact, Cube5CompoundDualExact,
                      Octahedron5CompoundExact.
  v1.03 (2000-01-01): URL updated
  v1.04 (2000-02-15): IcosahedronExact2
  v1.05 (2000-02-17): Orientations fixes
  v1.06 (2000-02-23): DodecahedronSTI orientation fixed
  v1.07 (2000-02-27): GreatStellatedDodecahedronExact cumulation sign fixed,
                      GreatIcosahedronExact oriented.
  v1.08 (2000-08-19): Added continuation characters to usage messages.
                      DodecahedronExactZ, Dodecahedron2Compound,
                      Dodecahedron5Compound.
  v1.09 (2000-08-22): Dodecahedron3Compound, DodecahedronTilted
  v1.10 (2000-08-23): Octahedron3Compound
  v1.11 (2000-11-13): PolyhedronOperations` added to BeginPackage
  v1.12 (2000-12-04): Fixed incorrect FiveCubes cached name.  Deleted
                      spurious quadrilaterals from Cube5Compound.
  v1.13 (2002-08-08): StellaOctangula.  Unnumericized and correctly oriented
                      Cube2Compound.
  v1.14 (2002-08-30): Cube5Compound now returns unprocessed version by default.
  v1.15 (2002-09-19): Platonic, PlatonicName
  v1.16 (2002-10-15): OctahedronExactZ, TetrahedronExactZ
  v1.17 (2002-11-06): Convexify fixed to new syntax.
  v1.20 (2002-11-10): Converted to new Polyhedron[] syntax and removed
                      construction code.  Added nets.
  v1.21 (2002-11-22): DodecahedronZ, Dodecahedron5Compound, Octahedron3Compound,
                      Octahedron4Compound
  v1.25 (2003-01-01): Dodecahedron6Compound.  Renamed Dodecahedron5Compound.
                      Added PolyhedronPolyhedra definitions for some compounds.
                      Cube6Compound, Cube7Compound.  Platonic symmetry orientations,
                      Octahedron2Compound, Icosahedron5Compound, Icosahedron6Compound
  v1.26 (2003-01-12): Cube*Compound nets, Dodecahedron6Compound net,
                      Tetrahedron5Compound net
  v1.27 (2003-01-19): Tetrahedron6Compound
  v1.28 (2003-01-20): Cube6Compound C3
  v1.29 (2003-01-27): Cube10Compound, Octahedron10Compound
  v1.30 (2003-02-04): DodecahedronIcosahedronCompound
  v1.31 (2003-02-08): Tetrahedron4Compound
  v1.32 (2003-02-19): Corrected orientations where needed, added radii
  v1.33 (2003-06-03): Platonics
  v1.34 (2003-08-10): Tetrahedron3Compound
  v1.35 (2003-08-17): Dodecahedron C2
  v1.36 (2003-09-04): Polyhedron[Octahehedron,"C4Diamond"],
                      Polyhedron[Octahehedron,"C4Square"]
  v1.37 (2003-10-19): context udpated
  v1.38 (2003-11-09): fixed default Polyhedron[Octahedron]
  v1.39 (2003-11-15): fixed NetFaces[Octahedron3Compound], Net[Octahedron2Compound,"C4"],
                      Net[Dodecahedron2Compound]
  v1.40 (2003-11-17): Added mirror image piece to Net[Tetrahedron4Compound,"C3"]
  v1.41 (2003-12-16): Added self-intersecting versions of the Kepler-Poinsot solids
  v1.42 (2003-12-30): Added Polyhedron[Icosahedron,"C2"]
  v1.43 (2004-09-08): Platonics is now in alphabetical order
  v1.44 (2005-04-08): Typo fixes
  v1.45 (2005-07-13): KeplerPoinsotSolids
  
  (c) 1999-2005 Eric W. Weisstein
*)

(*:Keywords:
*)

(*:Requirements:

  Requires the additional package PolyhedronOperations.m available from

    http://mathworld.wolfram.com/packages/

*)

(*:Discussion:

Contains definitions for contruction of all Platonic and Kepler-Poinsot
solids (and some compounds involving them) with analytic vertices 
and convexified corretly-oriented faces.

Polygons are all oriented so that the right-hand normal points away from the
centroid.

*)

(*:References: *)

(*:Limitations:
*)

BeginPackage["MathWorld`Platonic`",
	{
	"MathWorld`PolyhedronOperations`"
	}
]

Cube::usage =
"Polyhedron[Cube,<orientation>] gives a list of Polygons with analytic vertices \
representing a cube with unit side lengths.  The orientation may be specified \
with an optional second argument \"C3\" or \"C4\" (default) that specifies the rotational \
symmetry of the solid around the z-axis."

Cube2Compound::usage =
"Polyhedron[Cube2Compound,orientation] gives a list of Polygons with exact \
vertices representing the compound of 2 cubes, where orientation may be \"C3\" or \"C4\"."

Cube3Compound::usage =
"Polyhedron[Cube3Compound,orientation] gives a list of polygons with exact vertices \
comprising a compound of 3 cubes, where orientation may be \"C3\", \"C4\", or \"2C2\"."

Cube4Compound::usage =
"Polyhedron[Cube4Compound] gives a list of polygons comprising of a compound of \
4 cubes."

Cube5Compound::usage =
"Polyhedron[Cube5Compound] gives a list of polygons comprising a compound of \
5 cubes."

Cube6Compound::usage =
"Polyhedron[Cube6Compound,<orientation>] gives a list of polygons comprising a \
compound of 6 cubes for orientation \"C3\" or \"C4\"."

Cube7Compound::usage =
"Polyhedron[Cube7Compound] gives a list of polygons comprising a compound of \
7 cubes."

Cube10Compound::usage =
"Polyhedron[Cube10Compound] gives a list of polygons comprising a compound of \
10 cubes."

CubeOctahedronCompound::usage =
"Polyhedron[CubeOctahedronCompound] gives the compound of a cube and an octahedron \
by midpoint cumulation."

Dodecahedron::usage =
"Polyhedron[Dodecahedron,<orientation>] gives a list of Polygons with analytic vertices \
representing a dodecahedron with unit side lengths.  The orientation may be specified \
with an optional second argument \"C2\", \"C3\", or \"C5\" (default) that specifies the rotational \
symmetry of the solid around the z-axis."

Dodecahedron2Compound::usage =
"Polyhedron[Dodecahedron2Compound,orientation] gives a 2-dodecahedron compound with \
symmetry specified by orientation \"C2\", \"C3\", or \"C5\"."

Dodecahedron3Compound::usage =
"Polyhedron[Dodecahedron3Compound,n] gives the 3-dodecahedron compound."

Dodecahedron5Compound::usage =
"Polyhedron[Dodecahedron5Compound] gives the 5-dodecahedron compound."

Dodecahedron6Compound::usage =
"Polyhedron[Dodecahedron6Compound] gives the 6-dodecahedron compound."

DodecahedronIcosahedronCompound::usage =
"Polyhedron[DodecahedronIcosahedronCompound] gives a list of Polygons with \
analytic vertices representing a dodecahedron and icosahedron compound."

GreatDodecahedron::usage =
"Polyhedron[GreatDodecahedron] gives a list of mutually intersecting pentagons comprising \
the great dodecahedron.  Polyhedron[GreatDodecahedron,Split->True] gives a list of Polygons \
with analytic vertices representing the nonconvex hull of this Kepler-Poinsot solid."

GreatIcosahedron::usage =
"Polyhedron[GreatIcosahedron] gives a list of Polygons with analytic vertices \
representing this Kepler-Poinsot solid constructed from a unit icosahedron.  \
Polyhedron[GreatIcosahedron,Split->True] gives Polygons that have been split along \
face intersections and with the interior portion removed."

GreatStellatedDodecahedron::usage =
"Polyhedron[GreatStellatedDodecahedron] gives a list of Polygons representing \
this Kepler-Poinsot solid.  Polyhedron[GreatStellatedDodecahedron,Split->True] \
gives a version split at face intersections."

Icosahedron::usage =
"Polyhedron[Icosahedron,<orientation>] gives a list of Polygons with analytic vertices \
representing a icosahedron with unit side lengths.  The orientation may be specified \
with an optional second argument \"C2\", \"C3\", or \"C5\" (default) that specifies the rotational \
symmetry of the solid around the z-axis."

Icosahedron5Compound::usage =
"Polyhedron[Icosahedron5Compound] gives a list of Polygons with exact vertices representing \
a compound of 5 icosahedra."

Icosahedron6Compound::usage =
"Polyhedron[Icosahedron5Compound] gives a list of Polygons with exact vertices representing \
a compound of 6 icosahedra."

KeplerPoinsotSolids::usage =
"Platonics gives a list of the four Kepler-Poinsot solids in alphabetical order."

Octahedron::usage =
"Polyhedron[Octahedron,<orientation>] gives a list of Polygons with analytic vertices \
representing a octahedron with unit side lengths.  The orientation may be specified \
with an optional second argument \"C3\", \"C4Square\", or \"C4Diamond\" that specifies \
the rotational symmetry of the solid around the z-axis."

Octahedron2Compound::usage =
"Polyhedron[Octahedron2Compound,orientation] gives a list of Polygons with exact \
vertices representing the compound of 2 octahedra with orientation \"C3\" or \"C4\"."

Octahedron3Compound::usage =
"Polyhedron[Octahedron3Compound] gives a list of Polygons with exact vertices representing the \
compound of 3 octahedra."

Octahedron4Compound::usage =
"Polyhedron[Octahedron4Compound] gives a list of Polygons exact vertices representing the \
compound of 4 octahedra."

Octahedron5Compound::usage =
"Polyhedron[Octahedron5Compound] gives a list of Polygons with analytic \
vertices representing the compound of 5 octahedra."

Octahedron10Compound::usage =
"Polyhedron[Octahedron10Compound] gives a list of Polygons with analytic \
vertices representing the compound of 10 octahedra."

Platonic::usage =
"Platonic[n] gives an exact representation of the nth Platonic solid."

Platonics::usage =
"Platonics gives a list of the Platonic solids in alphabetical order."

SmallStellatedDodecahedron::usage =
"Polyhedron[SmallStellatedDodecahedron] gives a list of Polygons with \
analytic vertices representing this Kepler-Poinsot solid.  \
Polyhedron[SmallStellatedDodecahedron,Split->True] gives Polygons that have been \
split at face intersections."

StellaOctangula::usage =
"Polyhedron[StellaOctangula] gives a list of Polygons with analytic vertices \
representing a stella octangula constructed from tetrahedra with unit side lengths.  \
Polyhedron[StellaOctangula,Split->True] returns the polyedron split into its \
nonconvex hull."

Tetrahedron::usage =
"Polyhedron[Tetrahedron,<orientation>] gives a list of Polygons with analytic vertices \
representing a tetrahedron with unit side lengths.  The orientation may be specified \
with an optional second argument \"C2\" or \"C3\" (default) that specifies the rotational \
symmetry of the solid around the z-axis."

Tetrahedron2Compound::usage =
"Polyhedron[Tetrahedron2Compound,<chirality>] gives a list of Polygons with analytic \
vertices representing the 2-tetrahedron compound with handedness ±1.  \
If no handedness is specified, +1 (dextro) is assumed."

Tetrahedron3Compound::usage =
"Polyhedron[Tetrahedron3Compound] gives a list of Polygons with analytic \
vertices representing the 3-tetrahedron compound."

Tetrahedron4Compound::usage =
"Polyhedron[Tetrahedron4Compound,<orientation>] gives a list of Polygons with analytic vertices \
representing a compound of 4 tetrahedron with unit side lengths.  The orientation may be specified \
as \"C3\" (default)."

Tetrahedron5Compound::usage =
"Polyhedron[Tetrahedron5Compound,<chirality>] gives a list of Polygons with analytic \
vertices representing the 5-tetrahedron compound with handedness ±1.  \
If no handedness is specified, +1 (dextro) is assumed."

Tetrahedron6Compound::usage =
"Polyhedron[Tetrahedron6Compound] gives a list of Polygons with analytic \
vertices representing the 6-tetrahedron compound."

Tetrahedron10Compound::usage =
"Polyhedron[Tetrahedron10Compound] gives a list of Polygons with analytic \
vertices representing the 10-tetrahedron compound."

Begin["`Private`"]


(* Cube *)

PolyhedronName[Cube]:="cube"
PolyhedronDual[Cube]:=Octahedron
NetFaces[Cube]:={{5,4,8,9},{6,5,9,10},{2,1,4,5},{4,3,7,8},{9,8,11,12},{12,11,13,14}}
NetVertices[Cube]:=
{{-3/2,-1/2},{-3/2,1/2},{-1/2,-3/2},{-1/2,-1/2},{-1/2,1/2},{-1/2,3/2},{1/2,-3/2},{1/2,-1/2},{1/2,1/2},{1/2,3/2},
{3/2,-1/2},{3/2,1/2},{5/2,-1/2},{5/2,1/2}}
PolyhedronPolyhedra[Cube]:={Range[6]}
PolyhedronFaces[Cube]:=PolyhedronFaces[Cube,"C4"]
PolyhedronVertices[Cube]:=PolyhedronVertices[Cube,"C4"]

(* Cube C3 *)

PolyhedronDual[Cube,"C3"]:=Sequence[Octahedron,"C4"]
PolyhedronPolyhedra[Cube,"C3"]:={Range[6]}
PolyhedronFaces[Cube,"C3"]:=
{{1,4,3,5},{6,2,3,4},{7,5,3,2},{6,4,1,8},{1,5,7,8},{7,2,6,8}}
PolyhedronVertices[Cube,"C3"]:=
{{0,0,-Sqrt[3]/2},{0,0,Sqrt[3]/2},{-Sqrt[2/3],0,1/(2Sqrt[3])},
 {-(1/Sqrt[6]),-(1/Sqrt[2]),-1/(2Sqrt[3])},{-(1/Sqrt[6]),1/Sqrt[2],
  -1/(2Sqrt[3])},{1/Sqrt[6],-(1/Sqrt[2]),1/(2Sqrt[3])},
 {1/Sqrt[6],1/Sqrt[2],1/(2Sqrt[3])},{Sqrt[2/3],0,-1/(2Sqrt[3])}}
 
(* Cube C4 *)

PolyhedronDual[Cube,"C4"]:=Sequence[Octahedron,"C3"]
PolyhedronPolyhedra[Cube,"C4"]:={Range[6]}
PolyhedronFaces[Cube,"C4"]:={{8,4,2,6},{8,6,5,7},{8,7,3,4},{4,3,1,2},{1,3,7,5},{2,1,5,6}}
PolyhedronVertices[Cube,"C4"]:={{-1,-1,-1},{-1,-1,1},{-1,1,-1},{-1,1,1},
 {1,-1,-1},{1,-1,1},{1,1,-1},{1,1,1}}/2

(* Cube 2-Compound C3 *)

PolyhedronName[Cube2Compound,"C3"]:="cube 2-compound"
PolyhedronDual[Cube2Compound,"C3"]:=Octahedron2Compound
NetPieces[Cube2Compound,"C3"]:={{12,{1,2,3}}}
NetFaces[Cube2Compound,"C3"]:={{3,4,5},{3,4,1},{3,2,1}}
NetVertices[Cube2Compound,"C3"]:={{-1,0},{0,-1},{0,0},{0,2},{1,0}}/2
PolyhedronPolyhedra[Cube2Compound,"C3"]:={{1,2,3,4,5,6},{7,8,9,10,11,12}}
PolyhedronFaces[Cube2Compound,"C3"]:=
{{2,11,12,4},{8,3,11,2},{4,9,8,2},{3,8,9,1},{1,9,4,12},{1,12,11,3},{2,5,14,
    13},{10,7,5,2},{13,6,10,2},{7,10,6,1},{1,6,13,14},{1,14,5,7}}
PolyhedronVertices[Cube2Compound,"C3"]:=
{{0,0,-Sqrt[3]},{0,0,Sqrt[3]},
 {0,-2Sqrt[2/3],-(1/Sqrt[3])},{0,2Sqrt[2/3],1/Sqrt[3]},{0,-2Sqrt[2/3],1/Sqrt[3]},
 {0,2Sqrt[2/3],-(1/Sqrt[3])},{-Sqrt[2],-Sqrt[2/3],
  -(1/Sqrt[3])},{-Sqrt[2],-Sqrt[2/3],1/Sqrt[3]},
 {-Sqrt[2],Sqrt[2/3],-(1/Sqrt[3])},
 {-Sqrt[2],Sqrt[2/3],1/Sqrt[3]},{Sqrt[2],-Sqrt[2/3],
  1/Sqrt[3]},{Sqrt[2],Sqrt[2/3],-(1/Sqrt[3])},
 {Sqrt[2],Sqrt[2/3],1/Sqrt[3]},{Sqrt[2],-Sqrt[2/3],-(1/Sqrt[3])}}/2

(* Cube 2-Compound C4 *)

PolyhedronDual[Cube2Compound,"C4"]:=Octahedron2Compound
PolyhedronPolyhedra[Cube2Compound,"C4"]:={{1,2,3,4,5,6},{7,8,9,10,11,12}}
PolyhedronFaces[Cube2Compound,"C4"]:=
{{12,4,2,10},{12,10,9,11},{12,11,3,4},{4,3,1,2},{1,3,11,9},{2,1,9,10},{16,8,
    14,6},{16,6,5,15},{16,15,7,8},{8,7,13,14},{13,7,15,5},{14,13,5,6}}
PolyhedronVertices[Cube2Compound,"C4"]:=
{{-1/2,-1/2,-1/2},{-1/2,-1/2,1/2},{-1/2,1/2,-1/2},
 {-1/2,1/2,1/2},{0,-(1/Sqrt[2]),-1/2},
 {0,-(1/Sqrt[2]),1/2},{0,1/Sqrt[2],-1/2},
 {0,1/Sqrt[2],1/2},{1/2,-1/2,-1/2},{1/2,-1/2,1/2},
 {1/2,1/2,-1/2},{1/2,1/2,1/2},
 {-(1/Sqrt[2]),0,-1/2},{-(1/Sqrt[2]),0,1/2},
 {1/Sqrt[2],0,-1/2},{1/Sqrt[2],0,1/2}}

(* Cube 3-Compound 2C2 *)

PolyhedronName[Cube3Compound,"2C2"]:="cube 3-compound"
NetPieces[Cube3Compound,"2C2"]:={{24,{1,2,3}},{24,{4,5}}}
NetFaces[Cube3Compound,"2C2"]:={{6,1,9},{1,5,10,9},{1,4,7,6},{8,3,11},{2,8,11}}
NetVertices[Cube3Compound,"2C2"]:=
{{0,(-1+Sqrt[2])/2},{1,(-1+Sqrt[2])/4},{1,(-1+2Sqrt[2])/4},
 {-1/(2Sqrt[2]),(-2+3Sqrt[2])/4},{1/(2Sqrt[2]),(-2+3Sqrt[2])/4},
 {(1-Sqrt[2])/2,0},{1-Sqrt[2],(-1+Sqrt[2])/2},{(3-Sqrt[2])/2,1/4},
 {(-1+Sqrt[2])/2,0},{-1+Sqrt[2],(-1+Sqrt[2])/2},{(1+Sqrt[2])/2,1/4}}
PolyhedronPolyhedra[Cube3Compound,"2C2"]:=
{{1,2,3,4,5,6},{7,8,9,10,11,12},{13,14,15,16,17,18}}
PolyhedronFaces[Cube3Compound,"2C2"]:=
{{16,4,2,14},{16,14,15,13},{16,13,1,4},{4,1,3,2},{3,1,13,15},{2,3,15,14},{8,
    20,17,6},{8,6,21,24},{8,24,7,20},{20,7,5,17},{5,7,24,21},{17,5,21,6},{23,
    12,19,10},{23,10,9,22},{23,22,11,12},{12,11,18,19},{18,11,22,9},{19,18,9,
    10}}
PolyhedronVertices[Cube3Compound,"2C2"]:=
{{-1/2,0,-(1/Sqrt[2])},{-1/2,0,1/Sqrt[2]},{-1/2,-(1/Sqrt[2]),0},
 {-1/2,1/Sqrt[2],0},{0,-1/2,-(1/Sqrt[2])},{0,-1/2,1/Sqrt[2]},
 {0,1/2,-(1/Sqrt[2])},{0,1/2,1/Sqrt[2]},{0,-(1/Sqrt[2]),-1/2},
 {0,-(1/Sqrt[2]),1/2},{0,1/Sqrt[2],-1/2},{0,1/Sqrt[2],1/2},
 {1/2,0,-(1/Sqrt[2])},{1/2,0,1/Sqrt[2]},{1/2,-(1/Sqrt[2]),0},
 {1/2,1/Sqrt[2],0},{-(1/Sqrt[2]),-1/2,0},{-(1/Sqrt[2]),0,-1/2},
 {-(1/Sqrt[2]),0,1/2},{-(1/Sqrt[2]),1/2,0},{1/Sqrt[2],-1/2,0},
 {1/Sqrt[2],0,-1/2},{1/Sqrt[2],0,1/2},{1/Sqrt[2],1/2,0}}

(* Cube 3-Compound C3 *)

PolyhedronName[Cube3Compound,"C3"]:="cube 3-compound"
PolyhedronPolyhedra[Cube3Compound,"C3"]:=
{{1,2,3,4,5,6},{7,8,9,10,11,12},{13,14,15,16,17,18}}
PolyhedronFaces[Cube3Compound,"C3"]:=
{{1,5,3,6},{7,2,3,5},{8,6,3,2},{7,5,1,4},{1,6,8,4},{8,2,7,4},{1,9,12,16},{13,
    2,12,9},{20,16,12,2},{13,9,1,17},{1,16,20,17},{20,2,13,17},{1,10,14,
    18},{11,2,14,10},{19,18,14,2},{11,10,1,15},{1,18,19,15},{19,2,11,15}}
PolyhedronVertices[Cube3Compound,"C3"]:=
{{0,0,-Sqrt[3]/2},{0,0,Sqrt[3]/2},{-Sqrt[2/3],0,1/(2Sqrt[3])},
 {Sqrt[2/3],0,-1/(2Sqrt[3])},{-(1/Sqrt[6]),-(1/Sqrt[2]),-1/(2Sqrt[3])},
 {-(1/Sqrt[6]),1/Sqrt[2],-1/(2Sqrt[3])},{1/Sqrt[6],-(1/Sqrt[2]),1/(2Sqrt[3])},
 {1/Sqrt[6],1/Sqrt[2],1/(2Sqrt[3])},{Root[-1+54#1^2-216#1^4+216#1^6&,1,0],
  Root[-1+18#1^2-72#1^4+72#1^6&,3,0],-1/(2Sqrt[3])},
 {Root[-1+54#1^2-216#1^4+216#1^6&,1,0],Root[-1+18#1^2-72#1^4+72#1^6&,
   4,0],-1/(2Sqrt[3])},{Root[-1+54#1^2-216#1^4+216#1^6&,2,0],
  Root[-1+18#1^2-72#1^4+72#1^6&,2,0],1/(2Sqrt[3])},
 {Root[-1+54#1^2-216#1^4+216#1^6&,2,0],Root[-1+18#1^2-72#1^4+72#1^6&,
   5,0],1/(2Sqrt[3])},{Root[-1+54#1^2-216#1^4+216#1^6&,3,0],
  Root[-1+18#1^2-72#1^4+72#1^6&,1,0],1/(2Sqrt[3])},
 {Root[-1+54#1^2-216#1^4+216#1^6&,3,0],Root[-1+18#1^2-72#1^4+72#1^6&,
   6,0],1/(2Sqrt[3])},{Root[-1+54#1^2-216#1^4+216#1^6&,4,0],
  Root[-1+18#1^2-72#1^4+72#1^6&,1,0],-1/(2Sqrt[3])},
 {Root[-1+54#1^2-216#1^4+216#1^6&,4,0],Root[-1+18#1^2-72#1^4+72#1^6&,
   6,0],-1/(2Sqrt[3])},{Root[-1+54#1^2-216#1^4+216#1^6&,5,0],
  Root[-1+18#1^2-72#1^4+72#1^6&,2,0],-1/(2Sqrt[3])},
 {Root[-1+54#1^2-216#1^4+216#1^6&,5,0],Root[-1+18#1^2-72#1^4+72#1^6&,
   5,0],-1/(2Sqrt[3])},{Root[-1+54#1^2-216#1^4+216#1^6&,6,0],
  Root[-1+18#1^2-72#1^4+72#1^6&,3,0],1/(2Sqrt[3])},
 {Root[-1+54#1^2-216#1^4+216#1^6&,6,0],Root[-1+18#1^2-72#1^4+72#1^6&,
   4,0],1/(2Sqrt[3])}}

(* Cube 3-Compound C4 *)

PolyhedronName[Cube3Compound,"C4"]:="cube 3-compound"
PolyhedronPolyhedra[Cube3Compound,"C4"]:=
{{1,2,3,4,5,6},{7,8,9,10,11,12},{13,14,15,16,17,18}}
PolyhedronFaces[Cube3Compound,"C4"]:=
{{8,4,2,6},{8,6,5,7},{8,7,3,4},{4,3,1,2},{1,3,7,5},{2,1,5,6},{22,20,12,
    14},{22,14,13,21},{22,21,19,20},{20,19,11,12},{11,19,21,13},{12,11,13,
    14},{18,24,16,10},{18,10,9,17},{18,17,23,24},{24,23,15,16},{15,23,17,
    9},{16,15,9,10}}
PolyhedronVertices[Cube3Compound,"C4"]:=
{{-1/2,-1/2,-1/2},{-1/2,-1/2,1/2},{-1/2,1/2,-1/2},{-1/2,1/2,1/2},{1/2,-1/2,-1/2},
 {1/2,-1/2,1/2},{1/2,1/2,-1/2},{1/2,1/2,1/2},{(-1-Sqrt[3])/4,(1-Sqrt[3])/4,-1/2},
 {(-1-Sqrt[3])/4,(1-Sqrt[3])/4,1/2},{(-1-Sqrt[3])/4,(-1+Sqrt[3])/4,-1/2},
 {(-1-Sqrt[3])/4,(-1+Sqrt[3])/4,1/2},{(1-Sqrt[3])/4,(-1-Sqrt[3])/4,-1/2},
 {(1-Sqrt[3])/4,(-1-Sqrt[3])/4,1/2},{(1-Sqrt[3])/4,(1+Sqrt[3])/4,-1/2},
 {(1-Sqrt[3])/4,(1+Sqrt[3])/4,1/2},{(-1+Sqrt[3])/4,(-1-Sqrt[3])/4,-1/2},
 {(-1+Sqrt[3])/4,(-1-Sqrt[3])/4,1/2},{(-1+Sqrt[3])/4,(1+Sqrt[3])/4,-1/2},
 {(-1+Sqrt[3])/4,(1+Sqrt[3])/4,1/2},{(1+Sqrt[3])/4,(1-Sqrt[3])/4,-1/2},
 {(1+Sqrt[3])/4,(1-Sqrt[3])/4,1/2},{(1+Sqrt[3])/4,(-1+Sqrt[3])/4,-1/2},
 {(1+Sqrt[3])/4,(-1+Sqrt[3])/4,1/2}}

(* Cube 4-Compound *)

PolyhedronName[Cube4Compound]:="cube 4-compound"
NetPieces[Cube4Compound]:={{8,{1,2,3}},{24,{4,5,6}}}
NetFaces[Cube4Compound]:=
{{1,11,14,10},{16,13,14,12},{15,13,14,11},{4,9,8,6,7,5},{5,2,4},{4,3,9}}
NetVertices[Cube4Compound]:=
{{0,0},{19/20,0},{6/5,-1/4},{6/5,0},{6/5,1/2},{47/35,1/7},
 {76/55,5/11},{91/55,2/11},{17/10,0},{9/(77Sqrt[2]),-3/(7Sqrt[2])},
 {9/(77Sqrt[2]),3/(7Sqrt[2])},{75/(77Sqrt[2]),-3/(7Sqrt[2])},
 {75/(77Sqrt[2]),3/(7Sqrt[2])},{(3Sqrt[2])/11,0},
 {(3Sqrt[2])/11,(3Sqrt[2])/11},{(6Sqrt[2])/11,0}}
PolyhedronPolyhedra[Cube4Compound]:=
{{1,2,3,4,5,6},{7,8,9,10,11,12},{13,14,15,16,17,18},{19,20,21,22,23,24}}
PolyhedronFaces[Cube4Compound]:=
{{28,15,4,12},{28,12,18,29},{28,29,21,15},{15,21,5,4},{5,21,29,18},{4,5,18,
    12},{26,31,24,14},{26,14,2,9},{26,9,19,31},{31,19,7,24},{7,19,9,2},{24,7,
    2,14},{8,1,10,20},{8,20,32,23},{8,23,13,1},{1,13,25,10},{25,13,23,32},{10,
    25,32,20},{6,17,30,22},{6,22,16,3},{6,3,11,17},{17,11,27,30},{27,11,3,
    16},{30,27,16,22}}
PolyhedronVertices[Cube4Compound]:=
{{-5,-1,-1},{-5,-1,1},{-5,1,-1},{-5,1,1},{-3,-3,-3},{-3,-3,3},{-3,3,-3},{-3,3,
    3},{-1,-5,-1},{-1,-5,1},{-1,-1,-5},{-1,-1,5},{-1,1,-5},{-1,1,5},{-1,
    5,-1},{-1,5,1},{1,-5,-1},{1,-5,1},{1,-1,-5},{1,-1,5},{1,1,-5},{1,1,5},{1,
    5,-1},{1,5,1},{3,-3,-3},{3,-3,3},{3,3,-3},{3,3,3},{5,-1,-1},{5,-1,1},{5,
    1,-1},{5,1,1}}/6

(* Cube 5-Compound *)

PolyhedronName[Cube5Compound]:="cube 5-compound"
PolyhedronDual[Cube5Compound]:=Octahedron5Compound
NetPieces[Cube5Compound]:={{60,{1,2}},{60,{3,4}},{60,{5,6}}}
NetFaces[Cube5Compound]:={{1,3,12},{1,2,12},{6,5,9},{6,4,9},{7,11,8},{7,10,8}}
NetVertices[Cube5Compound]:=
{{0,0},{(-1+Sqrt[5])/4,(2-Sqrt[5])/2},{(-1+Sqrt[5])/4,(3-Sqrt[5])/4},
 {(-1+Sqrt[5])/2,(2-Sqrt[5])/2},{(-1+Sqrt[5])/2,(3-Sqrt[5])/4},
 {(3(-1+Sqrt[5]))/4,0},{(7(-1+Sqrt[5]))/8,0},{(-23+15Sqrt[5])/8,0},
 {(-9+5Sqrt[5])/4,0},{(-15+11Sqrt[5])/8,(-7+3Sqrt[5])/4},
 {(-15+11Sqrt[5])/8,(3-Sqrt[5])/4},{(3-Sqrt[5])/2,0}}
PolyhedronPolyhedra[Cube5Compound]:=
{{1,2,3,4,5,6},{7,8,9,10,11,12},{13,14,15,16,17,18},{19,20,21,22,23,24},{25,
    26,27,28,29,30}}
PolyhedronFaces[Cube5Compound]:=
{{1,9,18,14},{12,2,19,15},{1,15,19,9},{18,2,12,14},{1,14,12,15},{19,2,18,
    9},{13,17,10,1},{16,20,2,11},{10,20,16,1},{13,11,2,17},{16,11,13,1},{10,
    17,2,20},{3,5,10,14},{8,4,19,11},{3,11,19,5},{10,4,8,14},{3,14,8,11},{19,
    4,10,5},{13,9,6,3},{12,20,4,7},{6,20,12,3},{13,7,4,9},{12,7,13,3},{6,9,4,
    20},{5,6,16,15},{7,8,18,17},{5,17,18,6},{16,8,7,15},{5,15,7,17},{18,8,16,
    6}}
PolyhedronVertices[Cube5Compound]:=(*PolyhedronVertices[Dodecahedron,"C5"]*)
{{-Sqrt[1/2+1/(2Sqrt[5])],0,-Sqrt[1/4-1/(2Sqrt[5])]},
 {Sqrt[1/2+1/(2Sqrt[5])],0,Sqrt[1/4-1/(2Sqrt[5])]},
 {-Sqrt[1/2-1/(2Sqrt[5])],0,-Sqrt[1/4+1/(2Sqrt[5])]},
 {Sqrt[1/2-1/(2Sqrt[5])],0,Sqrt[1/4+1/(2Sqrt[5])]},
 {-Sqrt[1/4+1/(2Sqrt[5])],-1/2,Sqrt[1/4-1/(2Sqrt[5])]},
 {-Sqrt[1/4+1/(2Sqrt[5])],1/2,Sqrt[1/4-1/(2Sqrt[5])]},
 {Sqrt[1/4+1/(2Sqrt[5])],-1/2,-Sqrt[1/4-1/(2Sqrt[5])]},
 {Sqrt[1/4+1/(2Sqrt[5])],1/2,-Sqrt[1/4-1/(2Sqrt[5])]},
 {-Sqrt[1/8+1/(8Sqrt[5])],(1-Sqrt[5])/4,Sqrt[1/4+1/(2Sqrt[5])]},
 {-Sqrt[1/8+1/(8Sqrt[5])],(-1+Sqrt[5])/4,Sqrt[1/4+1/(2Sqrt[5])]},
 {Sqrt[1/8+1/(8Sqrt[5])],(1-Sqrt[5])/4,-Sqrt[1/4+1/(2Sqrt[5])]},
 {Sqrt[1/8+1/(8Sqrt[5])],(-1+Sqrt[5])/4,-Sqrt[1/4+1/(2Sqrt[5])]},
 {-Sqrt[1/8-1/(8Sqrt[5])],(-1-Sqrt[5])/4,-Sqrt[1/4-1/(2Sqrt[5])]},
 {-Sqrt[1/8-1/(8Sqrt[5])],(1+Sqrt[5])/4,-Sqrt[1/4-1/(2Sqrt[5])]},
 {-Sqrt[1/4-1/(2Sqrt[5])],-1/2,-Sqrt[1/4+1/(2Sqrt[5])]},
 {-Sqrt[1/4-1/(2Sqrt[5])],1/2,-Sqrt[1/4+1/(2Sqrt[5])]},
 {Sqrt[1/4-1/(2Sqrt[5])],-1/2,Sqrt[1/4+1/(2Sqrt[5])]},
 {Sqrt[1/4-1/(2Sqrt[5])],1/2,Sqrt[1/4+1/(2Sqrt[5])]},
 {Sqrt[1/8-1/(8Sqrt[5])],(-1-Sqrt[5])/4,Sqrt[1/4-1/(2Sqrt[5])]},
 {Sqrt[1/8-1/(8Sqrt[5])],(1+Sqrt[5])/4,Sqrt[1/4-1/(2Sqrt[5])]}}

(* Cube 6-Compound C3 *)

PolyhedronName[Cube6Compound,"C3"]:="cube 6-compound"
PolyhedronDual[Cube6Compound,"C3"]:=Octahedron6Compound
NetPieces[Cube6Compound,"C3"]:={{12,{1,2,3,4}},{24,{5,6,7}},{24,{8,9}},{24,{10,11}}}
NetFaces[Cube6Compound,"C3"]:=
{{1,3,20,4,22,6,29,8,32,12,10},{1,2,21,5,23,7,30,9,33,11,10},{26,18,1,3},{25,
    35,10,11},{13,28,34,27},{27,34,38,24},{27,13,31,37},{14,36,40},{14,39,
    40},{15,19,17},{15,16,17}}
NetVertices[Cube6Compound,"C3"]:=
{{0,0},{0,(4-3Sqrt[2])/2},{0,(-4+3Sqrt[2])/2},{1/4,(2-Sqrt[2])/4},{1/4,(-2+Sqrt[2])/4},
 {1/2,(5-3Sqrt[2])/7},{1/2,(-5+3Sqrt[2])/7},{3/4,(2-Sqrt[2])/4},{3/4,(-2+Sqrt[2])/4},{1,0},
 {1,(4-3Sqrt[2])/2},{1,(-4+3Sqrt[2])/2},{3/2,1/8},{9/4,0},{13/4,0},
 {(103-10Sqrt[2])/28,(1-2Sqrt[2])/14},{(3(33-4Sqrt[2]))/28,0},{(4-3Sqrt[2])/2,0},
 {(28-3Sqrt[2])/8,(2-Sqrt[2])/8},{(3-2Sqrt[2])/2,3-2Sqrt[2]},{(3-2Sqrt[2])/2,-3+2Sqrt[2]},
 {(3-Sqrt[2])/7,(8-5Sqrt[2])/14},{(3-Sqrt[2])/7,(-8+5Sqrt[2])/14},
 {(3(4-Sqrt[2]))/4,(-11+6Sqrt[2])/8},{(6-Sqrt[2])/4,(-2+Sqrt[2])/4},{(-2+Sqrt[2])/4,(2-Sqrt[2])/4},
 {(2+Sqrt[2])/2,(5-4Sqrt[2])/8},{(2+Sqrt[2])/2,(5-2Sqrt[2])/8},{(4+Sqrt[2])/7,(8-5Sqrt[2])/14},
 {(4+Sqrt[2])/7,(-8+5Sqrt[2])/14},{(10+Sqrt[2])/8,(3(1-Sqrt[2]))/8},{(-1+2Sqrt[2])/2,3-2Sqrt[2]},
 {(-1+2Sqrt[2])/2,-3+2Sqrt[2]},{(1+2Sqrt[2])/2,1/8},{(-2+3Sqrt[2])/2,0},
 {(16+3Sqrt[2])/8,(2-Sqrt[2])/8},{(-4+7Sqrt[2])/4,(-11+6Sqrt[2])/8},
 {(6+7Sqrt[2])/8,(3(1-Sqrt[2]))/8},{(51+10Sqrt[2])/28,(1-2Sqrt[2])/14},{(55+12Sqrt[2])/28,0}}

(*
NetPieces[Cube6Compound,"C3"]:={{12,{1,2,3,4}},{24,{7,8,9}},{48,{5,6}}}
NetFaces[Cube6Compound,"C3"]:=
{{1,3,16,4,18,6,25,8,28,12,10},{1,2,17,5,19,7,26,9,29,11,10},{22,15,1,3},{21,
    31,10,11},{14,32,36},{14,35,36},{13,24,30,23},{23,30,34,20},{23,13,27,33}}
NetVertices[Cube6Compound,"C3"]:=
{{0,0},{0,(4-3Sqrt[2])/2},{0,(-4+3Sqrt[2])/2},{1/4,(2-Sqrt[2])/4},{1/4,(-2+Sqrt[2])/4},
 {1/2,(5-3Sqrt[2])/7},{1/2,(-5+3Sqrt[2])/7},{3/4,(2-Sqrt[2])/4},{3/4,(-2+Sqrt[2])/4},{1,0},
 {1,(4-3Sqrt[2])/2},{1,(-4+3Sqrt[2])/2},{3/2,1/8},{9/4,0},{(4-3Sqrt[2])/2,0},
 {(3-2Sqrt[2])/2,3-2Sqrt[2]},{(3-2Sqrt[2])/2,-3+2Sqrt[2]},{(3-Sqrt[2])/7,(8-5Sqrt[2])/14},
 {(3-Sqrt[2])/7,(-8+5Sqrt[2])/14},{(3(4-Sqrt[2]))/4,(-11+6Sqrt[2])/8},{(6-Sqrt[2])/4,(-2+Sqrt[2])/4},
 {(-2+Sqrt[2])/4,(2-Sqrt[2])/4},{(2+Sqrt[2])/2,(5-4Sqrt[2])/8},{(2+Sqrt[2])/2,(5-2Sqrt[2])/8},
 {(4+Sqrt[2])/7,(8-5Sqrt[2])/14},{(4+Sqrt[2])/7,(-8+5Sqrt[2])/14},{(10+Sqrt[2])/8,(3(1-Sqrt[2]))/8},
 {(-1+2Sqrt[2])/2,3-2Sqrt[2]},{(-1+2Sqrt[2])/2,-3+2Sqrt[2]},{(1+2Sqrt[2])/2,1/8},
 {(-2+3Sqrt[2])/2,0},{(16+3Sqrt[2])/8,(2-Sqrt[2])/8},{(-4+7Sqrt[2])/4,(-11+6Sqrt[2])/8},
 {(6+7Sqrt[2])/8,(3(1-Sqrt[2]))/8},{(51+10Sqrt[2])/28,(1-2Sqrt[2])/14},{(55+12Sqrt[2])/28,0}}
*)

PolyhedronPolyhedra[Cube6Compound,"C3"]:=Partition[Range[6 6],6]
PolyhedronFaces[Cube6Compound,"C3"]:=
{{40,17,36,18},{40,18,24,47},{40,47,23,17},{17,23,43,36},{43,23,47,24},{36,43,
    24,18},{48,25,44,26},{48,26,16,39},{48,39,15,25},{25,15,35,44},{35,15,39,
    16},{44,35,16,26},{37,13,33,19},{37,19,28,46},{37,46,22,13},{13,22,42,
    33},{42,22,46,28},{33,42,28,19},{45,21,41,27},{45,27,20,38},{45,38,14,
    21},{21,14,34,41},{34,14,38,20},{41,34,20,27},{7,31,5,11},{7,11,10,4},{7,
    4,30,31},{31,30,2,5},{2,30,4,10},{5,2,10,11},{6,32,8,12},{6,12,9,1},{6,1,
    29,32},{32,29,3,8},{3,29,1,9},{8,3,9,12}}
PolyhedronVertices[Cube6Compound,"C3"]:=
{{0,(-2-Sqrt[2])/4,(2-Sqrt[2])/4},{0,(-2-Sqrt[2])/4,(-2+Sqrt[2])/4},
 {0,(2-Sqrt[2])/4,(-2-Sqrt[2])/4},{0,(2-Sqrt[2])/4,(2+Sqrt[2])/4},
 {0,(-2+Sqrt[2])/4,(-2-Sqrt[2])/4},{0,(-2+Sqrt[2])/4,(2+Sqrt[2])/4},
 {0,(2+Sqrt[2])/4,(2-Sqrt[2])/4},{0,(2+Sqrt[2])/4,(-2+Sqrt[2])/4},
 {-(1/Sqrt[2]),-1/(2Sqrt[2]),-1/(2Sqrt[2])},{-(1/Sqrt[2]),-1/(2Sqrt[2]),1/(2Sqrt[2])},
 {-(1/Sqrt[2]),1/(2Sqrt[2]),-1/(2Sqrt[2])},{-(1/Sqrt[2]),1/(2Sqrt[2]),1/(2Sqrt[2])},
 {-1/(2Sqrt[2]),-(1/Sqrt[2]),-1/(2Sqrt[2])},{-1/(2Sqrt[2]),-(1/Sqrt[2]),1/(2Sqrt[2])},
 {-1/(2Sqrt[2]),-1/(2Sqrt[2]),-(1/Sqrt[2])},{-1/(2Sqrt[2]),-1/(2Sqrt[2]),1/Sqrt[2]},
 {-1/(2Sqrt[2]),1/(2Sqrt[2]),-(1/Sqrt[2])},{-1/(2Sqrt[2]),1/(2Sqrt[2]),1/Sqrt[2]},
 {-1/(2Sqrt[2]),1/Sqrt[2],-1/(2Sqrt[2])},{-1/(2Sqrt[2]),1/Sqrt[2],1/(2Sqrt[2])},
 {1/(2Sqrt[2]),-(1/Sqrt[2]),-1/(2Sqrt[2])},{1/(2Sqrt[2]),-(1/Sqrt[2]),1/(2Sqrt[2])},
 {1/(2Sqrt[2]),-1/(2Sqrt[2]),-(1/Sqrt[2])},{1/(2Sqrt[2]),-1/(2Sqrt[2]),1/Sqrt[2]},
 {1/(2Sqrt[2]),1/(2Sqrt[2]),-(1/Sqrt[2])},{1/(2Sqrt[2]),1/(2Sqrt[2]),1/Sqrt[2]},
 {1/(2Sqrt[2]),1/Sqrt[2],-1/(2Sqrt[2])},{1/(2Sqrt[2]),1/Sqrt[2],1/(2Sqrt[2])},
 {1/Sqrt[2],-1/(2Sqrt[2]),-1/(2Sqrt[2])},{1/Sqrt[2],-1/(2Sqrt[2]),1/(2Sqrt[2])},
 {1/Sqrt[2],1/(2Sqrt[2]),-1/(2Sqrt[2])},{1/Sqrt[2],1/(2Sqrt[2]),1/(2Sqrt[2])},
 {(-2-Sqrt[2])/4,0,(2-Sqrt[2])/4},{(-2-Sqrt[2])/4,0,(-2+Sqrt[2])/4},
 {(-2-Sqrt[2])/4,(2-Sqrt[2])/4,0},{(-2-Sqrt[2])/4,(-2+Sqrt[2])/4,0},
 {(2-Sqrt[2])/4,0,(-2-Sqrt[2])/4},{(2-Sqrt[2])/4,0,(2+Sqrt[2])/4},
 {(2-Sqrt[2])/4,(-2-Sqrt[2])/4,0},{(2-Sqrt[2])/4,(2+Sqrt[2])/4,0},
 {(-2+Sqrt[2])/4,0,(-2-Sqrt[2])/4},{(-2+Sqrt[2])/4,0,(2+Sqrt[2])/4},
 {(-2+Sqrt[2])/4,(-2-Sqrt[2])/4,0},{(-2+Sqrt[2])/4,(2+Sqrt[2])/4,0},
 {(2+Sqrt[2])/4,0,(2-Sqrt[2])/4},{(2+Sqrt[2])/4,0,(-2+Sqrt[2])/4},
 {(2+Sqrt[2])/4,(2-Sqrt[2])/4,0},{(2+Sqrt[2])/4,(-2+Sqrt[2])/4,0}}

(* Cube 6-Compound C4 *)

PolyhedronName[Cube6Compound,"C4"]:="cube 6-compound"
PolyhedronDual[Cube6Compound,"C4"]:=Octahedron6Compound
PolyhedronPolyhedra[Cube6Compound,"C4"]:=Partition[Range[6 6],6]
PolyhedronFaces[Cube6Compound,"C4"]:=
{{24,30,46,40},{29,45,46,30},{39,40,46,45},{23,24,40,39},{23,39,45,29},{23,29,
    30,24},{6,14,15,7},{10,11,15,14},{3,7,15,11},{2,6,7,3},{2,3,11,10},{2,10,
    14,6},{17,26,28,19},{26,42,44,28},{19,28,44,35},{17,19,35,33},{33,35,44,
    42},{17,33,42,26},{1,9,12,4},{9,13,16,12},{4,12,16,8},{1,4,8,5},{5,8,16,
    13},{1,5,13,9},{22,38,48,32},{31,32,48,47},{21,22,32,31},{21,37,38,
    22},{21,31,47,37},{37,47,48,38},{34,41,43,36},{25,27,43,41},{20,36,43,
    27},{18,34,36,20},{18,20,27,25},{18,25,41,34}}
PolyhedronVertices[Cube6Compound,"C4"]:=
{{-1/2,(-1-Sqrt[3])/4,(1-Sqrt[3])/4},
 {-1/2,(-1-Sqrt[3])/4,(-1+Sqrt[3])/4},
 {-1/2,(1-Sqrt[3])/4,(-1-Sqrt[3])/4},
 {-1/2,(1-Sqrt[3])/4,(1+Sqrt[3])/4},
 {-1/2,(-1+Sqrt[3])/4,(-1-Sqrt[3])/4},
 {-1/2,(-1+Sqrt[3])/4,(1+Sqrt[3])/4},
 {-1/2,(1+Sqrt[3])/4,(1-Sqrt[3])/4},
 {-1/2,(1+Sqrt[3])/4,(-1+Sqrt[3])/4},
 {1/2,(-1-Sqrt[3])/4,(1-Sqrt[3])/4},
 {1/2,(-1-Sqrt[3])/4,(-1+Sqrt[3])/4},
 {1/2,(1-Sqrt[3])/4,(-1-Sqrt[3])/4},
 {1/2,(1-Sqrt[3])/4,(1+Sqrt[3])/4},
 {1/2,(-1+Sqrt[3])/4,(-1-Sqrt[3])/4},
 {1/2,(-1+Sqrt[3])/4,(1+Sqrt[3])/4},
 {1/2,(1+Sqrt[3])/4,(1-Sqrt[3])/4},
 {1/2,(1+Sqrt[3])/4,(-1+Sqrt[3])/4},
 {(-1-Sqrt[3])/4,-1/2,(1-Sqrt[3])/4},
 {(-1-Sqrt[3])/4,-1/2,(-1+Sqrt[3])/4},
 {(-1-Sqrt[3])/4,1/2,(1-Sqrt[3])/4},
 {(-1-Sqrt[3])/4,1/2,(-1+Sqrt[3])/4},
 {(-1-Sqrt[3])/4,(1-Sqrt[3])/4,-1/2},
 {(-1-Sqrt[3])/4,(1-Sqrt[3])/4,1/2},
 {(-1-Sqrt[3])/4,(-1+Sqrt[3])/4,-1/2},
 {(-1-Sqrt[3])/4,(-1+Sqrt[3])/4,1/2},
 {(1-Sqrt[3])/4,-1/2,(-1-Sqrt[3])/4},
 {(1-Sqrt[3])/4,-1/2,(1+Sqrt[3])/4},
 {(1-Sqrt[3])/4,1/2,(-1-Sqrt[3])/4},
 {(1-Sqrt[3])/4,1/2,(1+Sqrt[3])/4},
 {(1-Sqrt[3])/4,(-1-Sqrt[3])/4,-1/2},
 {(1-Sqrt[3])/4,(-1-Sqrt[3])/4,1/2},
 {(1-Sqrt[3])/4,(1+Sqrt[3])/4,-1/2},
 {(1-Sqrt[3])/4,(1+Sqrt[3])/4,1/2},
 {(-1+Sqrt[3])/4,-1/2,(-1-Sqrt[3])/4},
 {(-1+Sqrt[3])/4,-1/2,(1+Sqrt[3])/4},
 {(-1+Sqrt[3])/4,1/2,(-1-Sqrt[3])/4},
 {(-1+Sqrt[3])/4,1/2,(1+Sqrt[3])/4},
 {(-1+Sqrt[3])/4,(-1-Sqrt[3])/4,-1/2},
 {(-1+Sqrt[3])/4,(-1-Sqrt[3])/4,1/2},
 {(-1+Sqrt[3])/4,(1+Sqrt[3])/4,-1/2},
 {(-1+Sqrt[3])/4,(1+Sqrt[3])/4,1/2},
 {(1+Sqrt[3])/4,-1/2,(1-Sqrt[3])/4},
 {(1+Sqrt[3])/4,-1/2,(-1+Sqrt[3])/4},
 {(1+Sqrt[3])/4,1/2,(1-Sqrt[3])/4},
 {(1+Sqrt[3])/4,1/2,(-1+Sqrt[3])/4},
 {(1+Sqrt[3])/4,(1-Sqrt[3])/4,-1/2},
 {(1+Sqrt[3])/4,(1-Sqrt[3])/4,1/2},
 {(1+Sqrt[3])/4,(-1+Sqrt[3])/4,-1/2},
 {(1+Sqrt[3])/4,(-1+Sqrt[3])/4,1/2}}

(* Cube 7-Compound *)

PolyhedronName[Cube7Compound]:="cube 7-compound"
PolyhedronDual[Cube7Compound]:=Octahedron7Compound
PolyhedronPolyhedra[Cube7Compound]:=
{{1,2,3,4,5,6},{7,8,9,10,11,12},{13,14,15,16,17,18},{19,20,21,22,23,24},{25,
    26,27,28,29,30},{31,32,33,34,35,36},{37,38,39,40,41,42}}
PolyhedronFaces[Cube7Compound]:=
{{32,38,54,48},{37,53,54,38},{47,48,54,53},{31,32,48,47},{31,47,53,37},{31,37,
    38,32},{10,22,23,11},{18,19,23,22},{7,11,23,19},{6,10,11,7},{6,7,19,
    18},{6,18,22,10},{25,34,36,27},{34,50,52,36},{27,36,52,43},{25,27,43,
    41},{41,43,52,50},{25,41,50,34},{5,17,20,8},{17,21,24,20},{8,20,24,12},{5,
    8,12,9},{9,12,24,21},{5,9,21,17},{30,46,56,40},{39,40,56,55},{29,30,40,
    39},{29,45,46,30},{29,39,55,45},{45,55,56,46},{42,49,51,44},{33,35,51,
    49},{28,44,51,35},{26,42,44,28},{26,28,35,33},{26,33,49,42},{16,4,2,
    14},{16,14,13,15},{16,15,3,4},{4,3,1,2},{1,3,15,13},{2,1,13,14}}
PolyhedronVertices[Cube7Compound]:=
{{-1/2,-1/2,-1/2},{-1/2,-1/2,1/2},{-1/2,1/2,-1/2},{-1/2,1/2,1/2},
 {-1/2,(-1-Sqrt[3])/4,(1-Sqrt[3])/4},{-1/2,(-1-Sqrt[3])/4,
  (-1+Sqrt[3])/4},{-1/2,(1-Sqrt[3])/4,(-1-Sqrt[3])/4},
 {-1/2,(1-Sqrt[3])/4,(1+Sqrt[3])/4},{-1/2,(-1+Sqrt[3])/4,
  (-1-Sqrt[3])/4},{-1/2,(-1+Sqrt[3])/4,(1+Sqrt[3])/4},
 {-1/2,(1+Sqrt[3])/4,(1-Sqrt[3])/4},{-1/2,(1+Sqrt[3])/4,
  (-1+Sqrt[3])/4},{1/2,-1/2,-1/2},{1/2,-1/2,1/2},{1/2,1/2,-1/2},
 {1/2,1/2,1/2},{1/2,(-1-Sqrt[3])/4,(1-Sqrt[3])/4},
 {1/2,(-1-Sqrt[3])/4,(-1+Sqrt[3])/4},{1/2,(1-Sqrt[3])/4,
  (-1-Sqrt[3])/4},{1/2,(1-Sqrt[3])/4,(1+Sqrt[3])/4},
 {1/2,(-1+Sqrt[3])/4,(-1-Sqrt[3])/4},{1/2,(-1+Sqrt[3])/4,
  (1+Sqrt[3])/4},{1/2,(1+Sqrt[3])/4,(1-Sqrt[3])/4},
 {1/2,(1+Sqrt[3])/4,(-1+Sqrt[3])/4},{(-1-Sqrt[3])/4,-1/2,
  (1-Sqrt[3])/4},{(-1-Sqrt[3])/4,-1/2,(-1+Sqrt[3])/4},
 {(-1-Sqrt[3])/4,1/2,(1-Sqrt[3])/4},{(-1-Sqrt[3])/4,1/2,
  (-1+Sqrt[3])/4},{(-1-Sqrt[3])/4,(1-Sqrt[3])/4,-1/2},
 {(-1-Sqrt[3])/4,(1-Sqrt[3])/4,1/2},{(-1-Sqrt[3])/4,(-1+Sqrt[3])/4,
  -1/2},{(-1-Sqrt[3])/4,(-1+Sqrt[3])/4,1/2},
 {(1-Sqrt[3])/4,-1/2,(-1-Sqrt[3])/4},{(1-Sqrt[3])/4,-1/2,
  (1+Sqrt[3])/4},{(1-Sqrt[3])/4,1/2,(-1-Sqrt[3])/4},
 {(1-Sqrt[3])/4,1/2,(1+Sqrt[3])/4},{(1-Sqrt[3])/4,(-1-Sqrt[3])/4,
  -1/2},{(1-Sqrt[3])/4,(-1-Sqrt[3])/4,1/2},
 {(1-Sqrt[3])/4,(1+Sqrt[3])/4,-1/2},{(1-Sqrt[3])/4,(1+Sqrt[3])/4,1/2},
 {(-1+Sqrt[3])/4,-1/2,(-1-Sqrt[3])/4},{(-1+Sqrt[3])/4,-1/2,
  (1+Sqrt[3])/4},{(-1+Sqrt[3])/4,1/2,(-1-Sqrt[3])/4},
 {(-1+Sqrt[3])/4,1/2,(1+Sqrt[3])/4},{(-1+Sqrt[3])/4,(-1-Sqrt[3])/4,
  -1/2},{(-1+Sqrt[3])/4,(-1-Sqrt[3])/4,1/2},
 {(-1+Sqrt[3])/4,(1+Sqrt[3])/4,-1/2},{(-1+Sqrt[3])/4,(1+Sqrt[3])/4,
  1/2},{(1+Sqrt[3])/4,-1/2,(1-Sqrt[3])/4},
 {(1+Sqrt[3])/4,-1/2,(-1+Sqrt[3])/4},{(1+Sqrt[3])/4,1/2,
  (1-Sqrt[3])/4},{(1+Sqrt[3])/4,1/2,(-1+Sqrt[3])/4},
 {(1+Sqrt[3])/4,(1-Sqrt[3])/4,-1/2},{(1+Sqrt[3])/4,(1-Sqrt[3])/4,1/2},
 {(1+Sqrt[3])/4,(-1+Sqrt[3])/4,-1/2},{(1+Sqrt[3])/4,(-1+Sqrt[3])/4,
  1/2}}

(* Cube 10-Compound *)

PolyhedronName[Cube10Compound]:="cube 10-compound"
PolyhedronDual[Cube10Compound]:="octahedron 10-compound"
NetPieces[Cube10Compound]:={{20,{1,2,3}},{60,{4,5,6}},{60,{7,8}},{60,{9,10}}}
NetFaces[Cube10Compound]:=
{{1,20,27,32,18,22,19,31,28,3},{1,3,29,36,14,24,16,34,26,21},{
  1,21,25,33,17,23,15,35,30,2},{4,5,43,12,39,48},{4,6,42,11,40,48},{4,13,
    41,5},{7,10,45,47},{47,38,7},{46,37,8},{8,9,44,46}}
NetVertices[Cube10Compound]:=
{{0,0},{0,Root[-9+24#1+32#1^2-64#1^3+16#1^4&,2,0]},
 {0,Root[-9-24#1+32#1^2+64#1^3+16#1^4&,3,0]},{5/8,0},
 {5/8,(1-Sqrt[7-2Sqrt[10]])/2},{5/8,(-1+Sqrt[7-2Sqrt[10]])/2},{5/4,0},{7/4,0},
 {(3(1+Sqrt[2]))/4,Root[-1-16#1+14#1^2+24#1^3+4#1^4&,3,0]},
 {(1+3Sqrt[2])/4,Root[-1+16#1+14#1^2-24#1^3+4#1^4&,2,0]},
 {(5+2Sqrt[7-3Sqrt[5]])/8,Root[1+8#1-40#1^2-64#1^3+64#1^4&,2,0]},
 {(5+2Sqrt[7-3Sqrt[5]])/8,Root[1-8#1-40#1^2+64#1^3+64#1^4&,3,0]},
 {(1+4Sqrt[7-2Sqrt[10]])/8,0},{Root[81+576#1-310#1^2-352#1^3+4#1^4&,2,0],
  Root[9-12#1-218#1^2-152#1^3+4#1^4&,2,0]},
 {Root[81+576#1-310#1^2-352#1^3+4#1^4&,2,0],
  Root[9+12#1-218#1^2+152#1^3+4#1^4&,3,0]},
 {Root[9-12#1-218#1^2-152#1^3+4#1^4&,2,0],
  Root[81+576#1-310#1^2-352#1^3+4#1^4&,2,0]},
 {Root[9-12#1-218#1^2-152#1^3+4#1^4&,2,0],
  Root[81-576#1-310#1^2+352#1^3+4#1^4&,3,0]},
 {Root[9+12#1-218#1^2+152#1^3+4#1^4&,3,0],
  Root[81+576#1-310#1^2-352#1^3+4#1^4&,2,0]},
 {Root[81-576#1-310#1^2+352#1^3+4#1^4&,3,0],
  Root[9-12#1-218#1^2-152#1^3+4#1^4&,2,0]},
 {Root[-9+24#1+32#1^2-64#1^3+16#1^4&,2,0],0},
 {Root[-9-24#1+32#1^2+64#1^3+16#1^4&,3,0],0},
 {Root[-9+84#1-88#1^2-24#1^3+36#1^4&,2,0],
  Root[-9-84#1-88#1^2+24#1^3+36#1^4&,3,0]},
 {Root[-9-84#1-88#1^2+24#1^3+36#1^4&,3,0],
  Root[-9+84#1-88#1^2-24#1^3+36#1^4&,2,0]},
 {Root[-9-84#1-88#1^2+24#1^3+36#1^4&,3,0],
  Root[-9-84#1-88#1^2+24#1^3+36#1^4&,3,0]},
 {Root[-9-60#1-110#1^2+100#1^4&,3,0],Root[9-240#1+530#1^2-400#1^3+100#1^4&,1,
   0]},{Root[-9-60#1-110#1^2+100#1^4&,3,0],
  Root[9+240#1+530#1^2+400#1^3+100#1^4&,4,0]},
 {Root[-9+60#1-110#1^2+100#1^4&,2,0],Root[9+240#1+530#1^2+400#1^3+100#1^4&,4,
   0]},{Root[9-240#1+530#1^2-400#1^3+100#1^4&,1,0],
  Root[-9-60#1-110#1^2+100#1^4&,3,0]},
 {Root[9+240#1+530#1^2+400#1^3+100#1^4&,4,0],Root[-9-60#1-110#1^2+100#1^4&,3,
   0]},{Root[9+240#1+530#1^2+400#1^3+100#1^4&,4,0],
  Root[-9+60#1-110#1^2+100#1^4&,2,0]},
 {Root[1-36#1+190#1^2-288#1^3+124#1^4&,1,0],
  Root[9-12#1-98#1^2+128#1^3+124#1^4&,2,0]},
 {Root[9+12#1-98#1^2-128#1^3+124#1^4&,3,0],
  Root[1+36#1+190#1^2+288#1^3+124#1^4&,4,0]},
 {Root[9-12#1-98#1^2+128#1^3+124#1^4&,2,0],
  Root[1-36#1+190#1^2-288#1^3+124#1^4&,1,0]},
 {Root[9-12#1-98#1^2+128#1^3+124#1^4&,2,0],
  Root[1+36#1+190#1^2+288#1^3+124#1^4&,4,0]},
 {Root[1+36#1+190#1^2+288#1^3+124#1^4&,4,0],
  Root[9+12#1-98#1^2-128#1^3+124#1^4&,3,0]},
 {Root[1+36#1+190#1^2+288#1^3+124#1^4&,4,0],
  Root[9-12#1-98#1^2+128#1^3+124#1^4&,2,0]},
 {Root[-2759-11776#1+13856#1^2-4096#1^3+256#1^4&,2,0],
  Root[121-1716#1+1630#1^2-408#1^3+4#1^4&,1,0]},
 {Root[-5679-864#1+8096#1^2-3584#1^3+256#1^4&,2,0],
  Root[121+1716#1+1630#1^2+408#1^3+4#1^4&,4,0]},
 {Root[-340831-167968#1+936320#1^2-362496#1^3+4096#1^4&,2,0],
  Root[-9+60#1+104#1^2-200#1^3+4#1^4&,2,0]},
 {Root[-340831-167968#1+936320#1^2-362496#1^3+4096#1^4&,2,0],
  Root[-9-60#1+104#1^2+200#1^3+4#1^4&,3,0]},
 {Root[81-6048#1+32128#1^2-43008#1^3+4096#1^4&,3,0],
  Root[-1+12#1-38#1^2+32#1^3+4#1^4&,2,0]},
 {Root[4729-23776#1+37760#1^2-22528#1^3+4096#1^4&,2,0],
  Root[9+48#1-40#1^2-64#1^3+16#1^4&,2,0]},
 {Root[4729-23776#1+37760#1^2-22528#1^3+4096#1^4&,2,0],
  Root[9-48#1-40#1^2+64#1^3+16#1^4&,3,0]},
 {Root[-57591+12960#1+59040#1^2-38400#1^3+6400#1^4&,3,0],
  Root[-1-100#1-190#1^2+100#1^4&,3,0]},
 {Root[-40751+46400#1+11040#1^2-25600#1^3+6400#1^4&,3,0],
  Root[-1+100#1-190#1^2+100#1^4&,2,0]},
 {Root[-90641+74080#1+40544#1^2-43520#1^3+7936#1^4&,3,0],0},
 {Root[-48409+85952#1-12832#1^2-27648#1^3+7936#1^4&,3,0],0},
 {Root[-99511+239584#1-111232#1^2-67584#1^3+36864#1^4&,3,0],0}}

PolyhedronPolyhedra[Cube10Compound]:=Partition[Range[6 10],6]
PolyhedronFaces[Cube10Compound]:=
{{16,35,52,42},{16,42,38,53},{16,53,47,35},{35,47,1,52},{1,47,53,38},{52,1,38,
    42},{30,79,64,69},{30,69,62,77},{30,77,60,79},{79,60,27,64},{27,60,77,
    62},{64,27,62,69},{32,68,74,58},{32,58,73,67},{32,67,71,68},{68,71,25,
    74},{25,71,67,73},{74,25,73,58},{31,57,76,66},{31,66,72,65},{31,65,75,
    57},{57,75,26,76},{26,75,65,72},{76,26,72,66},{15,54,37,41},{15,41,51,
    36},{15,36,48,54},{54,48,2,37},{2,48,36,51},{37,2,51,41},{29,80,59,
    78},{29,78,61,70},{29,70,63,80},{80,63,28,59},{28,63,70,61},{59,28,61,
    78},{7,21,12,17},{7,17,20,9},{7,9,24,21},{21,24,6,12},{6,24,9,20},{12,6,
    20,17},{3,40,55,45},{3,45,33,50},{3,50,44,40},{40,44,14,55},{14,44,50,
    33},{55,14,33,45},{8,10,19,18},{8,18,11,22},{8,22,23,10},{10,23,5,19},{5,
    23,22,11},{19,5,11,18},{4,39,43,49},{4,49,34,46},{4,46,56,39},{39,56,13,
    43},{13,56,46,34},{43,13,34,49}}
PolyhedronVertices[Cube10Compound]:=
{{-1/2,-1/2,-1/2},{-1/2,-1/2,1/2},{-1/2,1/2,-1/2},{-1/2,1/2,1/2},
 {0,(-1-Sqrt[5])/4,(1-Sqrt[5])/4},{0,(-1-Sqrt[5])/4,(-1+Sqrt[5])/4},
 {0,(1+Sqrt[5])/4,(1-Sqrt[5])/4},{0,(1+Sqrt[5])/4,(-1+Sqrt[5])/4},
 {0,Root[1+44#1-100#1^2-48#1^3+144#1^4&,3,0],
  Root[1+44#1-100#1^2-48#1^3+144#1^4&,4,0]},
 {0,Root[1+44#1-100#1^2-48#1^3+144#1^4&,3,0],
  Root[1-44#1-100#1^2+48#1^3+144#1^4&,1,0]},
 {0,Root[1-44#1-100#1^2+48#1^3+144#1^4&,2,0],
  Root[1+44#1-100#1^2-48#1^3+144#1^4&,4,0]},
 {0,Root[1-44#1-100#1^2+48#1^3+144#1^4&,2,0],
  Root[1-44#1-100#1^2+48#1^3+144#1^4&,1,0]},{1/2,-1/2,-1/2},
 {1/2,-1/2,1/2},{1/2,1/2,-1/2},{1/2,1/2,1/2},
 {-(1/Sqrt[2]),Root[-1+12#1-28#1^2-48#1^3+144#1^4&,2,0],
  Root[-1+12#1-28#1^2-48#1^3+144#1^4&,1,0]},
 {-(1/Sqrt[2]),Root[-1+12#1-28#1^2-48#1^3+144#1^4&,2,0],
  Root[-1-12#1-28#1^2+48#1^3+144#1^4&,4,0]},
 {-(1/Sqrt[2]),Root[-1-12#1-28#1^2+48#1^3+144#1^4&,3,0],
  Root[-1+12#1-28#1^2-48#1^3+144#1^4&,1,0]},
 {-(1/Sqrt[2]),Root[-1-12#1-28#1^2+48#1^3+144#1^4&,3,0],
  Root[-1-12#1-28#1^2+48#1^3+144#1^4&,4,0]},
 {1/Sqrt[2],Root[-1+12#1-28#1^2-48#1^3+144#1^4&,2,0],
  Root[-1+12#1-28#1^2-48#1^3+144#1^4&,1,0]},
 {1/Sqrt[2],Root[-1+12#1-28#1^2-48#1^3+144#1^4&,2,0],
  Root[-1-12#1-28#1^2+48#1^3+144#1^4&,4,0]},
 {1/Sqrt[2],Root[-1-12#1-28#1^2+48#1^3+144#1^4&,3,0],
  Root[-1+12#1-28#1^2-48#1^3+144#1^4&,1,0]},
 {1/Sqrt[2],Root[-1-12#1-28#1^2+48#1^3+144#1^4&,3,0],
  Root[-1-12#1-28#1^2+48#1^3+144#1^4&,4,0]},
 {(-1-Sqrt[5])/4,(1-Sqrt[5])/4,0},{(-1-Sqrt[5])/4,(-1+Sqrt[5])/4,0},
 {(1-Sqrt[5])/4,0,(-1-Sqrt[5])/4},{(1-Sqrt[5])/4,0,(1+Sqrt[5])/4},
 {(-1+Sqrt[5])/4,0,(-1-Sqrt[5])/4},{(-1+Sqrt[5])/4,0,(1+Sqrt[5])/4},
 {(1+Sqrt[5])/4,(1-Sqrt[5])/4,0},{(1+Sqrt[5])/4,(-1+Sqrt[5])/4,0},
 {(1-Sqrt[10])/6,(-1-Sqrt[7+3Sqrt[5]])/6,(1-Sqrt[7-3Sqrt[5]])/6},
 {(1-Sqrt[10])/6,(-1-Sqrt[7+3Sqrt[5]])/6,(-1+Sqrt[7-3Sqrt[5]])/6},
 {(1-Sqrt[10])/6,(1+Sqrt[7+3Sqrt[5]])/6,(1-Sqrt[7-3Sqrt[5]])/6},
 {(1-Sqrt[10])/6,(1+Sqrt[7+3Sqrt[5]])/6,(-1+Sqrt[7-3Sqrt[5]])/6},
 {(-1+Sqrt[10])/6,(-1-Sqrt[7+3Sqrt[5]])/6,(1-Sqrt[7-3Sqrt[5]])/6},
 {(-1+Sqrt[10])/6,(-1-Sqrt[7+3Sqrt[5]])/6,(-1+Sqrt[7-3Sqrt[5]])/6},
 {(-1+Sqrt[10])/6,(1+Sqrt[7+3Sqrt[5]])/6,(1-Sqrt[7-3Sqrt[5]])/6},
 {(-1+Sqrt[10])/6,(1+Sqrt[7+3Sqrt[5]])/6,(-1+Sqrt[7-3Sqrt[5]])/6},
 {(1-Sqrt[7-3Sqrt[5]])/6,(1-Sqrt[10])/6,(-1-Sqrt[7+3Sqrt[5]])/6},
 {(1-Sqrt[7-3Sqrt[5]])/6,(1-Sqrt[10])/6,(1+Sqrt[7+3Sqrt[5]])/6},
 {(1-Sqrt[7-3Sqrt[5]])/6,(-1+Sqrt[10])/6,(-1-Sqrt[7+3Sqrt[5]])/6},
 {(1-Sqrt[7-3Sqrt[5]])/6,(-1+Sqrt[10])/6,(1+Sqrt[7+3Sqrt[5]])/6},
 {(-1+Sqrt[7-3Sqrt[5]])/6,(1-Sqrt[10])/6,(-1-Sqrt[7+3Sqrt[5]])/6},
 {(-1+Sqrt[7-3Sqrt[5]])/6,(1-Sqrt[10])/6,(1+Sqrt[7+3Sqrt[5]])/6},
 {(-1+Sqrt[7-3Sqrt[5]])/6,(-1+Sqrt[10])/6,(-1-Sqrt[7+3Sqrt[5]])/6},
 {(-1+Sqrt[7-3Sqrt[5]])/6,(-1+Sqrt[10])/6,(1+Sqrt[7+3Sqrt[5]])/6},
 {(-1-Sqrt[7+3Sqrt[5]])/6,(1-Sqrt[7-3Sqrt[5]])/6,(1-Sqrt[10])/6},
 {(-1-Sqrt[7+3Sqrt[5]])/6,(1-Sqrt[7-3Sqrt[5]])/6,(-1+Sqrt[10])/6},
 {(-1-Sqrt[7+3Sqrt[5]])/6,(-1+Sqrt[7-3Sqrt[5]])/6,(1-Sqrt[10])/6},
 {(-1-Sqrt[7+3Sqrt[5]])/6,(-1+Sqrt[7-3Sqrt[5]])/6,(-1+Sqrt[10])/6},
 {(1+Sqrt[7+3Sqrt[5]])/6,(1-Sqrt[7-3Sqrt[5]])/6,(1-Sqrt[10])/6},
 {(1+Sqrt[7+3Sqrt[5]])/6,(1-Sqrt[7-3Sqrt[5]])/6,(-1+Sqrt[10])/6},
 {(1+Sqrt[7+3Sqrt[5]])/6,(-1+Sqrt[7-3Sqrt[5]])/6,(1-Sqrt[10])/6},
 {(1+Sqrt[7+3Sqrt[5]])/6,(-1+Sqrt[7-3Sqrt[5]])/6,(-1+Sqrt[10])/6},
 {Root[1+44#1-100#1^2-48#1^3+144#1^4&,3,0],
  Root[1+44#1-100#1^2-48#1^3+144#1^4&,4,0],0},
 {Root[1+44#1-100#1^2-48#1^3+144#1^4&,3,0],
  Root[1-44#1-100#1^2+48#1^3+144#1^4&,1,0],0},
 {Root[1+44#1-100#1^2-48#1^3+144#1^4&,4,0],0,
  Root[1+44#1-100#1^2-48#1^3+144#1^4&,3,0]},
 {Root[1+44#1-100#1^2-48#1^3+144#1^4&,4,0],0,
  Root[1-44#1-100#1^2+48#1^3+144#1^4&,2,0]},
 {Root[-1+12#1-28#1^2-48#1^3+144#1^4&,1,0],-(1/Sqrt[2]),
  Root[-1+12#1-28#1^2-48#1^3+144#1^4&,2,0]},
 {Root[-1+12#1-28#1^2-48#1^3+144#1^4&,1,0],-(1/Sqrt[2]),
  Root[-1-12#1-28#1^2+48#1^3+144#1^4&,3,0]},
 {Root[-1+12#1-28#1^2-48#1^3+144#1^4&,1,0],1/Sqrt[2],
  Root[-1+12#1-28#1^2-48#1^3+144#1^4&,2,0]},
 {Root[-1+12#1-28#1^2-48#1^3+144#1^4&,1,0],1/Sqrt[2],
  Root[-1-12#1-28#1^2+48#1^3+144#1^4&,3,0]},
 {Root[-1+12#1-28#1^2-48#1^3+144#1^4&,2,0],
  Root[-1+12#1-28#1^2-48#1^3+144#1^4&,1,0],-(1/Sqrt[2])},
 {Root[-1+12#1-28#1^2-48#1^3+144#1^4&,2,0],
  Root[-1+12#1-28#1^2-48#1^3+144#1^4&,1,0],1/Sqrt[2]},
 {Root[-1+12#1-28#1^2-48#1^3+144#1^4&,2,0],
  Root[-1-12#1-28#1^2+48#1^3+144#1^4&,4,0],-(1/Sqrt[2])},
 {Root[-1+12#1-28#1^2-48#1^3+144#1^4&,2,0],
  Root[-1-12#1-28#1^2+48#1^3+144#1^4&,4,0],1/Sqrt[2]},
 {Root[1-44#1-100#1^2+48#1^3+144#1^4&,1,0],0,
  Root[1+44#1-100#1^2-48#1^3+144#1^4&,3,0]},
 {Root[1-44#1-100#1^2+48#1^3+144#1^4&,1,0],0,
  Root[1-44#1-100#1^2+48#1^3+144#1^4&,2,0]},
 {Root[1-44#1-100#1^2+48#1^3+144#1^4&,2,0],
  Root[1+44#1-100#1^2-48#1^3+144#1^4&,4,0],0},
 {Root[1-44#1-100#1^2+48#1^3+144#1^4&,2,0],
  Root[1-44#1-100#1^2+48#1^3+144#1^4&,1,0],0},
 {Root[-1-12#1-28#1^2+48#1^3+144#1^4&,3,0],
  Root[-1+12#1-28#1^2-48#1^3+144#1^4&,1,0],-(1/Sqrt[2])},
 {Root[-1-12#1-28#1^2+48#1^3+144#1^4&,3,0],
  Root[-1+12#1-28#1^2-48#1^3+144#1^4&,1,0],1/Sqrt[2]},
 {Root[-1-12#1-28#1^2+48#1^3+144#1^4&,3,0],
  Root[-1-12#1-28#1^2+48#1^3+144#1^4&,4,0],-(1/Sqrt[2])},
 {Root[-1-12#1-28#1^2+48#1^3+144#1^4&,3,0],
  Root[-1-12#1-28#1^2+48#1^3+144#1^4&,4,0],1/Sqrt[2]},
 {Root[-1-12#1-28#1^2+48#1^3+144#1^4&,4,0],-(1/Sqrt[2]),
  Root[-1+12#1-28#1^2-48#1^3+144#1^4&,2,0]},
 {Root[-1-12#1-28#1^2+48#1^3+144#1^4&,4,0],-(1/Sqrt[2]),
  Root[-1-12#1-28#1^2+48#1^3+144#1^4&,3,0]},
 {Root[-1-12#1-28#1^2+48#1^3+144#1^4&,4,0],1/Sqrt[2],
  Root[-1+12#1-28#1^2-48#1^3+144#1^4&,2,0]},
 {Root[-1-12#1-28#1^2+48#1^3+144#1^4&,4,0],1/Sqrt[2],
  Root[-1-12#1-28#1^2+48#1^3+144#1^4&,3,0]}}

(* Cube-Octahedron Compound *)

PolyhedronName[CubeOctahedronCompound]:="cube-octahedron compound"
NetFaces[CubeOctahedronCompound]:=
{{1,4,5},{1,5,2},{1,2,7},{1,7,6},{3,11,10},{3,10,8},{3,8,9}}
NetVertices[CubeOctahedronCompound]:=
{{0,0},{0,1/Sqrt[2]},{3/2,0},
 {-Sqrt[3/2]/2,-1/(2Sqrt[2])},{-Sqrt[3/2]/2,1/(2Sqrt[2])},
 {Sqrt[3/2]/2,-1/(2Sqrt[2])},{Sqrt[3/2]/2,1/(2Sqrt[2])},
 {(6-Sqrt[2])/4,-1/(2Sqrt[2])},{(6-Sqrt[2])/4,
  1/(2Sqrt[2])},{(6+Sqrt[2])/4,-1/(2Sqrt[2])},
 {(6+Sqrt[2])/4,1/(2Sqrt[2])}}
PolyhedronPolyhedra[CubeOctahedronCompound]:={Range[6],Range[7,14]}
PolyhedronFaces[CubeOctahedronCompound]:=
{{13,5,3,11},{13,11,10,12},{13,12,4,5},{5,4,2,3},{2,4,12,10},{3,2,10,11},{1,7,
    6},{8,1,6},{7,1,9},{8,9,1},{6,7,14},{6,14,8},{7,9,14},{8,14,9}}
PolyhedronVertices[CubeOctahedronCompound]:=
{{-1,0,0},{-1/2,-1/2,-1/2},{-1/2,-1/2,1/2},
 {-1/2,1/2,-1/2},{-1/2,1/2,1/2},{0,-1,0},{0,0,-1},
 {0,0,1},{0,1,0},{1/2,-1/2,-1/2},{1/2,-1/2,1/2},
 {1/2,1/2,-1/2},{1/2,1/2,1/2},{1,0,0}}
PolyhedronFaces[CubeOctahedronCompound,Split->True]:=
{{25,16,22},{9,6,16},{4,12,6},{20,22,12},{16,6,14},{6,12,14},{12,22,14},{22,
    16,14},{25,22,24},{20,19,22},{18,21,19},{23,24,21},{22,19,26},{19,21,
    26},{21,24,26},{24,22,26},{25,24,16},{23,15,24},{7,8,15},{9,16,8},{24,15,
    17},{15,8,17},{8,16,17},{16,24,17},{9,8,6},{7,5,8},{2,3,5},{4,6,3},{8,5,
    1},{5,3,1},{3,6,1},{6,8,1},{2,5,11},{7,15,5},{23,21,15},{18,11,21},{5,15,
    13},{15,21,13},{21,11,13},{11,5,13},{4,3,12},{2,11,3},{18,19,11},{20,12,
    19},{3,11,10},{11,19,10},{19,12,10},{12,3,10}}
PolyhedronVertices[CubeOctahedronCompound,Split->True]:=
{{-2,0,0},{-1,-1,-1},{-1,-1,0},{-1,-1,1},{-1,0,-1},{-1,0,1},{-1,1,-1},{-1,1,
    0},{-1,1,1},{0,-2,0},{0,-1,-1},{0,-1,1},{0,0,-2},{0,0,2},{0,1,-1},{0,1,
    1},{0,2,0},{1,-1,-1},{1,-1,0},{1,-1,1},{1,0,-1},{1,0,1},{1,1,-1},{1,1,
    0},{1,1,1},{2,0,0}}/2

(* Cube-Octahedron 5-Compound *)

(* Dodecahedron *)

PolyhedronName[Dodecahedron]:="dodecahedron"
PolyhedronDual[Dodecahedron]:=Icosahedron
NetFaces[Dodecahedron]:={{11,17,24,25,18},{38,33,25,24,32},{26,19,18,25,31},{5,4,11,18,12},{2,10,17,11,3},{23,30,24,17,16},{16,9,15,22,23},{1,6,14,15,7},{13,20,21,14,8},{34,35,28,21,27},{37,29,22,28,36},{28,22,15,14,21}}
NetVertices[Dodecahedron]:={{(-15Sqrt[2(5-Sqrt[5])]-5Sqrt[10(5-Sqrt[5])]-15Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/80,(-3(3+Sqrt[5]))/4},{(-10Sqrt[2(5-Sqrt[5])]-25Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/80,(-3-Sqrt[5])/4},{(-15Sqrt[2(5-Sqrt[5])]-5Sqrt[10(5-Sqrt[5])]-15Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/80,(1-Sqrt[5])/4},{(5Sqrt[2(5-Sqrt[5])]-5Sqrt[10(5-Sqrt[5])]-5Sqrt[2(5+Sqrt[5])]-13Sqrt[10(5+Sqrt[5])])/80,(-1+Sqrt[5])/4},{(10Sqrt[2(5-Sqrt[5])]-15Sqrt[2(5+Sqrt[5])]-13Sqrt[10(5+Sqrt[5])])/80,(3+Sqrt[5])/4},{(-15Sqrt[2(5-Sqrt[5])]-5Sqrt[10(5-Sqrt[5])]-25Sqrt[2(5+Sqrt[5])]+7Sqrt[10(5+Sqrt[5])])/80,(-5-2Sqrt[5])/2},{(-15Sqrt[2(5-Sqrt[5])]-5Sqrt[10(5-Sqrt[5])]-25Sqrt[2(5+Sqrt[5])]+7Sqrt[10(5+Sqrt[5])])/80,(-4-Sqrt[5])/2},{(-35Sqrt[2(5-Sqrt[5])]-15Sqrt[10(5-Sqrt[5])]+20Sqrt[2(5+Sqrt[5])]+2Sqrt[10(5+Sqrt[5])])/80,-3-Sqrt[5]},{(-35Sqrt[2(5-Sqrt[5])]-15Sqrt[10(5-Sqrt[5])]+20Sqrt[2(5+Sqrt[5])]+2Sqrt[10(5+Sqrt[5])])/80,(-3-Sqrt[5])/2},{(-5Sqrt[2(5-Sqrt[5])]+5Sqrt[10(5-Sqrt[5])]-15Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/80,(-1-Sqrt[5])/2},{-Sqrt[(5+Sqrt[5])/10],0},{(15Sqrt[2(5-Sqrt[5])]+5Sqrt[10(5-Sqrt[5])]-5Sqrt[2(5+Sqrt[5])]-13Sqrt[10(5+Sqrt[5])])/80,(1+Sqrt[5])/2},{(-20Sqrt[2(5-Sqrt[5])]-10Sqrt[10(5-Sqrt[5])]+5Sqrt[2(5+Sqrt[5])]+7Sqrt[10(5+Sqrt[5])])/80,(-13-5Sqrt[5])/4},{(-25Sqrt[2(5-Sqrt[5])]-15Sqrt[10(5-Sqrt[5])]+15Sqrt[2(5+Sqrt[5])]+7Sqrt[10(5+Sqrt[5])])/80,(-11-3Sqrt[5])/4},{(-25Sqrt[2(5-Sqrt[5])]-15Sqrt[10(5-Sqrt[5])]+15Sqrt[2(5+Sqrt[5])]+7Sqrt[10(5+Sqrt[5])])/80,(-7-3Sqrt[5])/4},{(-20Sqrt[2(5-Sqrt[5])]-10Sqrt[10(5-Sqrt[5])]+5Sqrt[2(5+Sqrt[5])]+7Sqrt[10(5+Sqrt[5])])/80,(-5-Sqrt[5])/4},{((-5+Sqrt[5])Sqrt[(5+Sqrt[5])/2])/20,(-1-Sqrt[5])/4},{((-5+Sqrt[5])Sqrt[(5+Sqrt[5])/2])/20,(1+Sqrt[5])/4},{(-10Sqrt[2(5-Sqrt[5])]-15Sqrt[2(5+Sqrt[5])]+7Sqrt[10(5+Sqrt[5])])/80,(5+Sqrt[5])/4},{(-5Sqrt[2(5-Sqrt[5])]-5Sqrt[10(5-Sqrt[5])]+20Sqrt[2(5+Sqrt[5])]+2Sqrt[10(5+Sqrt[5])])/80,(-7-2Sqrt[5])/2},{(-15Sqrt[2(5-Sqrt[5])]-5Sqrt[10(5-Sqrt[5])]+15Sqrt[2(5+Sqrt[5])]+7Sqrt[10(5+Sqrt[5])])/80,(-5-2Sqrt[5])/2},{(-15Sqrt[2(5-Sqrt[5])]-5Sqrt[10(5-Sqrt[5])]+15Sqrt[2(5+Sqrt[5])]+7Sqrt[10(5+Sqrt[5])])/80,(-4-Sqrt[5])/2},{(-5Sqrt[2(5-Sqrt[5])]-5Sqrt[10(5-Sqrt[5])]+20Sqrt[2(5+Sqrt[5])]+2Sqrt[10(5+Sqrt[5])])/80,(-2-Sqrt[5])/2},{(5+Sqrt[5])^(3/2)/(20Sqrt[2]),-1/2},{(5+Sqrt[5])^(3/2)/(20Sqrt[2]),1/2},{(5Sqrt[2(5-Sqrt[5])]+5Sqrt[10(5-Sqrt[5])]+2Sqrt[10(5+Sqrt[5])])/80,(2+Sqrt[5])/2},{(-15Sqrt[2(5-Sqrt[5])]-5Sqrt[10(5-Sqrt[5])]+5Sqrt[2(5+Sqrt[5])]+11Sqrt[10(5+Sqrt[5])])/40,(-11-5Sqrt[5])/4},{(5Sqrt[2(5-Sqrt[5])]-5Sqrt[10(5-Sqrt[5])]+15Sqrt[2(5+Sqrt[5])]+7Sqrt[10(5+Sqrt[5])])/80,(-3(3+Sqrt[5]))/4},{(-5Sqrt[2(5-Sqrt[5])]-5Sqrt[10(5-Sqrt[5])]+10Sqrt[2(5+Sqrt[5])]+6Sqrt[10(5+Sqrt[5])])/40,(-7-Sqrt[5])/4},{(10Sqrt[2(5-Sqrt[5])]+5Sqrt[2(5+Sqrt[5])]+7Sqrt[10(5+Sqrt[5])])/80,(-3-Sqrt[5])/4},{(20Sqrt[2(5-Sqrt[5])]+10Sqrt[10(5-Sqrt[5])]-15Sqrt[2(5+Sqrt[5])]+7Sqrt[10(5+Sqrt[5])])/80,(3+Sqrt[5])/4},{(Sqrt[(5+Sqrt[5])/2](15+Sqrt[5]))/20,(-1-Sqrt[5])/4},{(Sqrt[(5+Sqrt[5])/2](15+Sqrt[5]))/20,(1+Sqrt[5])/4},{(-25Sqrt[2(5-Sqrt[5])]-5Sqrt[10(5-Sqrt[5])]+20Sqrt[2(5+Sqrt[5])]+22Sqrt[10(5+Sqrt[5])])/80,-3-Sqrt[5]},{(-10Sqrt[2(5-Sqrt[5])]+5Sqrt[2(5+Sqrt[5])]+11Sqrt[10(5+Sqrt[5])])/40,-2-Sqrt[5]},{(Sqrt[(5+Sqrt[5])/2](5+3Sqrt[5]))/10,(-5-Sqrt[5])/2},{(-5Sqrt[2(5-Sqrt[5])]-5Sqrt[10(5-Sqrt[5])]+30Sqrt[2(5+Sqrt[5])]+12Sqrt[10(5+Sqrt[5])])/80,(-3-Sqrt[5])/2},{(Sqrt[(5+Sqrt[5])/2](5+3Sqrt[5]))/10,0}}
PolyhedronPolyhedra[Dodecahedron]:={Range[12]}
PolyhedronFaces[Dodecahedron]:=PolyhedronFaces[Dodecahedron,"C3"]
PolyhedronVertices[Dodecahedron]:=PolyhedronVertices[Dodecahedron,"C3"]

(* Dodecahedron C2 *)

PolyhedronName[Dodecahedron,"C2"]:="dodecahedron oriented with z-axis along a C2 axis of \
symmetry"
PolyhedronDual[Dodecahedron,"C2"]:=Sequence[Icosahedron,"C2"]
PolyhedronPolyhedra[Dodecahedron,"C2"]:={Range[12]}
PolyhedronFaces[Dodecahedron,"C2"]:=
{{14,10,9,12,2},{7,17,20,19,15},{15,19,16,4,3},{19,20,18,8,16},{20,17,5,6,
    18},{17,7,1,13,5},{7,15,3,11,1},{6,5,13,10,14},{13,1,11,9,10},{11,3,4,12,
    9},{4,16,8,2,12},{8,18,6,14,2}}
PolyhedronVertices[Dodecahedron,"C2"]:=
{{-1/2,0,(-3-Sqrt[5])/4},{-1/2,0,(3+Sqrt[5])/4},{0,(-3-Sqrt[5])/4,-1/2},
 {0,(-3-Sqrt[5])/4,1/2},{0,(3+Sqrt[5])/4,-1/2},{0,(3+Sqrt[5])/4,1/2},{1/2,0,(-3-Sqrt[5])/4},
 {1/2,0,(3+Sqrt[5])/4},{(-3-Sqrt[5])/4,-1/2,0},{(-3-Sqrt[5])/4,1/2,0},
 {(-1-Sqrt[5])/4,(-1-Sqrt[5])/4,(-1-Sqrt[5])/4},{(-1-Sqrt[5])/4,(-1-Sqrt[5])/4,(1+Sqrt[5])/4},
 {(-1-Sqrt[5])/4,(1+Sqrt[5])/4,(-1-Sqrt[5])/4},{(-1-Sqrt[5])/4,(1+Sqrt[5])/4,(1+Sqrt[5])/4},
 {(1+Sqrt[5])/4,(-1-Sqrt[5])/4,(-1-Sqrt[5])/4},{(1+Sqrt[5])/4,(-1-Sqrt[5])/4,(1+Sqrt[5])/4},
 {(1+Sqrt[5])/4,(1+Sqrt[5])/4,(-1-Sqrt[5])/4},{(1+Sqrt[5])/4,(1+Sqrt[5])/4,(1+Sqrt[5])/4},
 {(3+Sqrt[5])/4,-1/2,0},{(3+Sqrt[5])/4,1/2,0}}

(* Dodecahedron C3 *)

PolyhedronName[Dodecahedron,"C3"]:="dodecahedron oriented with z-axis along a C3 axis of \
symmetry"
PolyhedronDual[Dodecahedron,"C3"]:=Sequence[Icosahedron,"C3"]
PolyhedronPolyhedra[Dodecahedron,"C3"]:={Range[12]}
PolyhedronFaces[Dodecahedron,"C3"]:={{15,10,9,14,1},{2,6,12,11,5},{5,11,7,3,19},{11,12,8,16,7},{12,6,20,4,8},{6,2,
    13,18,20},{2,5,19,17,13},{4,20,18,10,15},{18,13,17,9,10},{17,19,3,14,
    9},{3,7,16,1,14},{16,8,4,15,1}}
PolyhedronVertices[Dodecahedron,"C3"]:=
{{0,0,Sqrt[9/8+(3Sqrt[5])/8]},{0,0,-Sqrt[(3(3+Sqrt[5]))/2]/2},
 {Sqrt[1/8-Sqrt[5]/24],(-3-Sqrt[5])/4,Sqrt[1/8+Sqrt[5]/24]},{Sqrt[1/8-Sqrt[5]/24],(3+Sqrt[5])/4,
  Sqrt[1/8+Sqrt[5]/24]},{Sqrt[1/8+Sqrt[5]/24],(-1-Sqrt[5])/4,-Sqrt[(5(3+Sqrt[5]))/6]/2},
 {Sqrt[1/8+Sqrt[5]/24],(1+Sqrt[5])/4,-Sqrt[(5(3+Sqrt[5]))/6]/2},
 {Sqrt[5/8+(5Sqrt[5])/24],(-1-Sqrt[5])/4,Sqrt[1/8+Sqrt[5]/24]},
 {Sqrt[5/8+(5Sqrt[5])/24],(1+Sqrt[5])/4,Sqrt[1/8+Sqrt[5]/24]},
 {-Sqrt[3/4+Sqrt[5]/3],-1/2,Sqrt[1/8+Sqrt[5]/24]},{-Sqrt[3/4+Sqrt[5]/3],1/2,Sqrt[1/8+Sqrt[5]/24]},
 {Sqrt[3/4+Sqrt[5]/3],-1/2,-Sqrt[(3+Sqrt[5])/6]/2},{Sqrt[3/4+Sqrt[5]/3],1/2,-Sqrt[(3+Sqrt[5])/6]/2},
 {-Sqrt[(3+Sqrt[5])/6],0,-Sqrt[(5(3+Sqrt[5]))/6]/2},{-Sqrt[(3+Sqrt[5])/6]/2,(-1-Sqrt[5])/4,
  Sqrt[5/8+(5Sqrt[5])/24]},{-Sqrt[(3+Sqrt[5])/6]/2,(1+Sqrt[5])/4,Sqrt[5/8+(5Sqrt[5])/24]},
 {Sqrt[(3+Sqrt[5])/6],0,Sqrt[5/8+(5Sqrt[5])/24]},{-Sqrt[(5(3+Sqrt[5]))/6]/2,(-1-Sqrt[5])/4,
  -Sqrt[(3+Sqrt[5])/6]/2},{-Sqrt[(5(3+Sqrt[5]))/6]/2,(1+Sqrt[5])/4,-Sqrt[(3+Sqrt[5])/6]/2},
 {Root[1-36#1^2+144#1^4&,2,0],(-3-Sqrt[5])/4,-Sqrt[(3+Sqrt[5])/6]/2},
 {Root[1-36#1^2+144#1^4&,2,0],(3+Sqrt[5])/4,-Sqrt[(3+Sqrt[5])/6]/2}}

(* Dodecahedron C5 *)

PolyhedronName[Dodecahedron,"C5"]:="dodecahedron oriented with z-axis along a C5 axis of \
symmetry"
PolyhedronDual[Dodecahedron,"C5"]:=Sequence[Icosahedron,"C5"]
PolyhedronPolyhedra[Dodecahedron,"C5"]:={Range[12]}
PolyhedronFaces[Dodecahedron,"C5"]:={{18,10,9,17,4},{3,16,12,11,15},{15,11,7,19,13},{11,12,8,2,7},
{12,16,14,20,
    8},{16,3,1,6,14},{3,15,13,5,1},{20,14,6,10,18},{6,1,5,9,10},{5,13,19,17,
    9},{19,7,2,4,17},{2,8,20,18,4}}
PolyhedronVertices[Dodecahedron,"C5"]:={
 {Root[1-10#1^2+5#1^4&,1,0],0,Root[1-20#1^2+80#1^4&,2,0]},
 {Root[1-10#1^2+5#1^4&,4,0],0,Root[1-20#1^2+80#1^4&,3,0]},
 {Root[1-5#1^2+5#1^4&,1,0],0,Root[1-100#1^2+80#1^4&,1,0]},
 {Root[1-5#1^2+5#1^4&,4,0],0,Root[1-100#1^2+80#1^4&,4,0]},
 {Root[1-100#1^2+80#1^4&,1,0],(-1-Sqrt[5])/4,Root[1-20#1^2+80#1^4&,3,0]},
 {Root[1-100#1^2+80#1^4&,1,0],(1+Sqrt[5])/4,Root[1-20#1^2+80#1^4&,3,0]},
 {Root[1-100#1^2+80#1^4&,4,0],(-1-Sqrt[5])/4,Root[1-20#1^2+80#1^4&,2,0]},
 {Root[1-100#1^2+80#1^4&,4,0],(1+Sqrt[5])/4,Root[1-20#1^2+80#1^4&,2,0]},
 {Root[1-40#1^2+80#1^4&,1,0],-1/2,Root[1-100#1^2+80#1^4&,4,0]},
 {Root[1-40#1^2+80#1^4&,1,0],1/2,Root[1-100#1^2+80#1^4&,4,0]},
 {Root[1-40#1^2+80#1^4&,4,0],-1/2,Root[1-100#1^2+80#1^4&,1,0]},
 {Root[1-40#1^2+80#1^4&,4,0],1/2,Root[1-100#1^2+80#1^4&,1,0]},
 {Root[1-20#1^2+80#1^4&,1,0],(-3-Sqrt[5])/4,Root[1-20#1^2+80#1^4&,2,0]},
 {Root[1-20#1^2+80#1^4&,1,0],(3+Sqrt[5])/4,Root[1-20#1^2+80#1^4&,2,0]},
 {Root[1-20#1^2+80#1^4&,2,0],(-1-Sqrt[5])/4,Root[1-100#1^2+80#1^4&,1,0]},
 {Root[1-20#1^2+80#1^4&,2,0],(1+Sqrt[5])/4,Root[1-100#1^2+80#1^4&,1,0]},
 {Root[1-20#1^2+80#1^4&,3,0],(-1-Sqrt[5])/4,Root[1-100#1^2+80#1^4&,4,0]},
 {Root[1-20#1^2+80#1^4&,3,0],(1+Sqrt[5])/4,Root[1-100#1^2+80#1^4&,4,0]},
 {Root[1-20#1^2+80#1^4&,4,0],(-3-Sqrt[5])/4,Root[1-20#1^2+80#1^4&,3,0]},
 {Root[1-20#1^2+80#1^4&,4,0],(3+Sqrt[5])/4,Root[1-20#1^2+80#1^4&,3,0]}}

(* Dodecahedron 2-Compound C2 *)

PolyhedronName[Dodecahedron2Compound,"C2"]:="dodecahedron 2-compound with 4-fold rotational symmetry"
NetPieces[Dodecahedron2Compound,"C2"]:={{24,{1,2,3}}}
NetFaces[Dodecahedron2Compound,"C2"]:={{1,2,3},{2,5,3},{2,4,1}}
NetVertices[Dodecahedron2Compound,"C2"]:=
{{0,0},{(1+Sqrt[5])/4,Sqrt[(5-Sqrt[5])/2]/2},{(1+Sqrt[5])/2,0},
 {(3+Sqrt[5])/8,Sqrt[25/32+(5Sqrt[5])/32]},
 {(1+3Sqrt[5])/8,Sqrt[25/32+(5Sqrt[5])/32]}}
 
PolyhedronPolyhedra[Dodecahedron2Compound,"C2"]:=
{{1,2,3,4,5,6,7,8,9,10,11,12},{13,14,15,16,17,18,19,20,21,22,23,24}}
PolyhedronFaces[Dodecahedron2Compound,"C2"]:=
{{24,15,14,23,1},{2,9,17,16,8},{8,16,12,3,31},{16,17,13,26,12},{17,9,32,4,
    13},{9,2,19,29,32},{2,8,31,28,19},{4,32,29,15,24},{29,19,28,14,15},{28,31,
    3,23,14},{3,12,26,1,23},{26,13,4,24,1},{29,21,20,28,30},{18,13,6,5,
    12},{12,5,23,22,7},{5,6,24,27,23},{6,13,10,25,24},{13,18,11,9,10},{18,12,
    7,8,11},{25,10,9,21,29},{9,11,8,20,21},{8,7,22,28,20},{22,23,27,30,
    28},{27,24,25,29,30}}
PolyhedronVertices[Dodecahedron2Compound,"C2"]:=
{{0,0,Sqrt[9/8+(3Sqrt[5])/8]},{0,0,-Sqrt[(3(3+Sqrt[5]))/2]/2},
 {Sqrt[1/8-Sqrt[5]/24],(-3-Sqrt[5])/4,Sqrt[1/8+Sqrt[5]/24]},{Sqrt[1/8-Sqrt[5]/24],(3+Sqrt[5])/4,
  Sqrt[1/8+Sqrt[5]/24]},{Sqrt[1/8+Sqrt[5]/24],-1/2,Sqrt[3/4+Sqrt[5]/3]},
 {Sqrt[1/8+Sqrt[5]/24],1/2,Sqrt[3/4+Sqrt[5]/3]},{Sqrt[1/8+Sqrt[5]/24],(-3-Sqrt[5])/4,
  Root[1-36#1^2+144#1^4&,2,0]},{Sqrt[1/8+Sqrt[5]/24],(-1-Sqrt[5])/4,-Sqrt[(5(3+Sqrt[5]))/6]/2},
 {Sqrt[1/8+Sqrt[5]/24],(1+Sqrt[5])/4,-Sqrt[(5(3+Sqrt[5]))/6]/2},
 {Sqrt[1/8+Sqrt[5]/24],(3+Sqrt[5])/4,Root[1-36#1^2+144#1^4&,2,0]},
 {Sqrt[5/8+(5Sqrt[5])/24],0,-Sqrt[(3+Sqrt[5])/6]},{Sqrt[5/8+(5Sqrt[5])/24],(-1-Sqrt[5])/4,
  Sqrt[1/8+Sqrt[5]/24]},{Sqrt[5/8+(5Sqrt[5])/24],(1+Sqrt[5])/4,Sqrt[1/8+Sqrt[5]/24]},
 {-Sqrt[3/4+Sqrt[5]/3],-1/2,Sqrt[1/8+Sqrt[5]/24]},{-Sqrt[3/4+Sqrt[5]/3],1/2,Sqrt[1/8+Sqrt[5]/24]},
 {Sqrt[3/4+Sqrt[5]/3],-1/2,-Sqrt[(3+Sqrt[5])/6]/2},{Sqrt[3/4+Sqrt[5]/3],1/2,-Sqrt[(3+Sqrt[5])/6]/2},
 {Sqrt[9/8+(3Sqrt[5])/8],0,0},{-Sqrt[(3+Sqrt[5])/6],0,-Sqrt[(5(3+Sqrt[5]))/6]/2},
 {-Sqrt[(3+Sqrt[5])/6]/2,-1/2,-Sqrt[3/4+Sqrt[5]/3]},{-Sqrt[(3+Sqrt[5])/6]/2,1/2,-Sqrt[3/4+Sqrt[5]/3]},
 {-Sqrt[(3+Sqrt[5])/6]/2,(-3-Sqrt[5])/4,Sqrt[1/8-Sqrt[5]/24]},{-Sqrt[(3+Sqrt[5])/6]/2,(-1-Sqrt[5])/4,
  Sqrt[5/8+(5Sqrt[5])/24]},{-Sqrt[(3+Sqrt[5])/6]/2,(1+Sqrt[5])/4,Sqrt[5/8+(5Sqrt[5])/24]},
 {-Sqrt[(3+Sqrt[5])/6]/2,(3+Sqrt[5])/4,Sqrt[1/8-Sqrt[5]/24]},
 {Sqrt[(3+Sqrt[5])/6],0,Sqrt[5/8+(5Sqrt[5])/24]},{-Sqrt[(5(3+Sqrt[5]))/6]/2,0,Sqrt[(3+Sqrt[5])/6]},
 {-Sqrt[(5(3+Sqrt[5]))/6]/2,(-1-Sqrt[5])/4,-Sqrt[(3+Sqrt[5])/6]/2},
 {-Sqrt[(5(3+Sqrt[5]))/6]/2,(1+Sqrt[5])/4,-Sqrt[(3+Sqrt[5])/6]/2},{-Sqrt[(3(3+Sqrt[5]))/2]/2,0,0},
 {Root[1-36#1^2+144#1^4&,2,0],(-3-Sqrt[5])/4,-Sqrt[(3+Sqrt[5])/6]/2},
 {Root[1-36#1^2+144#1^4&,2,0],(3+Sqrt[5])/4,-Sqrt[(3+Sqrt[5])/6]/2}}

(* Dodecahedron2Compound C3 *)

PolyhedronName[Dodecahedron2Compound,"C3"]:="dodecahedron 2-compound with 5-fold rotational symmetry"
PolyhedronPolyhedra[Dodecahedron2Compound,"C3"]:=
{{1,2,3,4,5,6,7,8,9,10,11,12},{13,14,15,16,17,18,19,20,21,22,23,24}}
PolyhedronFaces[Dodecahedron2Compound,"C3"]:=
{{27,17,15,25,1},{2,10,22,20,8},{8,20,11,3,36},{20,22,13,29,11},{22,10,38,5,
    13},{10,2,24,34,38},{2,8,36,32,24},{5,38,34,17,27},{34,24,32,15,17},{32,
    36,3,25,15},{3,11,29,1,25},{29,13,5,27,1},{9,37,33,23,1},{2,30,12,4,
    26},{26,4,35,31,16},{4,12,19,7,35},{12,30,14,21,19},{30,2,28,6,14},{2,26,
    16,18,28},{21,14,6,37,9},{6,28,18,33,37},{18,16,31,23,33},{31,35,7,1,
    23},{7,19,21,9,1}}
PolyhedronVertices[Dodecahedron2Compound,"C3"]:=
{{0,0,Sqrt[9/8+(3Sqrt[5])/8]},{0,0,-Sqrt[(3(3+Sqrt[5]))/2]/2},
 {Sqrt[1/8-Sqrt[5]/24],(-3-Sqrt[5])/4,Sqrt[1/8+Sqrt[5]/24]},{Sqrt[1/8-Sqrt[5]/24],(-3-Sqrt[5])/4,
  -Sqrt[(3+Sqrt[5])/6]/2},{Sqrt[1/8-Sqrt[5]/24],(3+Sqrt[5])/4,Sqrt[1/8+Sqrt[5]/24]},
 {Sqrt[1/8-Sqrt[5]/24],(3+Sqrt[5])/4,-Sqrt[(3+Sqrt[5])/6]/2},{Sqrt[1/8+Sqrt[5]/24],(-1-Sqrt[5])/4,
  Sqrt[5/8+(5Sqrt[5])/24]},{Sqrt[1/8+Sqrt[5]/24],(-1-Sqrt[5])/4,-Sqrt[(5(3+Sqrt[5]))/6]/2},
 {Sqrt[1/8+Sqrt[5]/24],(1+Sqrt[5])/4,Sqrt[5/8+(5Sqrt[5])/24]},
 {Sqrt[1/8+Sqrt[5]/24],(1+Sqrt[5])/4,-Sqrt[(5(3+Sqrt[5]))/6]/2},
 {Sqrt[5/8+(5Sqrt[5])/24],(-1-Sqrt[5])/4,Sqrt[1/8+Sqrt[5]/24]},
 {Sqrt[5/8+(5Sqrt[5])/24],(-1-Sqrt[5])/4,-Sqrt[(3+Sqrt[5])/6]/2},
 {Sqrt[5/8+(5Sqrt[5])/24],(1+Sqrt[5])/4,Sqrt[1/8+Sqrt[5]/24]},
 {Sqrt[5/8+(5Sqrt[5])/24],(1+Sqrt[5])/4,-Sqrt[(3+Sqrt[5])/6]/2},
 {-Sqrt[3/4+Sqrt[5]/3],-1/2,Sqrt[1/8+Sqrt[5]/24]},{-Sqrt[3/4+Sqrt[5]/3],-1/2,-Sqrt[(3+Sqrt[5])/6]/2},
 {-Sqrt[3/4+Sqrt[5]/3],1/2,Sqrt[1/8+Sqrt[5]/24]},{-Sqrt[3/4+Sqrt[5]/3],1/2,-Sqrt[(3+Sqrt[5])/6]/2},
 {Sqrt[3/4+Sqrt[5]/3],-1/2,Sqrt[1/8+Sqrt[5]/24]},{Sqrt[3/4+Sqrt[5]/3],-1/2,-Sqrt[(3+Sqrt[5])/6]/2},
 {Sqrt[3/4+Sqrt[5]/3],1/2,Sqrt[1/8+Sqrt[5]/24]},{Sqrt[3/4+Sqrt[5]/3],1/2,-Sqrt[(3+Sqrt[5])/6]/2},
 {-Sqrt[(3+Sqrt[5])/6],0,Sqrt[5/8+(5Sqrt[5])/24]},{-Sqrt[(3+Sqrt[5])/6],0,-Sqrt[(5(3+Sqrt[5]))/6]/2},
 {-Sqrt[(3+Sqrt[5])/6]/2,(-1-Sqrt[5])/4,Sqrt[5/8+(5Sqrt[5])/24]},
 {-Sqrt[(3+Sqrt[5])/6]/2,(-1-Sqrt[5])/4,-Sqrt[(5(3+Sqrt[5]))/6]/2},
 {-Sqrt[(3+Sqrt[5])/6]/2,(1+Sqrt[5])/4,Sqrt[5/8+(5Sqrt[5])/24]},
 {-Sqrt[(3+Sqrt[5])/6]/2,(1+Sqrt[5])/4,-Sqrt[(5(3+Sqrt[5]))/6]/2},
 {Sqrt[(3+Sqrt[5])/6],0,Sqrt[5/8+(5Sqrt[5])/24]},{Sqrt[(3+Sqrt[5])/6],0,-Sqrt[(5(3+Sqrt[5]))/6]/2},
 {-Sqrt[(5(3+Sqrt[5]))/6]/2,(-1-Sqrt[5])/4,Sqrt[1/8+Sqrt[5]/24]},
 {-Sqrt[(5(3+Sqrt[5]))/6]/2,(-1-Sqrt[5])/4,-Sqrt[(3+Sqrt[5])/6]/2},
 {-Sqrt[(5(3+Sqrt[5]))/6]/2,(1+Sqrt[5])/4,Sqrt[1/8+Sqrt[5]/24]},
 {-Sqrt[(5(3+Sqrt[5]))/6]/2,(1+Sqrt[5])/4,-Sqrt[(3+Sqrt[5])/6]/2},
 {Root[1-36#1^2+144#1^4&,2,0],(-3-Sqrt[5])/4,Sqrt[1/8+Sqrt[5]/24]},
 {Root[1-36#1^2+144#1^4&,2,0],(-3-Sqrt[5])/4,-Sqrt[(3+Sqrt[5])/6]/2},
 {Root[1-36#1^2+144#1^4&,2,0],(3+Sqrt[5])/4,Sqrt[1/8+Sqrt[5]/24]},
 {Root[1-36#1^2+144#1^4&,2,0],(3+Sqrt[5])/4,-Sqrt[(3+Sqrt[5])/6]/2}}

(* Dodecahedron2Compound C5 *)

PolyhedronPolyhedra[Dodecahedron2Compound,"C5"]:=
{{1,2,3,4,5,6,7,8,9,10,11,12},{13,14,15,16,17,18,19,20,21,22,23,24}}
PolyhedronName[Dodecahedron2Compound,"C5"]:="dodecahedron 2-compound with 10-fold rotational symmetry"
PolyhedronFaces[Dodecahedron2Compound,"C5"]:=
{{36,20,19,35,8},{6,32,24,23,31},{31,23,15,39,27},{23,24,16,4,15},{24,32,28,
    40,16},{32,6,2,12,28},{6,31,27,11,2},{40,28,12,20,36},{12,2,11,19,20},{11,
    27,39,35,19},{39,15,4,8,35},{4,16,40,36,8},{7,33,17,18,34},{29,21,22,30,
    5},{25,37,13,21,29},{13,3,14,22,21},{14,38,26,30,22},{26,10,1,5,30},{1,9,
    25,29,5},{34,18,10,26,38},{18,17,9,1,10},{17,33,37,25,9},{33,7,3,13,
    37},{7,34,38,14,3}}
PolyhedronVertices[Dodecahedron2Compound,"C5"]:=
{{Sqrt[1+2/Sqrt[5]],0,Root[1-20#1^2+80#1^4&,2,0]},
 {-Sqrt[1+2/Sqrt[5]],0,Root[1-20#1^2+80#1^4&,2,0]},{-Sqrt[1+2/Sqrt[5]],0,Sqrt[1/8-1/(8Sqrt[5])]},
 {Sqrt[1+2/Sqrt[5]],0,Sqrt[1/8-1/(8Sqrt[5])]},{Sqrt[(5+Sqrt[5])/10],0,
  Root[1-100#1^2+80#1^4&,1,0]},{Root[1-5#1^2+5#1^4&,1,0],0,
  Root[1-100#1^2+80#1^4&,1,0]},{Root[1-5#1^2+5#1^4&,1,0],0,Sqrt[5/8+11/(8Sqrt[5])]},
 {Sqrt[(5+Sqrt[5])/10],0,Sqrt[5/8+11/(8Sqrt[5])]},{Sqrt[5/8+11/(8Sqrt[5])],(-1-Sqrt[5])/4,
  Sqrt[1/8-1/(8Sqrt[5])]},{Sqrt[5/8+11/(8Sqrt[5])],(1+Sqrt[5])/4,Sqrt[1/8-1/(8Sqrt[5])]},
 {Root[1-100#1^2+80#1^4&,1,0],(-1-Sqrt[5])/4,Sqrt[1/8-1/(8Sqrt[5])]},
 {Root[1-100#1^2+80#1^4&,1,0],(1+Sqrt[5])/4,Sqrt[1/8-1/(8Sqrt[5])]},
 {Root[1-100#1^2+80#1^4&,1,0],(-1-Sqrt[5])/4,Root[1-20#1^2+80#1^4&,2,0]},
 {Root[1-100#1^2+80#1^4&,1,0],(1+Sqrt[5])/4,Root[1-20#1^2+80#1^4&,2,0]},
 {Sqrt[5/8+11/(8Sqrt[5])],(-1-Sqrt[5])/4,Root[1-20#1^2+80#1^4&,2,0]},
 {Sqrt[5/8+11/(8Sqrt[5])],(1+Sqrt[5])/4,Root[1-20#1^2+80#1^4&,2,0]},
 {Sqrt[1/4+1/(2Sqrt[5])],-1/2,Sqrt[5/8+11/(8Sqrt[5])]},{Sqrt[1/4+1/(2Sqrt[5])],1/2,
  Sqrt[5/8+11/(8Sqrt[5])]},{-Sqrt[1+2/Sqrt[5]]/2,-1/2,Sqrt[5/8+11/(8Sqrt[5])]},
 {-Sqrt[1+2/Sqrt[5]]/2,1/2,Sqrt[5/8+11/(8Sqrt[5])]},{-Sqrt[1+2/Sqrt[5]]/2,-1/2,
  Root[1-100#1^2+80#1^4&,1,0]},{-Sqrt[1+2/Sqrt[5]]/2,1/2,Root[1-100#1^2+80#1^4&,1,0]},
 {Sqrt[1/4+1/(2Sqrt[5])],-1/2,Root[1-100#1^2+80#1^4&,1,0]},
 {Sqrt[1/4+1/(2Sqrt[5])],1/2,Root[1-100#1^2+80#1^4&,1,0]},
 {Sqrt[1/8+1/(8Sqrt[5])],(-3-Sqrt[5])/4,Root[1-20#1^2+80#1^4&,2,0]},
 {Sqrt[1/8+1/(8Sqrt[5])],(3+Sqrt[5])/4,Root[1-20#1^2+80#1^4&,2,0]},
 {Root[1-20#1^2+80#1^4&,1,0],(-3-Sqrt[5])/4,Root[1-20#1^2+80#1^4&,2,0]},
 {Root[1-20#1^2+80#1^4&,1,0],(3+Sqrt[5])/4,Root[1-20#1^2+80#1^4&,2,0]},
 {Sqrt[1/8-1/(8Sqrt[5])],(-1-Sqrt[5])/4,Root[1-100#1^2+80#1^4&,1,0]},
 {Sqrt[1/8-1/(8Sqrt[5])],(1+Sqrt[5])/4,Root[1-100#1^2+80#1^4&,1,0]},
 {Root[1-20#1^2+80#1^4&,2,0],(-1-Sqrt[5])/4,Root[1-100#1^2+80#1^4&,1,0]},
 {Root[1-20#1^2+80#1^4&,2,0],(1+Sqrt[5])/4,Root[1-100#1^2+80#1^4&,1,0]},
 {Root[1-20#1^2+80#1^4&,2,0],(-1-Sqrt[5])/4,Sqrt[5/8+11/(8Sqrt[5])]},
 {Root[1-20#1^2+80#1^4&,2,0],(1+Sqrt[5])/4,Sqrt[5/8+11/(8Sqrt[5])]},
 {Sqrt[1/8-1/(8Sqrt[5])],(-1-Sqrt[5])/4,Sqrt[5/8+11/(8Sqrt[5])]},
 {Sqrt[1/8-1/(8Sqrt[5])],(1+Sqrt[5])/4,Sqrt[5/8+11/(8Sqrt[5])]},
 {Root[1-20#1^2+80#1^4&,1,0],(-3-Sqrt[5])/4,Sqrt[1/8-1/(8Sqrt[5])]},
 {Root[1-20#1^2+80#1^4&,1,0],(3+Sqrt[5])/4,Sqrt[1/8-1/(8Sqrt[5])]},
 {Sqrt[1/8+1/(8Sqrt[5])],(-3-Sqrt[5])/4,Sqrt[1/8-1/(8Sqrt[5])]},
 {Sqrt[1/8+1/(8Sqrt[5])],(3+Sqrt[5])/4,Sqrt[1/8-1/(8Sqrt[5])]}}

(* Dodecahedron 5-Compound *)

PolyhedronName[Dodecahedron5Compound]:="compound of 5 dodecahedra"
NetPieces[Dodecahedron5Compound]:={{30,{1,2,3,4}},{60,{5,6}},{60,{7,8}}}
NetFaces[Dodecahedron5Compound]:=
{{1,8,13,11,2,18,20,15,4},{1,9,14,12,3,19,21,16,4},{9,7,10,1},{16,17,22,
    4},{26,25,5},{24,25,5},{28,27,6},{23,27,6}}
NetVertices[Dodecahedron5Compound]:=
{{0,0},{1/2,-Sqrt[5/16-Sqrt[5]/8]},{1/2,Sqrt[5/16-Sqrt[5]/8]},{1,0},{7/4,-1/4},
 {2,-1/4},{-Sqrt[5]/8,Sqrt[5/64-Sqrt[5]/32]},
 {(9-5Sqrt[5])/44,-Sqrt[65/968-(19Sqrt[5])/968]},
 {(9-5Sqrt[5])/44,Sqrt[65/968-(19Sqrt[5])/968]},
 {(1-3Sqrt[5])/44,-Sqrt[145/968-(61Sqrt[5])/968]},
 {(5-Sqrt[5])/20,-Sqrt[5/8-11/(8Sqrt[5])]},{(5-Sqrt[5])/20,Sqrt[5/8-11/(8Sqrt[5])]},
 {(5-Sqrt[5])/20,-Sqrt[1/8-1/(8Sqrt[5])]},{(5-Sqrt[5])/20,Sqrt[1/8-1/(8Sqrt[5])]},
 {(5(7+Sqrt[5]))/44,-Sqrt[65/968-(19Sqrt[5])/968]},
 {(5(7+Sqrt[5]))/44,Sqrt[65/968-(19Sqrt[5])/968]},
 {(8+Sqrt[5])/8,Sqrt[5/64-Sqrt[5]/32]},{(15+Sqrt[5])/20,-Sqrt[5/8-11/(8Sqrt[5])]},
 {(15+Sqrt[5])/20,Sqrt[5/8-11/(8Sqrt[5])]},{(15+Sqrt[5])/20,-Sqrt[1/8-1/(8Sqrt[5])]},
 {(15+Sqrt[5])/20,Sqrt[1/8-1/(8Sqrt[5])]},{(43+3Sqrt[5])/44,
  -Sqrt[145/968-(61Sqrt[5])/968]},{(160-Sqrt[10(25-11Sqrt[5])])/80,
  (-15+11Sqrt[5])/80},{(140+Sqrt[10(25-11Sqrt[5])])/80,(-15+11Sqrt[5])/80},
 {(35-4Sqrt[5(5-2Sqrt[5])])/20,(-5+4Sqrt[5])/20},
 {(35-2Sqrt[5(5-2Sqrt[5])])/20,-1/4},{(10+Sqrt[5(5-2Sqrt[5])])/5,
  (-5+4Sqrt[5])/20},{(20+Sqrt[5(5-2Sqrt[5])])/10,-1/4}}
PolyhedronPolyhedra[Dodecahedron5Compound]:=
{{1,2,3,4,5,6,7,8,9,10,11,12},{13,14,15,16,17,18,19,20,21,22,23,24},{25,26,27,
    28,29,30,31,32,33,34,35,36},{37,38,39,40,41,42,43,44,45,46,47,48},{49,50,
    51,52,53,54,55,56,57,58,59,60}}
PolyhedronDual[Dodecahedron5Compound]:=Icosahedron5Compound
PolyhedronFaces[Dodecahedron5Compound]:=
{{48,34,33,47,3},{6,77,57,56,76},{76,56,31,29,99},{56,57,32,49,31},{57,77,100,
    30,32},{77,6,78,59,100},{6,76,99,58,78},{30,100,59,34,48},{59,78,58,33,
    34},{58,99,29,47,33},{29,31,49,3,47},{49,32,30,48,3},{25,20,22,23,2},{9,
    87,68,70,85},{85,70,15,11,95},{70,68,17,28,15},{68,87,97,13,17},{87,9,89,
    74,97},{9,85,95,72,89},{13,97,74,20,25},{74,89,72,22,20},{72,95,11,23,
    22},{11,15,28,2,23},{28,17,13,25,2},{52,44,46,50,5},{7,81,60,62,79},{79,
    62,39,35,91},{62,60,41,55,39},{60,81,93,37,41},{81,7,83,66,93},{7,79,91,
    64,83},{37,93,66,44,52},{66,83,64,46,44},{64,91,35,50,46},{35,39,55,5,
    50},{55,41,37,52,5},{53,45,43,51,4},{8,82,63,61,80},{80,61,40,36,92},{61,
    63,42,54,40},{63,82,94,38,42},{82,8,84,67,94},{8,80,92,65,84},{38,94,67,
    45,53},{67,84,65,43,45},{65,92,36,51,43},{36,40,54,4,51},{54,42,38,53,
    4},{26,21,19,24,1},{10,88,71,69,86},{86,69,16,12,96},{69,71,18,27,16},{71,
    88,98,14,18},{88,10,90,75,98},{10,86,96,73,90},{14,98,75,21,26},{75,90,73,
    19,21},{73,96,12,24,19},{12,16,27,1,24},{27,18,14,26,1}}
PolyhedronVertices[Dodecahedron5Compound]:=
{{Sqrt[1/16-1/(8Sqrt[5])],-1/4,Sqrt[1+2/Sqrt[5]]},{Sqrt[1/16-1/(8Sqrt[5])],1/4,Sqrt[1+2/Sqrt[5]]},
 {Sqrt[1/8-1/(8Sqrt[5])],0,Sqrt[1+2/Sqrt[5]]},{-Sqrt[(5+Sqrt[5])/10]/4,(1-Sqrt[5])/8,Sqrt[1+2/Sqrt[5]]},
 {-Sqrt[(5+Sqrt[5])/10]/4,(-1+Sqrt[5])/8,Sqrt[1+2/Sqrt[5]]},{Root[1-20#1^2+80#1^4&,2,0],0,
  -Sqrt[1+2/Sqrt[5]]},{Sqrt[(5+Sqrt[5])/10]/4,(1-Sqrt[5])/8,-Sqrt[1+2/Sqrt[5]]},
 {Sqrt[(5+Sqrt[5])/10]/4,(-1+Sqrt[5])/8,-Sqrt[1+2/Sqrt[5]]},
 {-Sqrt[1-2/Sqrt[5]]/4,-1/4,-Sqrt[1+2/Sqrt[5]]},{-Sqrt[1-2/Sqrt[5]]/4,1/4,-Sqrt[1+2/Sqrt[5]]},
 {Sqrt[29/32+61/(32Sqrt[5])],(1-Sqrt[5])/8,Sqrt[1/8+1/(8Sqrt[5])]},
 {-Sqrt[25/2+41/(2Sqrt[5])]/4,(-3-Sqrt[5])/8,Sqrt[1/8+1/(8Sqrt[5])]},
 {-Sqrt[25/2+41/(2Sqrt[5])]/4,(3+Sqrt[5])/8,Sqrt[1/8+1/(8Sqrt[5])]},
 {Sqrt[29/32+61/(32Sqrt[5])],(-1+Sqrt[5])/8,Sqrt[1/8+1/(8Sqrt[5])]},
 {Sqrt[5/8+11/(8Sqrt[5])],(1+Sqrt[5])/4,Sqrt[1/8-1/(8Sqrt[5])]},
 {Root[1-20#1^2+80#1^4&,1,0],(-3-Sqrt[5])/4,Sqrt[1/8-1/(8Sqrt[5])]},
 {Root[1-20#1^2+80#1^4&,1,0],(3+Sqrt[5])/4,Sqrt[1/8-1/(8Sqrt[5])]},
 {Sqrt[5/8+11/(8Sqrt[5])],(-1-Sqrt[5])/4,Sqrt[1/8-1/(8Sqrt[5])]},
 {-Sqrt[(65+19Sqrt[5])/10]/4,(5+Sqrt[5])/8,Sqrt[1/4+1/(2Sqrt[5])]},
 {-Sqrt[(65+19Sqrt[5])/10]/4,(-5-Sqrt[5])/8,Sqrt[1/4+1/(2Sqrt[5])]},
 {Sqrt[1/32-1/(32Sqrt[5])],(3(1+Sqrt[5]))/8,Sqrt[1/4+1/(2Sqrt[5])]},
 {Sqrt[1/32-1/(32Sqrt[5])],(-3(1+Sqrt[5]))/8,Sqrt[1/4+1/(2Sqrt[5])]},
 {Sqrt[1/4+1/(2Sqrt[5])],-1/2,Sqrt[5/8+11/(8Sqrt[5])]},{Root[1-5#1^2+5#1^4&,1,0],0,
  Sqrt[5/8+11/(8Sqrt[5])]},{Root[1-5#1^2+5#1^4&,1,0],0,Sqrt[5/8+11/(8Sqrt[5])]},
 {Sqrt[1/4+1/(2Sqrt[5])],1/2,Sqrt[5/8+11/(8Sqrt[5])]},{Sqrt[1/16+1/(8Sqrt[5])],(-2-Sqrt[5])/4,
  Sqrt[(5+Sqrt[5])/10]},{Sqrt[1/16+1/(8Sqrt[5])],(2+Sqrt[5])/4,Sqrt[(5+Sqrt[5])/10]},
 {Sqrt[1/8-1/(8Sqrt[5])],(-3-Sqrt[5])/4,Sqrt[1/8+1/(8Sqrt[5])]},
 {Sqrt[1/8-1/(8Sqrt[5])],(3+Sqrt[5])/4,Sqrt[1/8+1/(8Sqrt[5])]},
 {Sqrt[5/8+11/(8Sqrt[5])],(-1-Sqrt[5])/4,Sqrt[1/8-1/(8Sqrt[5])]},
 {Sqrt[5/8+11/(8Sqrt[5])],(1+Sqrt[5])/4,Sqrt[1/8-1/(8Sqrt[5])]},
 {Root[1-100#1^2+80#1^4&,1,0],-1/2,Sqrt[1/4+1/(2Sqrt[5])]},
 {Root[1-100#1^2+80#1^4&,1,0],1/2,Sqrt[1/4+1/(2Sqrt[5])]},
 {Sqrt[5/32+11/(32Sqrt[5])],(3(1+Sqrt[5]))/8,Sqrt[1/8+1/(8Sqrt[5])]},
 {-Sqrt[(85+31Sqrt[5])/10]/4,(5+Sqrt[5])/8,Sqrt[1/8+1/(8Sqrt[5])]},
 {-Sqrt[(85+31Sqrt[5])/10]/4,(-5-Sqrt[5])/8,Sqrt[1/8+1/(8Sqrt[5])]},
 {Sqrt[5/32+11/(32Sqrt[5])],(-3(1+Sqrt[5]))/8,Sqrt[1/8+1/(8Sqrt[5])]},
 {Root[1-20#1^2+80#1^4&,1,0],(3+Sqrt[5])/4,Sqrt[1/8-1/(8Sqrt[5])]},
 {-Sqrt[1+2/Sqrt[5]],0,Sqrt[1/8-1/(8Sqrt[5])]},{-Sqrt[1+2/Sqrt[5]],0,Sqrt[1/8-1/(8Sqrt[5])]},
 {Root[1-20#1^2+80#1^4&,1,0],(-3-Sqrt[5])/4,Sqrt[1/8-1/(8Sqrt[5])]},
 {Sqrt[5/16+1/(8Sqrt[5])],(2+Sqrt[5])/4,Sqrt[1/4+1/(2Sqrt[5])]},
 {Sqrt[5/16+1/(8Sqrt[5])],(-2-Sqrt[5])/4,Sqrt[1/4+1/(2Sqrt[5])]},
 {Sqrt[13/16+11/(8Sqrt[5])],1/4,Sqrt[1/4+1/(2Sqrt[5])]},{Sqrt[13/16+11/(8Sqrt[5])],-1/4,
  Sqrt[1/4+1/(2Sqrt[5])]},{Root[1-20#1^2+80#1^4&,2,0],(-1-Sqrt[5])/4,Sqrt[5/8+11/(8Sqrt[5])]},
 {Root[1-20#1^2+80#1^4&,2,0],(1+Sqrt[5])/4,Sqrt[5/8+11/(8Sqrt[5])]},
 {Sqrt[5/8+11/(8Sqrt[5])],0,Sqrt[(5+Sqrt[5])/10]},{Sqrt[1/4+1/(2Sqrt[5])],1/2,Sqrt[5/8+11/(8Sqrt[5])]},
 {Root[1-20#1^2+80#1^4&,2,0],(1+Sqrt[5])/4,Sqrt[5/8+11/(8Sqrt[5])]},
 {Root[1-20#1^2+80#1^4&,2,0],(-1-Sqrt[5])/4,Sqrt[5/8+11/(8Sqrt[5])]},
 {Sqrt[1/4+1/(2Sqrt[5])],-1/2,Sqrt[5/8+11/(8Sqrt[5])]},{Root[1-1040#1^2+1280#1^4&,1,0],
  (-3-Sqrt[5])/8,Sqrt[(5+Sqrt[5])/10]},{Root[1-1040#1^2+1280#1^4&,1,0],(3+Sqrt[5])/8,
  Sqrt[(5+Sqrt[5])/10]},{Sqrt[5/8+11/(8Sqrt[5])],-1/2,-Sqrt[1+2/Sqrt[5]]/2},
 {Sqrt[5/8+11/(8Sqrt[5])],1/2,-Sqrt[1+2/Sqrt[5]]/2},{Root[1-100#1^2+80#1^4&,1,0],(-1-Sqrt[5])/4,
  Root[1-20#1^2+80#1^4&,2,0]},{Root[1-100#1^2+80#1^4&,1,0],(1+Sqrt[5])/4,
  Root[1-20#1^2+80#1^4&,2,0]},{-Sqrt[13+22/Sqrt[5]]/4,1/4,-Sqrt[1+2/Sqrt[5]]/2},
 {-Sqrt[13+22/Sqrt[5]]/4,-1/4,-Sqrt[1+2/Sqrt[5]]/2},{-Sqrt[5+2/Sqrt[5]]/4,(2+Sqrt[5])/4,
  -Sqrt[1+2/Sqrt[5]]/2},{-Sqrt[5+2/Sqrt[5]]/4,(-2-Sqrt[5])/4,-Sqrt[1+2/Sqrt[5]]/2},
 {Sqrt[1+2/Sqrt[5]],0,Root[1-20#1^2+80#1^4&,2,0]},{Sqrt[1/8+1/(8Sqrt[5])],(3+Sqrt[5])/4,
  Root[1-20#1^2+80#1^4&,2,0]},{Sqrt[1/8+1/(8Sqrt[5])],(-3-Sqrt[5])/4,
  Root[1-20#1^2+80#1^4&,2,0]},{Sqrt[1+2/Sqrt[5]],0,Root[1-20#1^2+80#1^4&,2,0]},
 {Root[1-80#1^2+1280#1^4&,2,0],(3(1+Sqrt[5]))/8,-Sqrt[1+2/Sqrt[5]]/2},
 {Root[1-80#1^2+1280#1^4&,2,0],(-3(1+Sqrt[5]))/8,-Sqrt[1+2/Sqrt[5]]/2},
 {Sqrt[13/32+19/(32Sqrt[5])],(5+Sqrt[5])/8,-Sqrt[1+2/Sqrt[5]]/2},
 {Sqrt[13/32+19/(32Sqrt[5])],(-5-Sqrt[5])/8,-Sqrt[1+2/Sqrt[5]]/2},
 {Sqrt[1/8+1/(8Sqrt[5])],(-3-Sqrt[5])/4,Root[1-20#1^2+80#1^4&,2,0]},
 {Root[1-100#1^2+80#1^4&,1,0],(1+Sqrt[5])/4,Root[1-20#1^2+80#1^4&,2,0]},
 {Root[1-100#1^2+80#1^4&,1,0],(-1-Sqrt[5])/4,Root[1-20#1^2+80#1^4&,2,0]},
 {Sqrt[1/8+1/(8Sqrt[5])],(3+Sqrt[5])/4,Root[1-20#1^2+80#1^4&,2,0]},
 {Sqrt[1/8-1/(8Sqrt[5])],(-1-Sqrt[5])/4,Root[1-100#1^2+80#1^4&,1,0]},
 {Sqrt[1/8-1/(8Sqrt[5])],(1+Sqrt[5])/4,Root[1-100#1^2+80#1^4&,1,0]},
 {Root[1-100#1^2+80#1^4&,1,0],0,Root[1-5#1^2+5#1^4&,1,0]},
 {Sqrt[1/8-1/(8Sqrt[5])],(1+Sqrt[5])/4,Root[1-100#1^2+80#1^4&,1,0]},
 {-Sqrt[1+2/Sqrt[5]]/2,1/2,Root[1-100#1^2+80#1^4&,1,0]},
 {-Sqrt[1+2/Sqrt[5]]/2,-1/2,Root[1-100#1^2+80#1^4&,1,0]},
 {Sqrt[1/8-1/(8Sqrt[5])],(-1-Sqrt[5])/4,Root[1-100#1^2+80#1^4&,1,0]},
 {Sqrt[13/32+29/(32Sqrt[5])],(-3-Sqrt[5])/8,Root[1-5#1^2+5#1^4&,1,0]},
 {Sqrt[13/32+29/(32Sqrt[5])],(3+Sqrt[5])/8,Root[1-5#1^2+5#1^4&,1,0]},
 {Sqrt[(5+Sqrt[5])/10],0,Root[1-100#1^2+80#1^4&,1,0]},
 {-Sqrt[1+2/Sqrt[5]]/2,-1/2,Root[1-100#1^2+80#1^4&,1,0]},
 {-Sqrt[1+2/Sqrt[5]]/2,1/2,Root[1-100#1^2+80#1^4&,1,0]},
 {Sqrt[(5+Sqrt[5])/10],0,Root[1-100#1^2+80#1^4&,1,0]},{-Sqrt[1+2/Sqrt[5]]/4,(-2-Sqrt[5])/4,
  Root[1-5#1^2+5#1^4&,1,0]},{-Sqrt[1+2/Sqrt[5]]/4,(2+Sqrt[5])/4,Root[1-5#1^2+5#1^4&,1,0]},
 {Sqrt[17/32+31/(32Sqrt[5])],(5+Sqrt[5])/8,Root[1-20#1^2+80#1^4&,1,0]},
 {Root[1-400#1^2+1280#1^4&,1,0],(3(1+Sqrt[5]))/8,Root[1-20#1^2+80#1^4&,1,0]},
 {Root[1-400#1^2+1280#1^4&,1,0],(-3(1+Sqrt[5]))/8,Root[1-20#1^2+80#1^4&,1,0]},
 {Sqrt[17/32+31/(32Sqrt[5])],(-5-Sqrt[5])/8,Root[1-20#1^2+80#1^4&,1,0]},
 {Sqrt[25/32+41/(32Sqrt[5])],(-3-Sqrt[5])/8,Root[1-20#1^2+80#1^4&,1,0]},
 {-Sqrt[29/2+61/(2Sqrt[5])]/4,(1-Sqrt[5])/8,Root[1-20#1^2+80#1^4&,1,0]},
 {-Sqrt[29/2+61/(2Sqrt[5])]/4,(-1+Sqrt[5])/8,Root[1-20#1^2+80#1^4&,1,0]},
 {Sqrt[25/32+41/(32Sqrt[5])],(3+Sqrt[5])/8,Root[1-20#1^2+80#1^4&,1,0]},
 {Root[1-20#1^2+80#1^4&,2,0],(-3-Sqrt[5])/4,Root[1-20#1^2+80#1^4&,1,0]},
 {Root[1-20#1^2+80#1^4&,2,0],(3+Sqrt[5])/4,Root[1-20#1^2+80#1^4&,1,0]}}

(* Dodecahedron 6-Compound *)

PolyhedronName[Dodecahedron6Compound]:="compound of 6 dodecahedra"
PolyhedronDual[Dodecahedron6Compound]:=Icosahedron6Compound
NetPieces[Dodecahedron6Compound]:={{60,{1,2}},{60,{3,4,5}},{60,{6,7,8}}}
NetFaces[Dodecahedron6Compound]:=
{{7,8,10},{1,7,10,9},{2,18,16,12,3},{2,17,15,11,3},{17,14,2},{4,20,5},{4,21,
    5},{20,13,6,19,5}}
NetVertices[Dodecahedron6Compound]:=
{{0,0},{1,1/4},{1,(3-2Sqrt[5])/4},{2,-1/4},{2,(5-2Sqrt[5])/4},{2,(4-Sqrt[5])/4},
 {(3(5-3Sqrt[5]))/40,Sqrt[(5-Sqrt[5])/10]/4},
 {(3-Sqrt[5])/8,(3Sqrt[(1465-149Sqrt[5])/10])/116},{(3-Sqrt[5])/4,0},
 {(15-Sqrt[5])/40,Sqrt[(5-Sqrt[5])/10]/4},{(76-Sqrt[2(125-41Sqrt[5])])/76,
  (-16+13Sqrt[5])/76},{(76+Sqrt[2(125-41Sqrt[5])])/76,(-16+13Sqrt[5])/76},
 {(8-Sqrt[5-2Sqrt[5]])/4,(3-Sqrt[5])/2},{(10+Sqrt[5(5-2Sqrt[5])])/10,
  (5+2Sqrt[5])/20},{(8-Sqrt[2(5-Sqrt[5])])/8,(-1+Sqrt[5])/8},
 {(8+Sqrt[2(5-Sqrt[5])])/8,(-1+Sqrt[5])/8},{(20-Sqrt[10(5-Sqrt[5])])/20,
  3/(4Sqrt[5])},{(20+Sqrt[10(5-Sqrt[5])])/20,3/(4Sqrt[5])},
 {(152+Sqrt[2(85+Sqrt[5])])/76,(3(34-11Sqrt[5]))/76},
 {(76-Sqrt[65+22Sqrt[5]])/38,(93-34Sqrt[5])/76},{(76+Sqrt[65+22Sqrt[5]])/38,
  (93-34Sqrt[5])/76}}
PolyhedronPolyhedra[Dodecahedron6Compound]:=
{{1,2,3,4,5,6,7,8,9,10,11,12},{13,14,15,16,17,18,19,20,21,22,23,24},{25,26,27,
    28,29,30,31,32,33,34,35,36},{37,38,39,40,41,42,43,44,45,46,47,48},{49,50,
    51,52,53,54,55,56,57,58,59,60},{61,62,63,64,65,66,67,68,69,70,71,72}}
PolyhedronFaces[Dodecahedron6Compound]:=
{{67,68,6,69,5},{66,8,70,7,65},{41,39,95,65,7},{97,41,7,70,120},{120,70,8,42,
    98},{96,40,42,8,66},{96,66,65,95,119},{40,6,68,98,42},{96,119,69,6,
    40},{119,95,39,5,69},{97,67,5,39,41},{97,120,98,68,67},{110,50,26,21,
    89},{107,88,19,27,44},{27,85,79,9,44},{27,19,38,102,85},{19,88,78,12,
    38},{107,99,84,78,88},{107,44,9,35,99},{26,50,12,78,84},{21,26,84,99,
    35},{21,35,9,79,89},{110,89,79,85,102},{110,102,38,12,50},{63,2,77,51,
    34},{58,33,52,80,3},{80,117,62,1,3},{80,52,81,118,117},{52,33,59,4,
    81},{58,115,116,59,33},{58,3,1,76,115},{77,2,4,59,116},{51,77,116,115,
    76},{51,76,1,62,34},{63,34,62,117,118},{63,118,81,4,2},{23,36,100,83,
    25},{17,28,86,101,37},{101,109,48,11,37},{101,86,82,90,109},{86,28,46,10,
    82},{17,87,108,46,28},{17,37,11,75,87},{100,36,10,46,108},{83,100,108,87,
    75},{83,75,11,48,25},{23,25,48,109,90},{23,90,82,10,36},{47,30,91,57,
    15},{45,14,64,94,31},{94,113,55,22,31},{94,64,74,105,113},{64,14,54,20,
    74},{45,104,112,54,14},{45,31,22,71,104},{91,30,20,54,112},{57,91,112,104,
    71},{57,71,22,55,15},{47,15,55,113,105},{47,105,74,20,30},{106,49,29,18,
    73},{103,72,24,32,43},{32,93,61,13,43},{32,24,56,114,93},{24,72,60,16,
    56},{103,111,92,60,72},{103,43,13,53,111},{29,49,16,60,92},{18,29,92,111,
    53},{18,53,13,61,73},{106,73,61,93,114},{106,114,56,16,49}}
PolyhedronVertices[Dodecahedron6Compound]:=
{{-Sqrt[1/8-11/(40Sqrt[5])],(-3-Sqrt[5])/4,Sqrt[1/8+11/(40Sqrt[5])]},
 {-Sqrt[1/8-11/(40Sqrt[5])],(3+Sqrt[5])/4,Sqrt[1/8+11/(40Sqrt[5])]},
 {Sqrt[1/8-11/(40Sqrt[5])],(-3-Sqrt[5])/4,-Sqrt[1/8+11/(40Sqrt[5])]},
 {Sqrt[1/8-11/(40Sqrt[5])],(3+Sqrt[5])/4,-Sqrt[1/8+11/(40Sqrt[5])]},
 {-Sqrt[1/8-1/(8Sqrt[5])],(-1-Sqrt[5])/4,Sqrt[5/8+11/(8Sqrt[5])]},
 {-Sqrt[1/8-1/(8Sqrt[5])],(1+Sqrt[5])/4,Sqrt[5/8+11/(8Sqrt[5])]},
 {Sqrt[1/8-1/(8Sqrt[5])],(-1-Sqrt[5])/4,-Sqrt[5/8+11/(8Sqrt[5])]},
 {Sqrt[1/8-1/(8Sqrt[5])],(1+Sqrt[5])/4,-Sqrt[5/8+11/(8Sqrt[5])]},
 {-Sqrt[1/20-1/(10Sqrt[5])],(-5-4Sqrt[5])/10,Sqrt[1/40-1/(40Sqrt[5])]},
 {-Sqrt[1/20-1/(10Sqrt[5])],(5+4Sqrt[5])/10,Sqrt[1/40-1/(40Sqrt[5])]},
 {Sqrt[1/20-1/(10Sqrt[5])],(-5-4Sqrt[5])/10,-Sqrt[1/40-1/(40Sqrt[5])]},
 {Sqrt[1/20-1/(10Sqrt[5])],(5+4Sqrt[5])/10,-Sqrt[1/40-1/(40Sqrt[5])]},
 {-Sqrt[1/10-1/(10Sqrt[5])],(-5-3Sqrt[5])/10,-Sqrt[13/40+19/(40Sqrt[5])]},
 {-Sqrt[1/10-1/(10Sqrt[5])],(5+3Sqrt[5])/10,-Sqrt[13/40+19/(40Sqrt[5])]},
 {Sqrt[1/10-1/(10Sqrt[5])],(-5-3Sqrt[5])/10,Sqrt[13/40+19/(40Sqrt[5])]},
 {Sqrt[1/10-1/(10Sqrt[5])],(5+3Sqrt[5])/10,Sqrt[13/40+19/(40Sqrt[5])]},
 {-Sqrt[1/40-1/(40Sqrt[5])],(-5-Sqrt[5])/20,-Sqrt[41/40+71/(40Sqrt[5])]},
 {-Sqrt[1/40-1/(40Sqrt[5])],(3(-5-Sqrt[5]))/20,Sqrt[17/40+31/(40Sqrt[5])]},
 {-Sqrt[1/40-1/(40Sqrt[5])],(5+Sqrt[5])/20,-Sqrt[41/40+71/(40Sqrt[5])]},
 {-Sqrt[1/40-1/(40Sqrt[5])],(3(5+Sqrt[5]))/20,Sqrt[17/40+31/(40Sqrt[5])]},
 {Sqrt[1/40-1/(40Sqrt[5])],(-5-Sqrt[5])/20,Sqrt[41/40+71/(40Sqrt[5])]},
 {Sqrt[1/40-1/(40Sqrt[5])],(3(-5-Sqrt[5]))/20,-Sqrt[17/40+31/(40Sqrt[5])]},
 {Sqrt[1/40-1/(40Sqrt[5])],(5+Sqrt[5])/20,Sqrt[41/40+71/(40Sqrt[5])]},
 {Sqrt[1/40-1/(40Sqrt[5])],(3(5+Sqrt[5]))/20,-Sqrt[17/40+31/(40Sqrt[5])]},
 {-Sqrt[1/40+1/(40Sqrt[5])],(-5-3Sqrt[5])/20,Sqrt[37/40+59/(40Sqrt[5])]},
 {-Sqrt[1/40+1/(40Sqrt[5])],(5+3Sqrt[5])/20,Sqrt[37/40+59/(40Sqrt[5])]},
 {Sqrt[1/40+1/(40Sqrt[5])],(-5-3Sqrt[5])/20,-Sqrt[37/40+59/(40Sqrt[5])]},
 {Sqrt[1/40+1/(40Sqrt[5])],(5+3Sqrt[5])/20,-Sqrt[37/40+59/(40Sqrt[5])]},
 {-Sqrt[1/20+1/(10Sqrt[5])],-1/(2Sqrt[5]),Sqrt[41/40+71/(40Sqrt[5])]},
 {-Sqrt[1/20+1/(10Sqrt[5])],1/(2Sqrt[5]),Sqrt[41/40+71/(40Sqrt[5])]},
 {Sqrt[1/20+1/(10Sqrt[5])],-1/(2Sqrt[5]),-Sqrt[41/40+71/(40Sqrt[5])]},
 {Sqrt[1/20+1/(10Sqrt[5])],1/(2Sqrt[5]),-Sqrt[41/40+71/(40Sqrt[5])]},
 {-Sqrt[1/10+1/(10Sqrt[5])],0,-Sqrt[41/40+71/(40Sqrt[5])]},
 {Sqrt[1/10+1/(10Sqrt[5])],0,Sqrt[41/40+71/(40Sqrt[5])]},
 {-Sqrt[1/4+1/(10Sqrt[5])],(-5-2Sqrt[5])/10,Sqrt[17/40+31/(40Sqrt[5])]},
 {-Sqrt[1/4+1/(10Sqrt[5])],(5+2Sqrt[5])/10,Sqrt[17/40+31/(40Sqrt[5])]},
 {Sqrt[1/4+1/(10Sqrt[5])],(-5-2Sqrt[5])/10,-Sqrt[17/40+31/(40Sqrt[5])]},
 {Sqrt[1/4+1/(10Sqrt[5])],(5+2Sqrt[5])/10,-Sqrt[17/40+31/(40Sqrt[5])]},
 {-Sqrt[1/8+1/(8Sqrt[5])],(-3-Sqrt[5])/4,Sqrt[1/8-1/(8Sqrt[5])]},
 {-Sqrt[1/8+1/(8Sqrt[5])],(3+Sqrt[5])/4,Sqrt[1/8-1/(8Sqrt[5])]},
 {Sqrt[1/8+1/(8Sqrt[5])],(-3-Sqrt[5])/4,-Sqrt[1/8-1/(8Sqrt[5])]},
 {Sqrt[1/8+1/(8Sqrt[5])],(3+Sqrt[5])/4,-Sqrt[1/8-1/(8Sqrt[5])]},
 {-Sqrt[1/8+11/(40Sqrt[5])],(-5-Sqrt[5])/20,-Sqrt[37/40+59/(40Sqrt[5])]},
 {-Sqrt[1/8+11/(40Sqrt[5])],(3(-5-Sqrt[5]))/20,-Sqrt[13/40+19/(40Sqrt[5])]},
 {-Sqrt[1/8+11/(40Sqrt[5])],(5+Sqrt[5])/20,-Sqrt[37/40+59/(40Sqrt[5])]},
 {-Sqrt[1/8+11/(40Sqrt[5])],(3(5+Sqrt[5]))/20,-Sqrt[13/40+19/(40Sqrt[5])]},
 {Sqrt[1/8+11/(40Sqrt[5])],(-5-Sqrt[5])/20,Sqrt[37/40+59/(40Sqrt[5])]},
 {Sqrt[1/8+11/(40Sqrt[5])],(3(-5-Sqrt[5]))/20,Sqrt[13/40+19/(40Sqrt[5])]},
 {Sqrt[1/8+11/(40Sqrt[5])],(5+Sqrt[5])/20,Sqrt[37/40+59/(40Sqrt[5])]},
 {Sqrt[1/8+11/(40Sqrt[5])],(3(5+Sqrt[5]))/20,Sqrt[13/40+19/(40Sqrt[5])]},
 {-Sqrt[1/5+2/(5Sqrt[5])],0,Sqrt[37/40+59/(40Sqrt[5])]},
 {Sqrt[1/5+2/(5Sqrt[5])],0,-Sqrt[37/40+59/(40Sqrt[5])]},
 {-Sqrt[2/5+2/(5Sqrt[5])],(-5-3Sqrt[5])/10,Sqrt[1/40-1/(40Sqrt[5])]},
 {-Sqrt[2/5+2/(5Sqrt[5])],(5+3Sqrt[5])/10,Sqrt[1/40-1/(40Sqrt[5])]},
 {Sqrt[2/5+2/(5Sqrt[5])],(-5-3Sqrt[5])/10,-Sqrt[1/40-1/(40Sqrt[5])]},
 {Sqrt[2/5+2/(5Sqrt[5])],(5+3Sqrt[5])/10,-Sqrt[1/40-1/(40Sqrt[5])]},
 {-Sqrt[13/40+19/(40Sqrt[5])],(3(-5-Sqrt[5]))/20,Sqrt[1/8+11/(40Sqrt[5])]},
 {-Sqrt[13/40+19/(40Sqrt[5])],(-1-Sqrt[5])/4,-Sqrt[17/40+31/(40Sqrt[5])]},
 {-Sqrt[13/40+19/(40Sqrt[5])],(1+Sqrt[5])/4,-Sqrt[17/40+31/(40Sqrt[5])]},
 {-Sqrt[13/40+19/(40Sqrt[5])],(3(5+Sqrt[5]))/20,Sqrt[1/8+11/(40Sqrt[5])]},
 {Sqrt[13/40+19/(40Sqrt[5])],(3(-5-Sqrt[5]))/20,-Sqrt[1/8+11/(40Sqrt[5])]},
 {Sqrt[13/40+19/(40Sqrt[5])],(-1-Sqrt[5])/4,Sqrt[17/40+31/(40Sqrt[5])]},
 {Sqrt[13/40+19/(40Sqrt[5])],(1+Sqrt[5])/4,Sqrt[17/40+31/(40Sqrt[5])]},
 {Sqrt[13/40+19/(40Sqrt[5])],(3(5+Sqrt[5]))/20,-Sqrt[1/8+11/(40Sqrt[5])]},
 {-Sqrt[1/4+1/(2Sqrt[5])],-1/2,-Sqrt[5/8+11/(8Sqrt[5])]},
 {-Sqrt[1/4+1/(2Sqrt[5])],1/2,-Sqrt[5/8+11/(8Sqrt[5])]},
 {Sqrt[1/4+1/(2Sqrt[5])],-1/2,Sqrt[5/8+11/(8Sqrt[5])]},
 {Sqrt[1/4+1/(2Sqrt[5])],1/2,Sqrt[5/8+11/(8Sqrt[5])]},
 {-Sqrt[1/2+1/(2Sqrt[5])],0,Sqrt[5/8+11/(8Sqrt[5])]},
 {Sqrt[1/2+1/(2Sqrt[5])],0,-Sqrt[5/8+11/(8Sqrt[5])]},
 {-Sqrt[13/40+29/(40Sqrt[5])],(-5-7Sqrt[5])/20,-Sqrt[1/8+11/(40Sqrt[5])]},
 {-Sqrt[13/40+29/(40Sqrt[5])],(5+7Sqrt[5])/20,-Sqrt[1/8+11/(40Sqrt[5])]},
 {Sqrt[13/40+29/(40Sqrt[5])],(-5-7Sqrt[5])/20,Sqrt[1/8+11/(40Sqrt[5])]},
 {Sqrt[13/40+29/(40Sqrt[5])],(5+7Sqrt[5])/20,Sqrt[1/8+11/(40Sqrt[5])]},
 {-Sqrt[17/40+31/(40Sqrt[5])],(3(-5-Sqrt[5]))/20,-Sqrt[1/40-1/(40Sqrt[5])]},
 {-Sqrt[17/40+31/(40Sqrt[5])],(-1-Sqrt[5])/4,Sqrt[13/40+19/(40Sqrt[5])]},
 {-Sqrt[17/40+31/(40Sqrt[5])],(1+Sqrt[5])/4,Sqrt[13/40+19/(40Sqrt[5])]},
 {-Sqrt[17/40+31/(40Sqrt[5])],(3(5+Sqrt[5]))/20,-Sqrt[1/40-1/(40Sqrt[5])]},
 {Sqrt[17/40+31/(40Sqrt[5])],(3(-5-Sqrt[5]))/20,Sqrt[1/40-1/(40Sqrt[5])]},
 {Sqrt[17/40+31/(40Sqrt[5])],(-1-Sqrt[5])/4,-Sqrt[13/40+19/(40Sqrt[5])]},
 {Sqrt[17/40+31/(40Sqrt[5])],(1+Sqrt[5])/4,-Sqrt[13/40+19/(40Sqrt[5])]},
 {Sqrt[17/40+31/(40Sqrt[5])],(3(5+Sqrt[5]))/20,Sqrt[1/40-1/(40Sqrt[5])]},
 {-Sqrt[5/8+41/(40Sqrt[5])],(-5-3Sqrt[5])/20,Sqrt[13/40+19/(40Sqrt[5])]},
 {-Sqrt[5/8+41/(40Sqrt[5])],(5+3Sqrt[5])/20,Sqrt[13/40+19/(40Sqrt[5])]},
 {Sqrt[5/8+41/(40Sqrt[5])],(-5-3Sqrt[5])/20,-Sqrt[13/40+19/(40Sqrt[5])]},
 {Sqrt[5/8+41/(40Sqrt[5])],(5+3Sqrt[5])/20,-Sqrt[13/40+19/(40Sqrt[5])]},
 {-Sqrt[1/2+11/(10Sqrt[5])],-(1/Sqrt[5]),-Sqrt[17/40+31/(40Sqrt[5])]},
 {-Sqrt[1/2+11/(10Sqrt[5])],1/Sqrt[5],-Sqrt[17/40+31/(40Sqrt[5])]},
 {Sqrt[1/2+11/(10Sqrt[5])],-(1/Sqrt[5]),Sqrt[17/40+31/(40Sqrt[5])]},
 {Sqrt[1/2+11/(10Sqrt[5])],1/Sqrt[5],Sqrt[17/40+31/(40Sqrt[5])]},
 {-Sqrt[13/20+11/(10Sqrt[5])],-1/(2Sqrt[5]),Sqrt[17/40+31/(40Sqrt[5])]},
 {-Sqrt[13/20+11/(10Sqrt[5])],1/(2Sqrt[5]),Sqrt[17/40+31/(40Sqrt[5])]},
 {Sqrt[13/20+11/(10Sqrt[5])],-1/(2Sqrt[5]),-Sqrt[17/40+31/(40Sqrt[5])]},
 {Sqrt[13/20+11/(10Sqrt[5])],1/(2Sqrt[5]),-Sqrt[17/40+31/(40Sqrt[5])]},
 {-Sqrt[5/8+11/(8Sqrt[5])],(-1-Sqrt[5])/4,-Sqrt[1/8-1/(8Sqrt[5])]},
 {-Sqrt[5/8+11/(8Sqrt[5])],(1+Sqrt[5])/4,-Sqrt[1/8-1/(8Sqrt[5])]},
 {Sqrt[5/8+11/(8Sqrt[5])],(-1-Sqrt[5])/4,Sqrt[1/8-1/(8Sqrt[5])]},
 {Sqrt[5/8+11/(8Sqrt[5])],(1+Sqrt[5])/4,Sqrt[1/8-1/(8Sqrt[5])]},
 {-Sqrt[37/40+59/(40Sqrt[5])],(-5-Sqrt[5])/20,Sqrt[1/8+11/(40Sqrt[5])]},
 {-Sqrt[37/40+59/(40Sqrt[5])],(5+Sqrt[5])/20,Sqrt[1/8+11/(40Sqrt[5])]},
 {Sqrt[37/40+59/(40Sqrt[5])],(-5-Sqrt[5])/20,-Sqrt[1/8+11/(40Sqrt[5])]},
 {Sqrt[37/40+59/(40Sqrt[5])],(5+Sqrt[5])/20,-Sqrt[1/8+11/(40Sqrt[5])]},
 {-Sqrt[29/40+61/(40Sqrt[5])],(5-Sqrt[5])/20,-Sqrt[13/40+19/(40Sqrt[5])]},
 {-Sqrt[29/40+61/(40Sqrt[5])],(-5+Sqrt[5])/20,-Sqrt[13/40+19/(40Sqrt[5])]},
 {Sqrt[29/40+61/(40Sqrt[5])],(5-Sqrt[5])/20,Sqrt[13/40+19/(40Sqrt[5])]},
 {Sqrt[29/40+61/(40Sqrt[5])],(-5+Sqrt[5])/20,Sqrt[13/40+19/(40Sqrt[5])]},
 {-Sqrt[4/5+8/(5Sqrt[5])],-(1/Sqrt[5]),-Sqrt[1/8+11/(40Sqrt[5])]},
 {-Sqrt[4/5+8/(5Sqrt[5])],1/Sqrt[5],-Sqrt[1/8+11/(40Sqrt[5])]},
 {Sqrt[4/5+8/(5Sqrt[5])],-(1/Sqrt[5]),Sqrt[1/8+11/(40Sqrt[5])]},
 {Sqrt[4/5+8/(5Sqrt[5])],1/Sqrt[5],Sqrt[1/8+11/(40Sqrt[5])]},
 {-Sqrt[41/40+71/(40Sqrt[5])],(-5-Sqrt[5])/20,Sqrt[1/40-1/(40Sqrt[5])]},
 {-Sqrt[41/40+71/(40Sqrt[5])],(5+Sqrt[5])/20,Sqrt[1/40-1/(40Sqrt[5])]},
 {Sqrt[41/40+71/(40Sqrt[5])],(-5-Sqrt[5])/20,-Sqrt[1/40-1/(40Sqrt[5])]},
 {Sqrt[41/40+71/(40Sqrt[5])],(5+Sqrt[5])/20,-Sqrt[1/40-1/(40Sqrt[5])]},
 {-Sqrt[17/20+19/(10Sqrt[5])],-1/2,-Sqrt[1/40-1/(40Sqrt[5])]},
 {-Sqrt[17/20+19/(10Sqrt[5])],1/2,-Sqrt[1/40-1/(40Sqrt[5])]},
 {Sqrt[17/20+19/(10Sqrt[5])],-1/2,Sqrt[1/40-1/(40Sqrt[5])]},
 {Sqrt[17/20+19/(10Sqrt[5])],1/2,Sqrt[1/40-1/(40Sqrt[5])]},
 {-Sqrt[(5+2Sqrt[5])/5],0,Sqrt[1/8-1/(8Sqrt[5])]},
 {Sqrt[(5+2Sqrt[5])/5],0,-Sqrt[1/8-1/(8Sqrt[5])]}}

(* Dodecahedron-Icosahedron Compound *)

PolyhedronName[DodecahedronIcosahedronCompound]:="dodecahedron-icosahedron compound"
NetPieces[DodecahedronIcosahedronCompound]:={{0,{1,2,3,4,5}},{0,{6,7,8}}}
NetFaces[DodecahedronIcosahedronCompound]:=
{{1,10,11},{1,11,9},{1,9,3},{1,5,6},{1,6,4},{13,2,12},{7,2,13},{8,2,7}}
NetVertices[DodecahedronIcosahedronCompound]:=
{{0,0},{3/2,-1/4},{(-1-Sqrt[5])/8,-(Sqrt[3](1+Sqrt[5]))/8},
 {(-1-Sqrt[5])/8,(Sqrt[3](1+Sqrt[5]))/8},{(-1-Sqrt[5])/8,-Sqrt[(3(3+Sqrt[5]))/2]/4},
 {(-1-Sqrt[5])/4,0},{(11-Sqrt[5])/8,(-2-Sqrt[10-2Sqrt[5]])/8},
 {(13-Sqrt[5])/8,(-2+Sqrt[2(5+Sqrt[5])])/8},{(1+Sqrt[5])/8,-(Sqrt[3](1+Sqrt[5]))/8},
 {(1+Sqrt[5])/8,Sqrt[(3(3+Sqrt[5]))/2]/4},{(1+Sqrt[5])/4,0},
 {(11+Sqrt[5])/8,(-2+Sqrt[2(5+Sqrt[5])])/8},{(13+Sqrt[5])/8,(-2-Sqrt[10-2Sqrt[5]])/8}}

PolyhedronPolyhedra[DodecahedronIcosahedronCompound]:={Range[12],Range[13,32]}
PolyhedronFaces[DodecahedronIcosahedronCompound]:=
{{7,26,25,5,2},{1,11,28,27,9},{9,27,23,3,31},{27,28,24,19,23},{28,11,32,4,
    24},{11,1,17,22,32},{1,9,31,21,17},{4,32,22,26,7},{22,17,21,25,26},{21,31,
    3,5,25},{3,23,19,2,5},{19,24,4,7,2},{18,10,12},{6,8,20},{10,13,15},{16,14,
    12},{20,15,6},{8,16,20},{15,30,10},{30,16,12},{18,29,13},{18,14,29},{30,
    15,20},{16,30,20},{8,6,29},{18,13,10},{12,14,18},{12,10,30},{13,29,6},{29,
    14,8},{13,6,15},{16,8,14}}
PolyhedronVertices[DodecahedronIcosahedronCompound]:=
{{0,0,-Sqrt[9/8+(3Sqrt[5])/8]},{0,0,Sqrt[9/8+(3Sqrt[5])/8]},
 {Sqrt[1/8-Sqrt[5]/24],(-3-Sqrt[5])/4,Sqrt[1/8+Sqrt[5]/24]},
 {Sqrt[1/8-Sqrt[5]/24],(3+Sqrt[5])/4,Sqrt[1/8+Sqrt[5]/24]},
 {-Sqrt[1/8+Sqrt[5]/24],(-1-Sqrt[5])/4,Sqrt[5/8+(5Sqrt[5])/24]},
 {-Sqrt[1/8+Sqrt[5]/24],(-1-Sqrt[5])/4,-Sqrt[3/4+Sqrt[5]/3]},
 {-Sqrt[1/8+Sqrt[5]/24],(1+Sqrt[5])/4,Sqrt[5/8+(5Sqrt[5])/24]},
 {-Sqrt[1/8+Sqrt[5]/24],(1+Sqrt[5])/4,-Sqrt[3/4+Sqrt[5]/3]},
 {Sqrt[1/8+Sqrt[5]/24],(-1-Sqrt[5])/4,-Sqrt[5/8+(5Sqrt[5])/24]},
 {Sqrt[1/8+Sqrt[5]/24],(-1-Sqrt[5])/4,Sqrt[3/4+Sqrt[5]/3]},
 {Sqrt[1/8+Sqrt[5]/24],(1+Sqrt[5])/4,-Sqrt[5/8+(5Sqrt[5])/24]},
 {Sqrt[1/8+Sqrt[5]/24],(1+Sqrt[5])/4,Sqrt[3/4+Sqrt[5]/3]},
 {-Sqrt[7/24+Sqrt[5]/8],(-3-Sqrt[5])/4,1/(2Sqrt[3])},{-Sqrt[7/24+Sqrt[5]/8],(3+Sqrt[5])/4,
  1/(2Sqrt[3])},{Sqrt[7/24+Sqrt[5]/8],(-3-Sqrt[5])/4,-1/(2Sqrt[3])},
 {Sqrt[7/24+Sqrt[5]/8],(3+Sqrt[5])/4,-1/(2Sqrt[3])},{-Sqrt[1/2+Sqrt[5]/6],0,
  -Sqrt[5/8+(5Sqrt[5])/24]},{-Sqrt[1/2+Sqrt[5]/6],0,Sqrt[3/4+Sqrt[5]/3]},
 {Sqrt[1/2+Sqrt[5]/6],0,Sqrt[5/8+(5Sqrt[5])/24]},{Sqrt[1/2+Sqrt[5]/6],0,-Sqrt[3/4+Sqrt[5]/3]},
 {-Sqrt[5/8+(5Sqrt[5])/24],(-1-Sqrt[5])/4,-Sqrt[1/8+Sqrt[5]/24]},
 {-Sqrt[5/8+(5Sqrt[5])/24],(1+Sqrt[5])/4,-Sqrt[1/8+Sqrt[5]/24]},
 {Sqrt[5/8+(5Sqrt[5])/24],(-1-Sqrt[5])/4,Sqrt[1/8+Sqrt[5]/24]},
 {Sqrt[5/8+(5Sqrt[5])/24],(1+Sqrt[5])/4,Sqrt[1/8+Sqrt[5]/24]},
 {-Sqrt[3/4+Sqrt[5]/3],-1/2,Sqrt[1/8+Sqrt[5]/24]},{-Sqrt[3/4+Sqrt[5]/3],1/2,
  Sqrt[1/8+Sqrt[5]/24]},{Sqrt[3/4+Sqrt[5]/3],-1/2,-Sqrt[1/8+Sqrt[5]/24]},
 {Sqrt[3/4+Sqrt[5]/3],1/2,-Sqrt[1/8+Sqrt[5]/24]},{-Sqrt[7/6+Sqrt[5]/2],0,-1/(2Sqrt[3])},
 {Sqrt[7/6+Sqrt[5]/2],0,1/(2Sqrt[3])},{Root[1-36#1^2+144#1^4&,2,0],(-3-Sqrt[5])/4,
  -Sqrt[1/8+Sqrt[5]/24]},{Root[1-36#1^2+144#1^4&,2,0],(3+Sqrt[5])/4,-Sqrt[1/8+Sqrt[5]/24]}}

PolyhedronFaces[DodecahedronIcosahedronCompound,Split->True]:=
{{18,34,12},{55,54,34},{53,33,54},{16,11,33},{2,12,11},{34,54,38},{54,33,
    38},{33,11,38},{11,12,38},{12,34,38},{1,14,13},{23,36,14},{58,57,36},{56,
    35,57},{21,13,35},{14,36,40},{36,57,40},{57,35,40},{35,13,40},{13,14,
    40},{21,35,7},{56,51,35},{43,27,51},{9,3,27},{61,7,3},{35,51,31},{51,27,
    31},{27,3,31},{3,7,31},{7,35,31},{56,57,51},{58,52,57},{44,48,52},{39,47,
    48},{43,51,47},{57,52,60},{52,48,60},{48,47,60},{47,51,60},{51,57,60},{58,
    36,52},{23,8,36},{62,4,8},{10,28,4},{44,52,28},{36,8,32},{8,4,32},{4,28,
    32},{28,52,32},{52,36,32},{23,14,8},{1,15,14},{37,46,15},{42,26,46},{62,8,
    26},{14,15,19},{15,46,19},{46,26,19},{26,8,19},{8,14,19},{1,13,15},{21,7,
    13},{61,25,7},{41,45,25},{37,15,45},{13,7,17},{7,25,17},{25,45,17},{45,15,
    17},{15,13,17},{10,4,6},{62,26,4},{42,50,26},{55,34,50},{18,6,34},{4,26,
    30},{26,50,30},{50,34,30},{34,6,30},{6,4,30},{42,46,50},{37,45,46},{41,49,
    45},{53,54,49},{55,50,54},{46,45,59},{45,49,59},{49,54,59},{54,50,59},{50,
    46,59},{41,25,49},{61,3,25},{9,5,3},{16,33,5},{53,49,33},{25,3,29},{3,5,
    29},{5,33,29},{33,49,29},{49,25,29},{9,27,5},{43,47,27},{39,20,47},{2,11,
    20},{16,5,11},{27,47,22},{47,20,22},{20,11,22},{11,5,22},{5,27,22},{39,48,
    20},{44,28,48},{10,6,28},{18,12,6},{2,20,12},{48,28,24},{28,6,24},{6,12,
    24},{12,20,24},{20,48,24}}
PolyhedronVertices[DodecahedronIcosahedronCompound,Split->True]:=
{{0,0,-Sqrt[9/8+(3Sqrt[5])/8]},{0,0,Sqrt[9/8+(3Sqrt[5])/8]},{0,(-3-Sqrt[5])/4,0},
 {0,(3+Sqrt[5])/4,0},{-1/(4Sqrt[3]),(-2-Sqrt[5])/4,Sqrt[7/24+Sqrt[5]/8]},
 {-1/(4Sqrt[3]),(2+Sqrt[5])/4,Sqrt[7/24+Sqrt[5]/8]},{1/(4Sqrt[3]),(-2-Sqrt[5])/4,
  -Sqrt[7/24+Sqrt[5]/8]},{1/(4Sqrt[3]),(2+Sqrt[5])/4,-Sqrt[7/24+Sqrt[5]/8]},
 {Sqrt[1/8-Sqrt[5]/24],(-3-Sqrt[5])/4,Sqrt[1/8+Sqrt[5]/24]},
 {Sqrt[1/8-Sqrt[5]/24],(3+Sqrt[5])/4,Sqrt[1/8+Sqrt[5]/24]},
 {-Sqrt[1/32+Sqrt[5]/96],(-1-Sqrt[5])/8,Sqrt[3/4+Sqrt[5]/3]},
 {-Sqrt[1/32+Sqrt[5]/96],(1+Sqrt[5])/8,Sqrt[3/4+Sqrt[5]/3]},
 {Sqrt[1/32+Sqrt[5]/96],(-1-Sqrt[5])/8,-Sqrt[3/4+Sqrt[5]/3]},
 {Sqrt[1/32+Sqrt[5]/96],(1+Sqrt[5])/8,-Sqrt[3/4+Sqrt[5]/3]},
 {-Sqrt[1/8+Sqrt[5]/24],0,-Sqrt[3/4+Sqrt[5]/3]},{-Sqrt[1/8+Sqrt[5]/24],(-1-Sqrt[5])/4,
  Sqrt[5/8+(5Sqrt[5])/24]},{-Sqrt[1/8+Sqrt[5]/24],(-1-Sqrt[5])/4,-Sqrt[3/4+Sqrt[5]/3]},
 {-Sqrt[1/8+Sqrt[5]/24],(1+Sqrt[5])/4,Sqrt[5/8+(5Sqrt[5])/24]},
 {-Sqrt[1/8+Sqrt[5]/24],(1+Sqrt[5])/4,-Sqrt[3/4+Sqrt[5]/3]},
 {Sqrt[1/8+Sqrt[5]/24],0,Sqrt[3/4+Sqrt[5]/3]},{Sqrt[1/8+Sqrt[5]/24],(-1-Sqrt[5])/4,
  -Sqrt[5/8+(5Sqrt[5])/24]},{Sqrt[1/8+Sqrt[5]/24],(-1-Sqrt[5])/4,Sqrt[3/4+Sqrt[5]/3]},
 {Sqrt[1/8+Sqrt[5]/24],(1+Sqrt[5])/4,-Sqrt[5/8+(5Sqrt[5])/24]},
 {Sqrt[1/8+Sqrt[5]/24],(1+Sqrt[5])/4,Sqrt[3/4+Sqrt[5]/3]},
 {-Sqrt[3/16+Sqrt[5]/12],(-2-Sqrt[5])/4,-Sqrt[1/8+Sqrt[5]/24]},
 {-Sqrt[3/16+Sqrt[5]/12],(2+Sqrt[5])/4,-Sqrt[1/8+Sqrt[5]/24]},
 {Sqrt[3/16+Sqrt[5]/12],(-2-Sqrt[5])/4,Sqrt[1/8+Sqrt[5]/24]},
 {Sqrt[3/16+Sqrt[5]/12],(2+Sqrt[5])/4,Sqrt[1/8+Sqrt[5]/24]},
 {-Sqrt[7/24+Sqrt[5]/8],(-3-Sqrt[5])/4,1/(2Sqrt[3])},{-Sqrt[7/24+Sqrt[5]/8],(3+Sqrt[5])/4,
  1/(2Sqrt[3])},{Sqrt[7/24+Sqrt[5]/8],(-3-Sqrt[5])/4,-1/(2Sqrt[3])},
 {Sqrt[7/24+Sqrt[5]/8],(3+Sqrt[5])/4,-1/(2Sqrt[3])},{-Sqrt[35/96+(5Sqrt[5])/32],(-3-Sqrt[5])/8,
  Sqrt[7/24+Sqrt[5]/8]},{-Sqrt[35/96+(5Sqrt[5])/32],(3+Sqrt[5])/8,Sqrt[7/24+Sqrt[5]/8]},
 {Sqrt[35/96+(5Sqrt[5])/32],(-3-Sqrt[5])/8,-Sqrt[7/24+Sqrt[5]/8]},
 {Sqrt[35/96+(5Sqrt[5])/32],(3+Sqrt[5])/8,-Sqrt[7/24+Sqrt[5]/8]},
 {-Sqrt[1/2+Sqrt[5]/6],0,-Sqrt[5/8+(5Sqrt[5])/24]},{-Sqrt[1/2+Sqrt[5]/6],0,
  Sqrt[3/4+Sqrt[5]/3]},{Sqrt[1/2+Sqrt[5]/6],0,Sqrt[5/8+(5Sqrt[5])/24]},
 {Sqrt[1/2+Sqrt[5]/6],0,-Sqrt[3/4+Sqrt[5]/3]},{-Sqrt[5/8+(5Sqrt[5])/24],(-1-Sqrt[5])/4,
  -Sqrt[1/8+Sqrt[5]/24]},{-Sqrt[5/8+(5Sqrt[5])/24],(1+Sqrt[5])/4,-Sqrt[1/8+Sqrt[5]/24]},
 {Sqrt[5/8+(5Sqrt[5])/24],(-1-Sqrt[5])/4,Sqrt[1/8+Sqrt[5]/24]},
 {Sqrt[5/8+(5Sqrt[5])/24],(1+Sqrt[5])/4,Sqrt[1/8+Sqrt[5]/24]},
 {-Sqrt[47/96+(7Sqrt[5])/32],(-1-Sqrt[5])/8,-Sqrt[7/24+Sqrt[5]/8]},
 {-Sqrt[47/96+(7Sqrt[5])/32],(1+Sqrt[5])/8,-Sqrt[7/24+Sqrt[5]/8]},
 {Sqrt[47/96+(7Sqrt[5])/32],(-1-Sqrt[5])/8,Sqrt[7/24+Sqrt[5]/8]},
 {Sqrt[47/96+(7Sqrt[5])/32],(1+Sqrt[5])/8,Sqrt[7/24+Sqrt[5]/8]},
 {-Sqrt[21/32+(9Sqrt[5])/32],(-3-Sqrt[5])/8,0},{-Sqrt[21/32+(9Sqrt[5])/32],(3+Sqrt[5])/8,0},
 {Sqrt[21/32+(9Sqrt[5])/32],(-3-Sqrt[5])/8,0},{Sqrt[21/32+(9Sqrt[5])/32],(3+Sqrt[5])/8,0},
 {-Sqrt[3/4+Sqrt[5]/3],-1/2,Sqrt[1/8+Sqrt[5]/24]},{-Sqrt[3/4+Sqrt[5]/3],0,Sqrt[1/8+Sqrt[5]/24]},
 {-Sqrt[3/4+Sqrt[5]/3],1/2,Sqrt[1/8+Sqrt[5]/24]},{Sqrt[3/4+Sqrt[5]/3],-1/2,
  -Sqrt[1/8+Sqrt[5]/24]},{Sqrt[3/4+Sqrt[5]/3],0,-Sqrt[1/8+Sqrt[5]/24]},
 {Sqrt[3/4+Sqrt[5]/3],1/2,-Sqrt[1/8+Sqrt[5]/24]},{-Sqrt[7/6+Sqrt[5]/2],0,-1/(2Sqrt[3])},
 {Sqrt[7/6+Sqrt[5]/2],0,1/(2Sqrt[3])},{Root[1-36#1^2+144#1^4&,2,0],(-3-Sqrt[5])/4,
  -Sqrt[1/8+Sqrt[5]/24]},{Root[1-36#1^2+144#1^4&,2,0],(3+Sqrt[5])/4,-Sqrt[1/8+Sqrt[5]/24]}}

(* Great Dodecahedron *)

PolyhedronName[GreatDodecahedron]:="great dodecahedron"
PolyhedronDual[GreatDodecahedron]:=SmallStellatedDodecahedron

PolyhedronFaces[GreatDodecahedron]:=
{{11,1,3,9,7},{12,8,10,3,1},{4,8,6,1,11},{4,12,1,5,7},{6,10,9,5,1},{8,6,3,9,
    2},{7,2,10,3,5},{4,2,9,5,11},{4,12,6,10,2},{8,2,7,11,12},{11,12,6,3,5},{4,
    8,10,9,7}}
PolyhedronVertices[GreatDodecahedron]:=
{{0,0,-5/Sqrt[50-10Sqrt[5]]},{0,0,5/Sqrt[50-10Sqrt[5]]},
 {-Sqrt[2/(5-Sqrt[5])],0,-(1/Sqrt[10-2Sqrt[5]])},
 {Sqrt[2/(5-Sqrt[5])],0,1/Sqrt[10-2Sqrt[5]]},
 {-(-1+Sqrt[5])/(2Sqrt[10-2Sqrt[5]]),
  -Sqrt[(5+Sqrt[5])/(5-Sqrt[5])]/2,-(1/Sqrt[10-2Sqrt[5]])},
 {-(-1+Sqrt[5])/(2Sqrt[10-2Sqrt[5]]),
  Sqrt[(5+Sqrt[5])/(5-Sqrt[5])]/2,-(1/Sqrt[10-2Sqrt[5]])},
 {(-1+Sqrt[5])/(2Sqrt[10-2Sqrt[5]]),
  -Sqrt[(5+Sqrt[5])/(5-Sqrt[5])]/2,1/Sqrt[10-2Sqrt[5]]},
 {(-1+Sqrt[5])/(2Sqrt[10-2Sqrt[5]]),
  Sqrt[(5+Sqrt[5])/(5-Sqrt[5])]/2,1/Sqrt[10-2Sqrt[5]]},
 {-(1+Sqrt[5])/(2Sqrt[10-2Sqrt[5]]),-1/2,
  1/Sqrt[10-2Sqrt[5]]},{-(1+Sqrt[5])/(2Sqrt[10-2Sqrt[5]]),
  1/2,1/Sqrt[10-2Sqrt[5]]},{(1+Sqrt[5])/(2Sqrt[10-2Sqrt[5]]),
  -1/2,-(1/Sqrt[10-2Sqrt[5]])},
 {(1+Sqrt[5])/(2Sqrt[10-2Sqrt[5]]),1/2,-(1/Sqrt[10-2Sqrt[5]])}}

PolyhedronFaces[GreatDodecahedron,Split->True]:=
{{24,2,8},{24,8,10},{24,10,2},{13,2,10},{13,10,9},{13,9,2},{23,2,9},{23,9,
    7},{23,7,2},{19,2,7},{19,7,4},{19,4,2},{20,2,4},{20,4,8},{20,8,2},{27,11,
    5},{27,5,1},{27,1,11},{14,12,11},{14,11,1},{14,1,12},{28,6,12},{28,12,
    1},{28,1,6},{18,3,6},{18,6,1},{18,1,3},{17,5,3},{17,3,1},{17,1,5},{21,8,
    6},{21,6,10},{21,10,8},{15,10,3},{15,3,9},{15,9,10},{22,9,5},{22,5,7},{22,
    7,9},{26,7,11},{26,11,4},{26,4,7},{25,4,12},{25,12,8},{25,8,4},{32,11,
    7},{32,7,5},{32,5,11},{16,12,4},{16,4,11},{16,11,12},{31,6,8},{31,8,
    12},{31,12,6},{29,3,10},{29,10,6},{29,6,3},{30,5,9},{30,9,3},{30,3,5}}
PolyhedronVertices[GreatDodecahedron,Split->True]:=
{{0,0,-5/Sqrt[50-10Sqrt[5]]},{0,0,5/Sqrt[50-10Sqrt[5]]},{Root[1-5#1^2+5#1^4&,1,0],0,
  -(1/Sqrt[10-2Sqrt[5]])},{Sqrt[(5+Sqrt[5])/10],0,1/Sqrt[10-2Sqrt[5]]},
 {Root[1-20#1^2+80#1^4&,2,0],(-1-Sqrt[5])/4,-(1/Sqrt[10-2Sqrt[5]])},
 {Root[1-20#1^2+80#1^4&,2,0],(1+Sqrt[5])/4,-(1/Sqrt[10-2Sqrt[5]])},
 {Sqrt[1/8-1/(8Sqrt[5])],(-1-Sqrt[5])/4,1/Sqrt[10-2Sqrt[5]]},
 {Sqrt[1/8-1/(8Sqrt[5])],(1+Sqrt[5])/4,1/Sqrt[10-2Sqrt[5]]},
 {-Sqrt[1+2/Sqrt[5]]/2,-1/2,1/Sqrt[10-2Sqrt[5]]},{-Sqrt[1+2/Sqrt[5]]/2,1/2,1/Sqrt[10-2Sqrt[5]]},
 {Sqrt[1/4+1/(2Sqrt[5])],-1/2,-(1/Sqrt[10-2Sqrt[5]])},{Sqrt[1/4+1/(2Sqrt[5])],1/2,
  -(1/Sqrt[10-2Sqrt[5]])},{-Sqrt[1-2/Sqrt[5]],0,Sqrt[1/8+1/(8Sqrt[5])]},
 {Sqrt[1-2/Sqrt[5]],0,Root[1-20#1^2+80#1^4&,1,0]},{Root[1-5#1^2+5#1^4&,2,0],0,
  Sqrt[5/8-11/(8Sqrt[5])]},{Root[1-5#1^2+5#1^4&,3,0],0,Root[1-100#1^2+80#1^4&,2,0]},
 {Root[1-20#1^2+80#1^4&,2,0],(-3+Sqrt[5])/4,-(1/Sqrt[10-2Sqrt[5]])},
 {Root[1-20#1^2+80#1^4&,2,0],(3-Sqrt[5])/4,-(1/Sqrt[10-2Sqrt[5]])},
 {Sqrt[1/8-1/(8Sqrt[5])],(-3+Sqrt[5])/4,1/Sqrt[10-2Sqrt[5]]},
 {Sqrt[1/8-1/(8Sqrt[5])],(3-Sqrt[5])/4,1/Sqrt[10-2Sqrt[5]]},
 {-Sqrt[1-2/Sqrt[5]]/2,1/2,Sqrt[5/8-11/(8Sqrt[5])]},{-Sqrt[1-2/Sqrt[5]]/2,-1/2,Sqrt[5/8-11/(8Sqrt[5])]},
 {Root[1-100#1^2+80#1^4&,2,0],(1-Sqrt[5])/4,Sqrt[1/8+1/(8Sqrt[5])]},
 {Root[1-100#1^2+80#1^4&,2,0],(-1+Sqrt[5])/4,Sqrt[1/8+1/(8Sqrt[5])]},
 {Sqrt[1/8+1/(8Sqrt[5])],(-1+Sqrt[5])/4,Sqrt[5/8-11/(8Sqrt[5])]},
 {Sqrt[1/8+1/(8Sqrt[5])],(1-Sqrt[5])/4,Sqrt[5/8-11/(8Sqrt[5])]},
 {Sqrt[5/8-11/(8Sqrt[5])],(1-Sqrt[5])/4,Root[1-20#1^2+80#1^4&,1,0]},
 {Sqrt[5/8-11/(8Sqrt[5])],(-1+Sqrt[5])/4,Root[1-20#1^2+80#1^4&,1,0]},
 {Root[1-20#1^2+80#1^4&,1,0],(-1+Sqrt[5])/4,-(1/Sqrt[50+22Sqrt[5]])},
 {Root[1-20#1^2+80#1^4&,1,0],(1-Sqrt[5])/4,-(1/Sqrt[50+22Sqrt[5]])},
 {Sqrt[1/4-1/(2Sqrt[5])],1/2,Root[1-100#1^2+80#1^4&,2,0]},
 {Sqrt[1/4-1/(2Sqrt[5])],-1/2,Root[1-100#1^2+80#1^4&,2,0]}}

(* Great Icosahedron *)

PolyhedronName[GreatIcosahedron]:="great icosahedron"
PolyhedronDual[GreatIcosahedron]:=GreatStellatedDodecahedron
PolyhedronFaces[GreatIcosahedron]:=
{{1,7,5},{1,7,6},{1,5,12},{1,6,11},{1,11,12},{2,8,3},{2,8,4},{2,3,10},{2,4,
    9},{2,9,10},{8,3,12},{8,4,11},{8,11,12},{7,5,10},{7,6,9},{7,9,10},{3,5,
    10},{3,5,12},{4,6,9},{4,6,11}}
PolyhedronVertices[GreatIcosahedron]:=
{{0,0,-5/Sqrt[50-10Sqrt[5]]},{0,0,5/Sqrt[50-10Sqrt[5]]},
 {Sqrt[1/4+1/(2Sqrt[5])],-1/2,-(1/Sqrt[10-2Sqrt[5]])},
 {Sqrt[1/4+1/(2Sqrt[5])],1/2,-(1/Sqrt[10-2Sqrt[5]])},
 {-Sqrt[1+2/Sqrt[5]]/2,-1/2,1/Sqrt[10-2Sqrt[5]]},
 {-Sqrt[1+2/Sqrt[5]]/2,1/2,1/Sqrt[10-2Sqrt[5]]},
 {Sqrt[(5+Sqrt[5])/10],0,1/Sqrt[10-2Sqrt[5]]},
 {Root[1-5#1^2+5#1^4&,1,0],0,-(1/Sqrt[10-2Sqrt[5]])},
 {Root[1-20#1^2+80#1^4&,2,0],(-1-Sqrt[5])/4,
  -(1/Sqrt[10-2Sqrt[5]])},{Root[1-20#1^2+80#1^4&,2,0],
  (1+Sqrt[5])/4,-(1/Sqrt[10-2Sqrt[5]])},
 {Root[1-20#1^2+80#1^4&,3,0],(-1-Sqrt[5])/4,
  1/Sqrt[10-2Sqrt[5]]},{Root[1-20#1^2+80#1^4&,3,0],
  (1+Sqrt[5])/4,1/Sqrt[10-2Sqrt[5]]}}

PolyhedronFaces[GreatIcosahedron,Split->True]:=
{{79,87,2},{2,87,47},{79,47,87},{47,14,2},{2,14,48},{47,48,14},{48,90,2},{2,
    90,80},{48,80,90},{80,8,2},{2,8,51},{80,51,8},{51,7,2},{2,7,79},{51,79,
    7},{52,9,1},{1,9,81},{52,81,9},{81,83,1},{1,83,43},{81,43,83},{43,11,
    1},{1,11,44},{43,44,11},{44,86,1},{1,86,82},{44,82,86},{82,10,1},{1,10,
    52},{82,52,10},{82,4,46},{46,4,44},{82,44,4},{44,22,46},{46,22,66},{44,66,
    22},{66,27,46},{46,27,18},{66,18,27},{18,76,46},{46,76,20},{18,20,76},{20,
    89,46},{46,89,82},{20,82,89},{44,33,73},{73,33,43},{44,43,33},{43,39,
    73},{73,39,65},{43,65,39},{65,61,73},{73,61,91},{65,91,61},{91,62,73},{73,
    62,66},{91,66,62},{66,40,73},{73,40,44},{66,44,40},{43,3,45},{45,3,
    81},{43,81,3},{81,88,45},{45,88,19},{81,19,88},{19,75,45},{45,75,17},{19,
    17,75},{17,26,45},{45,26,65},{17,65,26},{65,21,45},{45,21,43},{65,43,
    21},{81,29,71},{71,29,52},{81,52,29},{52,56,71},{71,56,92},{52,92,56},{92,
    59,71},{71,59,67},{92,67,59},{67,37,71},{71,37,19},{67,19,37},{19,15,
    71},{71,15,81},{19,81,15},{52,32,72},{72,32,82},{52,82,32},{82,16,72},{72,
    16,20},{82,20,16},{20,38,72},{72,38,68},{20,68,38},{68,60,72},{72,60,
    92},{68,92,60},{92,55,72},{72,55,52},{92,52,55},{17,77,49},{49,77,19},{17,
    19,77},{19,30,49},{49,30,67},{19,67,30},{67,23,49},{49,23,47},{67,47,
    23},{47,5,49},{49,5,79},{47,79,5},{79,84,49},{49,84,17},{79,17,84},{67,63,
    74},{74,63,92},{67,92,63},{92,64,74},{74,64,68},{92,68,64},{68,42,74},{74,
    42,48},{68,48,42},{48,36,74},{74,36,47},{48,47,36},{47,41,74},{74,41,
    67},{47,67,41},{68,31,50},{50,31,20},{68,20,31},{20,78,50},{50,78,18},{20,
    18,78},{18,85,50},{50,85,80},{18,80,85},{80,6,50},{50,6,48},{80,48,6},{48,
    24,50},{50,24,68},{48,68,24},{18,35,70},{70,35,66},{18,66,35},{66,58,
    70},{70,58,91},{66,91,58},{91,53,70},{70,53,51},{91,51,53},{51,28,70},{70,
    28,80},{51,80,28},{80,13,70},{70,13,18},{80,18,13},{91,57,69},{69,57,
    65},{91,65,57},{65,34,69},{69,34,17},{65,17,34},{17,12,69},{69,12,79},{17,
    79,12},{79,25,69},{69,25,51},{79,51,25},{51,54,69},{69,54,91},{51,91,54}}
PolyhedronVertices[GreatIcosahedron,Split->True]:=
{{0, 0, -Sqrt[25/8 + (11*Sqrt[5])/8]}, {0, 0, Sqrt[25/8 + (11*Sqrt[5])/8]}, 
 {-Sqrt[1/40 + 1/(40*Sqrt[5])], (-5 - 3*Sqrt[5])/20, -Sqrt[13/40 + 19/(40*Sqrt[5])]}, 
 {-Sqrt[1/40 + 1/(40*Sqrt[5])], (5 + 3*Sqrt[5])/20, -Sqrt[13/40 + 19/(40*Sqrt[5])]}, 
 {Sqrt[1/40 + 1/(40*Sqrt[5])], (-5 - 3*Sqrt[5])/20, Sqrt[13/40 + 19/(40*Sqrt[5])]}, 
 {Sqrt[1/40 + 1/(40*Sqrt[5])], (5 + 3*Sqrt[5])/20, Sqrt[13/40 + 19/(40*Sqrt[5])]}, 
 {-Sqrt[1/20 + 1/(10*Sqrt[5])], -1/(2*Sqrt[5]), Sqrt[17/40 + 31/(40*Sqrt[5])]}, 
 {-Sqrt[1/20 + 1/(10*Sqrt[5])], 1/(2*Sqrt[5]), Sqrt[17/40 + 31/(40*Sqrt[5])]}, 
 {Sqrt[1/20 + 1/(10*Sqrt[5])], -1/(2*Sqrt[5]), -Sqrt[17/40 + 31/(40*Sqrt[5])]}, 
 {Sqrt[1/20 + 1/(10*Sqrt[5])], 1/(2*Sqrt[5]), -Sqrt[17/40 + 31/(40*Sqrt[5])]}, 
 {-Sqrt[1/10 + 1/(10*Sqrt[5])], 0, -Sqrt[17/40 + 31/(40*Sqrt[5])]}, 
 {-Sqrt[1/10 + 1/(10*Sqrt[5])], (-5 - Sqrt[5])/10, Sqrt[1/8 + 11/(40*Sqrt[5])]}, 
 {-Sqrt[1/10 + 1/(10*Sqrt[5])], (5 + Sqrt[5])/10, Sqrt[1/8 + 11/(40*Sqrt[5])]}, 
 {Sqrt[1/10 + 1/(10*Sqrt[5])], 0, Sqrt[17/40 + 31/(40*Sqrt[5])]}, 
 {Sqrt[1/10 + 1/(10*Sqrt[5])], (-5 - Sqrt[5])/10, -Sqrt[1/8 + 11/(40*Sqrt[5])]}, 
 {Sqrt[1/10 + 1/(10*Sqrt[5])], (5 + Sqrt[5])/10, -Sqrt[1/8 + 11/(40*Sqrt[5])]}, 
 {-Sqrt[1/8 + 1/(8*Sqrt[5])], (-3 - Sqrt[5])/4, Root[1 - 20*#1^2 + 80*#1^4 & , 3, 0]}, 
 {-Sqrt[1/8 + 1/(8*Sqrt[5])], (3 + Sqrt[5])/4, Root[1 - 20*#1^2 + 80*#1^4 & , 3, 0]}, 
 {Sqrt[1/8 + 1/(8*Sqrt[5])], (-3 - Sqrt[5])/4, Root[1 - 20*#1^2 + 80*#1^4 & , 2, 0]}, 
 {Sqrt[1/8 + 1/(8*Sqrt[5])], (3 + Sqrt[5])/4, Root[1 - 20*#1^2 + 80*#1^4 & , 2, 0]}, 
 {-Sqrt[9/40 + 9/(40*Sqrt[5])], (-5 - 3*Sqrt[5])/20, -Sqrt[1/8 + 11/(40*Sqrt[5])]}, 
 {-Sqrt[9/40 + 9/(40*Sqrt[5])], (5 + 3*Sqrt[5])/20, -Sqrt[1/8 + 11/(40*Sqrt[5])]}, 
 {Sqrt[9/40 + 9/(40*Sqrt[5])], (-5 - 3*Sqrt[5])/20, Sqrt[1/8 + 11/(40*Sqrt[5])]}, 
 {Sqrt[9/40 + 9/(40*Sqrt[5])], (5 + 3*Sqrt[5])/20, Sqrt[1/8 + 11/(40*Sqrt[5])]}, 
 {-Sqrt[1/8 + 11/(40*Sqrt[5])], (-5 - Sqrt[5])/20, Sqrt[13/40 + 19/(40*Sqrt[5])]}, 
 {-Sqrt[1/8 + 11/(40*Sqrt[5])], (-1 - Sqrt[5])/4, -Sqrt[(5 - Sqrt[5])/2]/10}, 
 {-Sqrt[1/8 + 11/(40*Sqrt[5])], (1 + Sqrt[5])/4, -Sqrt[(5 - Sqrt[5])/2]/10}, 
 {-Sqrt[1/8 + 11/(40*Sqrt[5])], (5 + Sqrt[5])/20, Sqrt[13/40 + 19/(40*Sqrt[5])]}, 
 {Sqrt[1/8 + 11/(40*Sqrt[5])], (-5 - Sqrt[5])/20, -Sqrt[13/40 + 19/(40*Sqrt[5])]}, 
 {Sqrt[1/8 + 11/(40*Sqrt[5])], (-1 - Sqrt[5])/4, Sqrt[(5 - Sqrt[5])/2]/10}, 
 {Sqrt[1/8 + 11/(40*Sqrt[5])], (1 + Sqrt[5])/4, Sqrt[(5 - Sqrt[5])/2]/10}, 
 {Sqrt[1/8 + 11/(40*Sqrt[5])], (5 + Sqrt[5])/20, -Sqrt[13/40 + 19/(40*Sqrt[5])]}, 
 {-Sqrt[1/5 + 2/(5*Sqrt[5])], 0, -Sqrt[13/40 + 19/(40*Sqrt[5])]}, 
 {-Sqrt[1/5 + 2/(5*Sqrt[5])], (-5 - Sqrt[5])/10, Sqrt[(5 - Sqrt[5])/2]/10}, 
 {-Sqrt[1/5 + 2/(5*Sqrt[5])], (5 + Sqrt[5])/10, Sqrt[(5 - Sqrt[5])/2]/10}, 
 {Sqrt[1/5 + 2/(5*Sqrt[5])], 0, Sqrt[13/40 + 19/(40*Sqrt[5])]}, 
 {Sqrt[1/5 + 2/(5*Sqrt[5])], (-5 - Sqrt[5])/10, -Sqrt[(5 - Sqrt[5])/2]/10}, 
 {Sqrt[1/5 + 2/(5*Sqrt[5])], (5 + Sqrt[5])/10, -Sqrt[(5 - Sqrt[5])/2]/10}, 
 {-Sqrt[13/40 + 19/(40*Sqrt[5])], (-5 - Sqrt[5])/20, -Sqrt[1/8 + 11/(40*Sqrt[5])]}, 
 {-Sqrt[13/40 + 19/(40*Sqrt[5])], (5 + Sqrt[5])/20, -Sqrt[1/8 + 11/(40*Sqrt[5])]}, 
 {Sqrt[13/40 + 19/(40*Sqrt[5])], (-5 - Sqrt[5])/20, Sqrt[1/8 + 11/(40*Sqrt[5])]}, 
 {Sqrt[13/40 + 19/(40*Sqrt[5])], (5 + Sqrt[5])/20, Sqrt[1/8 + 11/(40*Sqrt[5])]}, 
 {-Sqrt[1/4 + 1/(2*Sqrt[5])], -1/2, Root[1 - 100*#1^2 + 80*#1^4 & , 1, 0]}, 
 {-Sqrt[1/4 + 1/(2*Sqrt[5])], 1/2, Root[1 - 100*#1^2 + 80*#1^4 & , 1, 0]}, 
 {-Sqrt[1/4 + 1/(2*Sqrt[5])], (-2 - Sqrt[5])/2, -Sqrt[5/8 + 11/(8*Sqrt[5])]}, 
 {-Sqrt[1/4 + 1/(2*Sqrt[5])], (2 + Sqrt[5])/2, -Sqrt[5/8 + 11/(8*Sqrt[5])]}, 
 {Sqrt[1/4 + 1/(2*Sqrt[5])], -1/2, Root[1 - 100*#1^2 + 80*#1^4 & , 4, 0]}, 
 {Sqrt[1/4 + 1/(2*Sqrt[5])], 1/2, Root[1 - 100*#1^2 + 80*#1^4 & , 4, 0]}, 
 {Sqrt[1/4 + 1/(2*Sqrt[5])], (-2 - Sqrt[5])/2, Sqrt[5/8 + 11/(8*Sqrt[5])]}, 
 {Sqrt[1/4 + 1/(2*Sqrt[5])], (2 + Sqrt[5])/2, Sqrt[5/8 + 11/(8*Sqrt[5])]}, 
 {-Sqrt[1/2 + 1/(2*Sqrt[5])], 0, Root[1 - 100*#1^2 + 80*#1^4 & , 4, 0]}, 
 {Sqrt[1/2 + 1/(2*Sqrt[5])], 0, Root[1 - 100*#1^2 + 80*#1^4 & , 1, 0]}, 
 {-Sqrt[13/40 + 29/(40*Sqrt[5])], (5 - Sqrt[5])/20, Sqrt[1/8 + 11/(40*Sqrt[5])]}, 
 {-Sqrt[13/40 + 29/(40*Sqrt[5])], (-5 + Sqrt[5])/20, Sqrt[1/8 + 11/(40*Sqrt[5])]}, 
 {Sqrt[13/40 + 29/(40*Sqrt[5])], (5 - Sqrt[5])/20, -Sqrt[1/8 + 11/(40*Sqrt[5])]}, 
 {Sqrt[13/40 + 29/(40*Sqrt[5])], (-5 + Sqrt[5])/20, -Sqrt[1/8 + 11/(40*Sqrt[5])]}, 
 {-Sqrt[17/40 + 31/(40*Sqrt[5])], (-5 - Sqrt[5])/20, Sqrt[(5 - Sqrt[5])/2]/10}, 
 {-Sqrt[17/40 + 31/(40*Sqrt[5])], (5 + Sqrt[5])/20, Sqrt[(5 - Sqrt[5])/2]/10}, 
 {Sqrt[17/40 + 31/(40*Sqrt[5])], (-5 - Sqrt[5])/20, -Sqrt[(5 - Sqrt[5])/2]/10}, 
 {Sqrt[17/40 + 31/(40*Sqrt[5])], (5 + Sqrt[5])/20, -Sqrt[(5 - Sqrt[5])/2]/10}, 
 {-Sqrt[9/20 + 9/(10*Sqrt[5])], -1/(2*Sqrt[5]), -Sqrt[(5 - Sqrt[5])/2]/10}, 
 {-Sqrt[9/20 + 9/(10*Sqrt[5])], 1/(2*Sqrt[5]), -Sqrt[(5 - Sqrt[5])/2]/10}, 
 {Sqrt[9/20 + 9/(10*Sqrt[5])], -1/(2*Sqrt[5]), Sqrt[(5 - Sqrt[5])/2]/10}, 
 {Sqrt[9/20 + 9/(10*Sqrt[5])], 1/(2*Sqrt[5]), Sqrt[(5 - Sqrt[5])/2]/10}, 
 {-Sqrt[5/8 + 11/(8*Sqrt[5])], (-1 - Sqrt[5])/4, Root[1 - 20*#1^2 + 80*#1^4 & , 2, 0]}, 
 {-Sqrt[5/8 + 11/(8*Sqrt[5])], (1 + Sqrt[5])/4, Root[1 - 20*#1^2 + 80*#1^4 & , 2, 0]}, 
 {Sqrt[5/8 + 11/(8*Sqrt[5])], (-1 - Sqrt[5])/4, Root[1 - 20*#1^2 + 80*#1^4 & , 3, 0]}, 
 {Sqrt[5/8 + 11/(8*Sqrt[5])], (1 + Sqrt[5])/4, Root[1 - 20*#1^2 + 80*#1^4 & , 3, 0]}, 
 {-Sqrt[13/8 + 29/(8*Sqrt[5])], (-3 - Sqrt[5])/4, Sqrt[5/8 + 11/(8*Sqrt[5])]}, 
 {-Sqrt[13/8 + 29/(8*Sqrt[5])], (3 + Sqrt[5])/4, Sqrt[5/8 + 11/(8*Sqrt[5])]}, 
 {Sqrt[13/8 + 29/(8*Sqrt[5])], (-3 - Sqrt[5])/4, -Sqrt[5/8 + 11/(8*Sqrt[5])]}, 
 {Sqrt[13/8 + 29/(8*Sqrt[5])], (3 + Sqrt[5])/4, -Sqrt[5/8 + 11/(8*Sqrt[5])]}, 
 {-Sqrt[5/2 + 11/(2*Sqrt[5])], 0, -Sqrt[5/8 + 11/(8*Sqrt[5])]}, 
 {Sqrt[5/2 + 11/(2*Sqrt[5])], 0, Sqrt[5/8 + 11/(8*Sqrt[5])]}, {-Sqrt[5 - 2*Sqrt[5]]/10, (-5 - 2*Sqrt[5])/10, 
  -Sqrt[(5 - Sqrt[5])/2]/10}, {-Sqrt[5 - 2*Sqrt[5]]/10, (5 + 2*Sqrt[5])/10, -Sqrt[(5 - Sqrt[5])/2]/10}, 
 {Sqrt[5 - 2*Sqrt[5]]/10, (-5 - 2*Sqrt[5])/10, Sqrt[(5 - Sqrt[5])/2]/10}, 
 {Sqrt[5 - 2*Sqrt[5]]/10, (5 + 2*Sqrt[5])/10, Sqrt[(5 - Sqrt[5])/2]/10}, 
 {-Sqrt[(5 - Sqrt[5])/10]/2, (-1 - Sqrt[5])/4, Root[1 - 100*#1^2 + 80*#1^4 & , 4, 0]}, 
 {-Sqrt[(5 - Sqrt[5])/10]/2, (1 + Sqrt[5])/4, Root[1 - 100*#1^2 + 80*#1^4 & , 4, 0]}, 
 {Sqrt[(5 - Sqrt[5])/10]/2, (-1 - Sqrt[5])/4, Root[1 - 100*#1^2 + 80*#1^4 & , 1, 0]}, 
 {Sqrt[(5 - Sqrt[5])/10]/2, (1 + Sqrt[5])/4, Root[1 - 100*#1^2 + 80*#1^4 & , 1, 0]}, 
 {-Sqrt[(5 - Sqrt[5])/2]/10, (-5 - Sqrt[5])/20, -Sqrt[17/40 + 31/(40*Sqrt[5])]}, 
 {-Sqrt[(5 - Sqrt[5])/2]/10, (-1 - Sqrt[5])/4, Sqrt[1/8 + 11/(40*Sqrt[5])]}, 
 {-Sqrt[(5 - Sqrt[5])/2]/10, (1 + Sqrt[5])/4, Sqrt[1/8 + 11/(40*Sqrt[5])]}, 
 {-Sqrt[(5 - Sqrt[5])/2]/10, (5 + Sqrt[5])/20, -Sqrt[17/40 + 31/(40*Sqrt[5])]}, 
 {Sqrt[(5 - Sqrt[5])/2]/10, (-5 - Sqrt[5])/20, Sqrt[17/40 + 31/(40*Sqrt[5])]}, 
 {Sqrt[(5 - Sqrt[5])/2]/10, (-1 - Sqrt[5])/4, -Sqrt[1/8 + 11/(40*Sqrt[5])]}, 
 {Sqrt[(5 - Sqrt[5])/2]/10, (1 + Sqrt[5])/4, -Sqrt[1/8 + 11/(40*Sqrt[5])]}, 
 {Sqrt[(5 - Sqrt[5])/2]/10, (5 + Sqrt[5])/20, Sqrt[17/40 + 31/(40*Sqrt[5])]}, 
 {-Sqrt[(5 + 2*Sqrt[5])/5], 0, Root[1 - 20*#1^2 + 80*#1^4 & , 3, 0]}, 
 {Sqrt[(5 + 2*Sqrt[5])/5], 0, Root[1 - 20*#1^2 + 80*#1^4 & , 2, 0]}}

(* Great Stellated Dodecahedron *)

PolyhedronName[GreatStellatedDodecahedron]:="great stellated dodecahedron"
PolyhedronDual[GreatStellatedDodecahedron]:=GreatIcosahedron

PolyhedronFaces[GreatStellatedDodecahedron]:=
{{4,3,6,1,5},{7,10,5,1,13},{8,13,1,6,9},{11,15,2,16,18},{12,17,16,2,14},{20,
    14,2,15,19},{4,3,18,16,17},{11,18,3,6,9},{12,10,5,4,17},{20,7,13,8,
    19},{12,10,7,20,14},{11,15,19,8,9}}
PolyhedronVertices[GreatStellatedDodecahedron]:=
{{0,0,Sqrt[9/8+(3Sqrt[5])/8]},{0,0,-Sqrt[(3(3+Sqrt[5]))/2]/2},
 {Sqrt[1/8-Sqrt[5]/24],(-3-Sqrt[5])/4,Sqrt[1/8+Sqrt[5]/24]},
 {Sqrt[1/8-Sqrt[5]/24],(3+Sqrt[5])/4,Sqrt[1/8+Sqrt[5]/24]},
 {Sqrt[1/8+Sqrt[5]/24],(-1-Sqrt[5])/4,-Sqrt[(5(3+Sqrt[5]))/6]/2},
 {Sqrt[1/8+Sqrt[5]/24],(1+Sqrt[5])/4,-Sqrt[(5(3+Sqrt[5]))/6]/2},
 {Sqrt[5/8+(5Sqrt[5])/24],(-1-Sqrt[5])/4,Sqrt[1/8+Sqrt[5]/24]},
 {Sqrt[5/8+(5Sqrt[5])/24],(1+Sqrt[5])/4,Sqrt[1/8+Sqrt[5]/24]},
 {-Sqrt[3/4+Sqrt[5]/3],-1/2,Sqrt[1/8+Sqrt[5]/24]},
 {-Sqrt[3/4+Sqrt[5]/3],1/2,Sqrt[1/8+Sqrt[5]/24]},
 {Sqrt[3/4+Sqrt[5]/3],-1/2,-Sqrt[(3+Sqrt[5])/6]/2},
 {Sqrt[3/4+Sqrt[5]/3],1/2,-Sqrt[(3+Sqrt[5])/6]/2},
 {-Sqrt[(3+Sqrt[5])/6],0,-Sqrt[(5(3+Sqrt[5]))/6]/2},
 {-Sqrt[(3+Sqrt[5])/6]/2,(-1-Sqrt[5])/4,Sqrt[5/8+(5Sqrt[5])/24]},
 {-Sqrt[(3+Sqrt[5])/6]/2,(1+Sqrt[5])/4,Sqrt[5/8+(5Sqrt[5])/24]},
 {Sqrt[(3+Sqrt[5])/6],0,Sqrt[5/8+(5Sqrt[5])/24]},
 {-Sqrt[(5(3+Sqrt[5]))/6]/2,(-1-Sqrt[5])/4,-Sqrt[(3+Sqrt[5])/6]/2},
 {-Sqrt[(5(3+Sqrt[5]))/6]/2,(1+Sqrt[5])/4,-Sqrt[(3+Sqrt[5])/6]/2},
 {Root[1-36#1^2+144#1^4&,2,0],(-3-Sqrt[5])/4,
  -Sqrt[(3+Sqrt[5])/6]/2},{Root[1-36#1^2+144#1^4&,2,0],
  (3+Sqrt[5])/4,-Sqrt[(3+Sqrt[5])/6]/2}}

PolyhedronFaces[GreatStellatedDodecahedron,Split->True]:=
{{24,2,8},{24,8,10},{24,10,2},{13,2,10},{13,10,9},{13,9,2},{23,2,9},{23,9,
    7},{23,7,2},{19,2,7},{19,7,4},{19,4,2},{20,2,4},{20,4,8},{20,8,2},{27,11,
    5},{27,5,1},{27,1,11},{14,12,11},{14,11,1},{14,1,12},{28,6,12},{28,12,
    1},{28,1,6},{18,3,6},{18,6,1},{18,1,3},{17,5,3},{17,3,1},{17,1,5},{22,8,
    6},{22,6,10},{22,10,8},{15,10,3},{15,3,9},{15,9,10},{21,9,5},{21,5,7},{21,
    7,9},{26,7,11},{26,11,4},{26,4,7},{25,4,12},{25,12,8},{25,8,4},{31,11,
    7},{31,7,5},{31,5,11},{16,12,4},{16,4,11},{16,11,12},{32,6,8},{32,8,
    12},{32,12,6},{29,3,10},{29,10,6},{29,6,3},{30,5,9},{30,9,3},{30,3,5}}
PolyhedronVertices[GreatStellatedDodecahedron,Split->True]:=
{{0,0,-5/Sqrt[50-10Sqrt[5]]},{0,0,5/Sqrt[50-10Sqrt[5]]},{Root[1-5#1^2+5#1^4&,1,0],0,
  -(1/Sqrt[10-2Sqrt[5]])},{Sqrt[(5+Sqrt[5])/10],0,1/Sqrt[10-2Sqrt[5]]},
 {Root[1-20#1^2+80#1^4&,2,0],(-1-Sqrt[5])/4,-(1/Sqrt[10-2Sqrt[5]])},
 {Root[1-20#1^2+80#1^4&,2,0],(1+Sqrt[5])/4,-(1/Sqrt[10-2Sqrt[5]])},
 {Sqrt[1/8-1/(8Sqrt[5])],(-1-Sqrt[5])/4,1/Sqrt[10-2Sqrt[5]]},
 {Sqrt[1/8-1/(8Sqrt[5])],(1+Sqrt[5])/4,1/Sqrt[10-2Sqrt[5]]},
 {-Sqrt[1+2/Sqrt[5]]/2,-1/2,1/Sqrt[10-2Sqrt[5]]},{-Sqrt[1+2/Sqrt[5]]/2,1/2,1/Sqrt[10-2Sqrt[5]]},
 {Sqrt[1/4+1/(2Sqrt[5])],-1/2,-(1/Sqrt[10-2Sqrt[5]])},{Sqrt[1/4+1/(2Sqrt[5])],1/2,
  -(1/Sqrt[10-2Sqrt[5]])},{-Sqrt[1+2/Sqrt[5]],0,Sqrt[13/8+29/(8Sqrt[5])]},
 {Sqrt[1+2/Sqrt[5]],0,Root[1-260#1^2+80#1^4&,1,0]},{Root[1-25#1^2+5#1^4&,1,0],0,
  Sqrt[1/8+1/(8Sqrt[5])]},{Sqrt[5/2+11/(2Sqrt[5])],0,Root[1-20#1^2+80#1^4&,1,0]},
 {-Sqrt[10+22/Sqrt[5]]/4,(-1-Sqrt[5])/4,Root[1-260#1^2+80#1^4&,1,0]},
 {-Sqrt[10+22/Sqrt[5]]/4,(1+Sqrt[5])/4,Root[1-260#1^2+80#1^4&,1,0]},
 {Sqrt[10+22/Sqrt[5]]/4,(-1-Sqrt[5])/4,Sqrt[13/8+29/(8Sqrt[5])]},
 {Sqrt[10+22/Sqrt[5]]/4,(1+Sqrt[5])/4,Sqrt[13/8+29/(8Sqrt[5])]},
 {-Sqrt[1+2/Sqrt[5]]/2,-1-Sqrt[5]/2,1/Sqrt[10-2Sqrt[5]]},{-Sqrt[1+2/Sqrt[5]]/2,(2+Sqrt[5])/2,
  1/Sqrt[10-2Sqrt[5]]},{Root[1-20#1^2+80#1^4&,1,0],(-3-Sqrt[5])/4,Sqrt[13/8+29/(8Sqrt[5])]},
 {Root[1-20#1^2+80#1^4&,1,0],(3+Sqrt[5])/4,Sqrt[13/8+29/(8Sqrt[5])]},
 {Sqrt[13/8+29/(8Sqrt[5])],(3+Sqrt[5])/4,1/Sqrt[10-2Sqrt[5]]},
 {Sqrt[13/8+29/(8Sqrt[5])],(-3-Sqrt[5])/4,1/Sqrt[10-2Sqrt[5]]},
 {((3+Sqrt[5])/10)^(1/4)/2,(-3-Sqrt[5])/4,Root[1-260#1^2+80#1^4&,1,0]},
 {((3+Sqrt[5])/10)^(1/4)/2,(3+Sqrt[5])/4,Root[1-260#1^2+80#1^4&,1,0]},
 {Root[1-260#1^2+80#1^4&,1,0],(3+Sqrt[5])/4,-(1/Sqrt[10-2Sqrt[5]])},
 {Root[1-260#1^2+80#1^4&,1,0],(-3-Sqrt[5])/4,-(1/Sqrt[10-2Sqrt[5]])},
 {Sqrt[1/4+1/(2Sqrt[5])],-1-Sqrt[5]/2,-(1/Sqrt[10-2Sqrt[5]])},
 {Sqrt[1/4+1/(2Sqrt[5])],(2+Sqrt[5])/2,-(1/Sqrt[10-2Sqrt[5]])}}

(* Icosahedron *)

PolyhedronName[Icosahedron]:="icosahedron"
PolyhedronDual[Icosahedron]:=Dodecahedron
NetFaces[Icosahedron]:={{1,6,7},{6,12,13},{2,7,8},{7,13,14},{3,8,9},{8,14,15},{4,9,10},{9,15,16},{5,10,11},{10,16,17},{13,7,6},{18,13,12},{14,8,7},{19,14,13},{15,9,8},{20,15,14},{16,10,9},{21,16,15},{17,11,10},{22,17,16}}
NetVertices[Icosahedron]:={{-(1/Sqrt[3]),0},{-(1/Sqrt[3]),1},{-(1/Sqrt[3]),2},{-(1/Sqrt[3]),3},{-(1/Sqrt[3]),4},{1/(2Sqrt[3]),-1/2},{1/(2Sqrt[3]),1/2},{1/(2Sqrt[3]),3/2},{1/(2Sqrt[3]),5/2},{1/(2Sqrt[3]),7/2},{1/(2Sqrt[3]),9/2},{2/Sqrt[3],-1},{2/Sqrt[3],0},{2/Sqrt[3],1},{2/Sqrt[3],2},{2/Sqrt[3],3},{2/Sqrt[3],4},{7/(2Sqrt[3]),-1/2},{7/(2Sqrt[3]),1/2},{7/(2Sqrt[3]),3/2},{7/(2Sqrt[3]),5/2},{7/(2Sqrt[3]),7/2}}
PolyhedronPolyhedra[Icosahedron]:={Range[20]}
PolyhedronFaces[Icosahedron]:=PolyhedronFaces[Icosahedron,"C5"]
PolyhedronVertices[Icosahedron]:=PolyhedronVertices[Icosahedron,"C5"]

(* Icosahedron C2 *)

PolyhedronName[Icosahedron,"C2"]:="icosahedron oriented with the z-axis along a C2 symmetry \
axis"
PolyhedronDual[Icosahedron,"C2"]:=Sequence[Dodecahedron,"C2"]
PolyhedronPolyhedra[Icosahedron,"C2"]:={Range[20]}
PolyhedronFaces[Icosahedron,"C2"]:=
{{2,6,10},{2,10,9},{2,9,4},{2,4,8},{2,
8,6},{11,3,7},{12,11,7},{5,12,7},{1,5,7},{3,1,7},{6,5,10},{
10,1,9},{9,3,4},{4,11,8},{8,12,6},{11,4,3},{12,8,11},{
5,6,12},{1,10,5},{3,9,1}}
PolyhedronVertices[Icosahedron,"C2"]:=
{{-1/2,0,(-1-Sqrt[5])/4},{-1/2,0,(1+Sqrt[5])/4},
{0,(-1-Sqrt[5])/4,-1/2},{0,(-1-Sqrt[5])/4,1/2},
{0,(1+Sqrt[5])/4,-1/2},{0,(1+Sqrt[5])/4,1/2},
{1/2,0,(-1-Sqrt[5])/4},{1/2,0,(1+Sqrt[5])/4},
{(-1-Sqrt[5])/4,-1/2,0},{(-1-Sqrt[5])/4,1/2,0},
{(1+Sqrt[5])/4,-1/2,0},{(1+Sqrt[5])/4,1/2,0}}

(* Icosahedron C3 *)

PolyhedronName[Icosahedron,"C3"]:="icosahedron oriented with the z-axis along a C3 symmetry \
axis"
PolyhedronDual[Icosahedron,"C3"]:=Sequence[Dodecahedron,"C5"]
PolyhedronPolyhedra[Icosahedron,"C3"]:={Range[20]}
PolyhedronFaces[Icosahedron,"C3"]:=
{{6,3,2},{2,3,11},{6,2,9},{3,6,10},{9,12,6},{6,12,10},{10,8,3},{3,8,11},{11,7,
    2},{2,7,9},{5,4,1},{12,4,5},{8,5,1},{7,1,4},{1,11,8},{7,11,1},{4,9,7},{12,
    9,4},{5,10,12},{8,10,5}}
PolyhedronVertices[Icosahedron,"C3"]:=
{{-(1/Sqrt[3]),0,-Sqrt[7/24+Sqrt[5]/8]},{-1/(2Sqrt[3]),-1/2,Sqrt[7/24+Sqrt[5]/8]},
 {-1/(2Sqrt[3]),1/2,Sqrt[7/24+Sqrt[5]/8]},{1/(2Sqrt[3]),-1/2,-Sqrt[7/24+Sqrt[5]/8]},
 {1/(2Sqrt[3]),1/2,-Sqrt[7/24+Sqrt[5]/8]},{1/Sqrt[3],0,Sqrt[7/24+Sqrt[5]/8]},
 {-Sqrt[1/8+Sqrt[5]/24],(-1-Sqrt[5])/4,-Sqrt[1/8-Sqrt[5]/24]},
 {-Sqrt[1/8+Sqrt[5]/24],(1+Sqrt[5])/4,-Sqrt[1/8-Sqrt[5]/24]},
 {Sqrt[1/8+Sqrt[5]/24],(-1-Sqrt[5])/4,Sqrt[1/8-Sqrt[5]/24]},
 {Sqrt[1/8+Sqrt[5]/24],(1+Sqrt[5])/4,Sqrt[1/8-Sqrt[5]/24]},
 {-Sqrt[1/2+Sqrt[5]/6],0,Sqrt[1/8-Sqrt[5]/24]},{Sqrt[1/2+Sqrt[5]/6],0,-Sqrt[1/8-Sqrt[5]/24]}}

(* Icosahedron C5 *)

PolyhedronName[Icosahedron,"C5"]:="icosahedron oriented with the z-axis along a C5 symmetry \
axis"
PolyhedronDual[Icosahedron,"C5"]:=Sequence[Dodecahedron,"C3"]
PolyhedronPolyhedra[Icosahedron,"C5"]:={Range[20]}
PolyhedronFaces[Icosahedron,"C5"]:={{2,12,8},{2,8,7},{2,7,11},{2,11,4},{2,4,12},{5,9,1},{6,5,1},{10,6,1},{3,10,
    1},{9,3,1},{12,10,8},{8,3,7},{7,9,11},{11,5,4},{4,6,12},{5,11,9},{6,4,
    5},{10,12,6},{3,8,10},{9,7,3}}
PolyhedronVertices[Icosahedron,"C5"]:={{0,0,-5/Sqrt[50-10Sqrt[5]]},
 {0,0,5/Sqrt[50-10Sqrt[5]]},
 {-Sqrt[2/(5-Sqrt[5])],0,-(1/Sqrt[10-2Sqrt[5]])},
 {Sqrt[2/(5-Sqrt[5])],0,1/Sqrt[10-2Sqrt[5]]},
 {(1+Sqrt[5])/(2Sqrt[10-2Sqrt[5]]),-1/2, -(1/Sqrt[10-2Sqrt[5]])},
 {(1+Sqrt[5])/(2Sqrt[10-2Sqrt[5]]),1/2,-(1/Sqrt[10-2Sqrt[5]])},
 {-(1+Sqrt[5])/(2Sqrt[10-2Sqrt[5]]),-1/2,1/Sqrt[10-2Sqrt[5]]},
 {-(1+Sqrt[5])/(2Sqrt[10-2Sqrt[5]]),1/2,1/Sqrt[10-2Sqrt[5]]},
 {-(-1+Sqrt[5])/(2Sqrt[10-2Sqrt[5]]),-Sqrt[(5+Sqrt[5])/(5-Sqrt[5])]/2,-(1/Sqrt[10-2Sqrt[5]])},
 {-(-1+Sqrt[5])/(2Sqrt[10-2Sqrt[5]]),Sqrt[(5+Sqrt[5])/(5-Sqrt[5])]/2,-(1/Sqrt[10-2Sqrt[5]])},
 {(-1+Sqrt[5])/(2Sqrt[10-2Sqrt[5]]),-Sqrt[(5+Sqrt[5])/(5-Sqrt[5])]/2, 1/Sqrt[10-2Sqrt[5]]},
 {(-1+Sqrt[5])/(2Sqrt[10-2Sqrt[5]]),Sqrt[(5+Sqrt[5])/(5-Sqrt[5])]/2,1/Sqrt[10-2Sqrt[5]]}}

(* Icosahedron 2-Compound *)

(* Icosahedron 5-Compound *)

PolyhedronName[Icosahedron5Compound]:="5-icosahedron compound"
PolyhedronDual[Icosahedron5Compound]:=Dodecahedron5Compound
PolyhedronPolyhedra[Icosahedron5Compound]:=
{{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20},{21,22,23,24,25,26,27,
    28,29,30,31,32,33,34,35,36,37,38,39,40},{41,42,43,44,45,46,47,48,49,50,51,
    52,53,54,55,56,57,58,59,60},{61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,
    76,77,78,79,80},{81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,
    100}}
PolyhedronFaces[Icosahedron5Compound]:=
{{21,28,29},{28,22,27},{26,27,23},{24,30,26},{30,25,29},{60,27,26},{29,28,
    60},{30,60,26},{27,60,28},{60,30,29},{21,59,22},{24,23,59},{21,25,59},{22,
    59,23},{59,25,24},{23,24,26},{22,23,27},{21,22,28},{29,25,21},{30,24,
    25},{17,31,11},{6,34,16},{6,51,41},{16,34,54},{41,51,17},{6,41,34},{54,50,
    11},{41,17,50},{50,54,34},{34,41,50},{31,38,11},{31,17,51},{38,47,16},{38,
    31,47},{11,50,17},{6,47,51},{16,47,6},{51,47,31},{54,38,16},{38,54,
    11},{15,12,40},{57,5,3},{18,5,57},{2,46,12},{18,46,2},{3,40,57},{46,40,
    12},{18,57,46},{56,43,35},{3,43,15},{46,57,40},{56,12,15},{43,56,15},{18,
    2,35},{40,3,15},{43,5,35},{12,56,2},{35,5,18},{35,2,56},{43,3,5},{39,9,
    14},{8,58,4},{19,58,8},{9,45,1},{1,45,19},{39,4,58},{9,39,45},{19,45,
    58},{44,55,36},{14,44,4},{58,45,39},{9,55,14},{55,44,14},{19,36,1},{4,39,
    14},{36,8,44},{55,9,1},{19,8,36},{1,36,55},{8,4,44},{32,20,10},{7,13,
    33},{7,42,52},{33,13,53},{52,42,20},{33,42,7},{49,53,10},{49,20,42},{53,
    49,33},{49,42,33},{10,37,32},{20,32,52},{13,48,37},{48,32,37},{20,49,
    10},{7,52,48},{7,48,13},{48,52,32},{37,53,13},{53,37,10}}
PolyhedronVertices[Icosahedron5Compound]:=
{{-Sqrt[1/16-1/(8Sqrt[5])],(-2-Sqrt[5])/4,Sqrt[5/8+11/(8Sqrt[5])]},
 {-Sqrt[1/16-1/(8Sqrt[5])],(2+Sqrt[5])/4,Sqrt[5/8+11/(8Sqrt[5])]},
 {Sqrt[1/16-1/(8Sqrt[5])],(-2-Sqrt[5])/4,-Sqrt[5/8+11/(8Sqrt[5])]},
 {Sqrt[1/16-1/(8Sqrt[5])],(2+Sqrt[5])/4,-Sqrt[5/8+11/(8Sqrt[5])]},
 {-Sqrt[1/32+1/(32Sqrt[5])],(-5-3Sqrt[5])/8,Sqrt[1/8+1/(8Sqrt[5])]},
 {-Sqrt[1/32+1/(32Sqrt[5])],(-3-Sqrt[5])/8,Sqrt[(5+2Sqrt[5])/5]},
 {-Sqrt[1/32+1/(32Sqrt[5])],(3+Sqrt[5])/8,Sqrt[(5+2Sqrt[5])/5]},
 {-Sqrt[1/32+1/(32Sqrt[5])],(5+3Sqrt[5])/8,Sqrt[1/8+1/(8Sqrt[5])]},
 {Sqrt[1/32+1/(32Sqrt[5])],(-5-3Sqrt[5])/8,-Sqrt[1/8+1/(8Sqrt[5])]},
 {Sqrt[1/32+1/(32Sqrt[5])],(-3-Sqrt[5])/8,-Sqrt[(5+2Sqrt[5])/5]},
 {Sqrt[1/32+1/(32Sqrt[5])],(3+Sqrt[5])/8,-Sqrt[(5+2Sqrt[5])/5]},
 {Sqrt[1/32+1/(32Sqrt[5])],(5+3Sqrt[5])/8,-Sqrt[1/8+1/(8Sqrt[5])]},
 {-Sqrt[5/32+11/(32Sqrt[5])],(-5-Sqrt[5])/8,Sqrt[5/8+11/(8Sqrt[5])]},
 {-Sqrt[5/32+11/(32Sqrt[5])],(-1-Sqrt[5])/8,-Sqrt[(5+2Sqrt[5])/5]},
 {-Sqrt[5/32+11/(32Sqrt[5])],(1+Sqrt[5])/8,-Sqrt[(5+2Sqrt[5])/5]},
 {-Sqrt[5/32+11/(32Sqrt[5])],(5+Sqrt[5])/8,Sqrt[5/8+11/(8Sqrt[5])]},
 {Sqrt[5/32+11/(32Sqrt[5])],(-5-Sqrt[5])/8,-Sqrt[5/8+11/(8Sqrt[5])]},
 {Sqrt[5/32+11/(32Sqrt[5])],(-1-Sqrt[5])/8,Sqrt[(5+2Sqrt[5])/5]},
 {Sqrt[5/32+11/(32Sqrt[5])],(1+Sqrt[5])/8,Sqrt[(5+2Sqrt[5])/5]},
 {Sqrt[5/32+11/(32Sqrt[5])],(5+Sqrt[5])/8,-Sqrt[5/8+11/(8Sqrt[5])]},
 {-Sqrt[1/4+1/(2Sqrt[5])],0,Sqrt[(5+2Sqrt[5])/5]},
 {-Sqrt[1/4+1/(2Sqrt[5])],(-3-Sqrt[5])/4,Sqrt[1/8+1/(8Sqrt[5])]},
 {-Sqrt[1/4+1/(2Sqrt[5])],(-1-Sqrt[5])/4,-Sqrt[5/8+11/(8Sqrt[5])]},
 {-Sqrt[1/4+1/(2Sqrt[5])],(1+Sqrt[5])/4,-Sqrt[5/8+11/(8Sqrt[5])]},
 {-Sqrt[1/4+1/(2Sqrt[5])],(3+Sqrt[5])/4,Sqrt[1/8+1/(8Sqrt[5])]},
 {Sqrt[1/4+1/(2Sqrt[5])],0,-Sqrt[(5+2Sqrt[5])/5]},
 {Sqrt[1/4+1/(2Sqrt[5])],(-3-Sqrt[5])/4,-Sqrt[1/8+1/(8Sqrt[5])]},
 {Sqrt[1/4+1/(2Sqrt[5])],(-1-Sqrt[5])/4,Sqrt[5/8+11/(8Sqrt[5])]},
 {Sqrt[1/4+1/(2Sqrt[5])],(1+Sqrt[5])/4,Sqrt[5/8+11/(8Sqrt[5])]},
 {Sqrt[1/4+1/(2Sqrt[5])],(3+Sqrt[5])/4,-Sqrt[1/8+1/(8Sqrt[5])]},
 {-Sqrt[17/32+31/(32Sqrt[5])],(-1-Sqrt[5])/8,-Sqrt[5/8+11/(8Sqrt[5])]},
 {-Sqrt[17/32+31/(32Sqrt[5])],(1+Sqrt[5])/8,-Sqrt[5/8+11/(8Sqrt[5])]},
 {Sqrt[17/32+31/(32Sqrt[5])],(-1-Sqrt[5])/8,Sqrt[5/8+11/(8Sqrt[5])]},
 {Sqrt[17/32+31/(32Sqrt[5])],(1+Sqrt[5])/8,Sqrt[5/8+11/(8Sqrt[5])]},
 {-Sqrt[9/16+9/(8Sqrt[5])],-1/4,Sqrt[5/8+11/(8Sqrt[5])]},
 {-Sqrt[9/16+9/(8Sqrt[5])],1/4,Sqrt[5/8+11/(8Sqrt[5])]},
 {-Sqrt[9/16+9/(8Sqrt[5])],(-2-Sqrt[5])/4,-Sqrt[1/8+1/(8Sqrt[5])]},
 {-Sqrt[9/16+9/(8Sqrt[5])],(2+Sqrt[5])/4,-Sqrt[1/8+1/(8Sqrt[5])]},
 {Sqrt[9/16+9/(8Sqrt[5])],-1/4,-Sqrt[5/8+11/(8Sqrt[5])]},
 {Sqrt[9/16+9/(8Sqrt[5])],1/4,-Sqrt[5/8+11/(8Sqrt[5])]},
 {Sqrt[9/16+9/(8Sqrt[5])],(-2-Sqrt[5])/4,Sqrt[1/8+1/(8Sqrt[5])]},
 {Sqrt[9/16+9/(8Sqrt[5])],(2+Sqrt[5])/4,Sqrt[1/8+1/(8Sqrt[5])]},
 {-Sqrt[29/32+61/(32Sqrt[5])],(-3-Sqrt[5])/8,-Sqrt[1/8+1/(8Sqrt[5])]},
 {-Sqrt[29/32+61/(32Sqrt[5])],(3+Sqrt[5])/8,-Sqrt[1/8+1/(8Sqrt[5])]},
 {Sqrt[29/32+61/(32Sqrt[5])],(-3-Sqrt[5])/8,Sqrt[1/8+1/(8Sqrt[5])]},
 {Sqrt[29/32+61/(32Sqrt[5])],(3+Sqrt[5])/8,Sqrt[1/8+1/(8Sqrt[5])]},
 {-Sqrt[17/16+19/(8Sqrt[5])],-1/4,Sqrt[1/8+1/(8Sqrt[5])]},
 {-Sqrt[17/16+19/(8Sqrt[5])],1/4,Sqrt[1/8+1/(8Sqrt[5])]},
 {Sqrt[17/16+19/(8Sqrt[5])],-1/4,-Sqrt[1/8+1/(8Sqrt[5])]},
 {Sqrt[17/16+19/(8Sqrt[5])],1/4,-Sqrt[1/8+1/(8Sqrt[5])]},
 {-Sqrt[5/32+Sqrt[5]/32],(-5-3Sqrt[5])/8,0},
 {-Sqrt[5/32+Sqrt[5]/32],(5+3Sqrt[5])/8,0},{Sqrt[5/32+Sqrt[5]/32],
  (-5-3Sqrt[5])/8,0},{Sqrt[5/32+Sqrt[5]/32],(5+3Sqrt[5])/8,0},
 {-Sqrt[25/32+(11Sqrt[5])/32],(-5-Sqrt[5])/8,0},
 {-Sqrt[25/32+(11Sqrt[5])/32],(5+Sqrt[5])/8,0},
 {Sqrt[25/32+(11Sqrt[5])/32],(-5-Sqrt[5])/8,0},
 {Sqrt[25/32+(11Sqrt[5])/32],(5+Sqrt[5])/8,0},{-Sqrt[5/4+Sqrt[5]/2],0,0},
 {Sqrt[5/4+Sqrt[5]/2],0,0}}

(* Icosahedron 6-Compound *)

PolyhedronName[Icosahedron6Compound]:="6-icosahedron compound"
PolyhedronDual[Icosahedron6Compound]:=Dodecahedron6Compound
PolyhedronPolyhedra[Icosahedron6Compound]:=
{{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20},{21,22,23,24,25,26,27,
    28,29,30,31,32,33,34,35,36,37,38,39,40},{41,42,43,44,45,46,47,48,49,50,51,
    52,53,54,55,56,57,58,59,60},{61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,
    76,77,78,79,80},{81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,
    100},{101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,
    118,119,120}}
PolyhedronFaces[Icosahedron6Compound]:=
{{2,52,24},{2,26,54},{55,19,1},{21,57,1},{24,52,19},{54,26,21},{19,55,24},{57,
    21,26},{1,19,69},{69,21,1},{2,24,72},{72,26,2},{2,54,52},{57,55,1},{52,69,
    19},{21,69,54},{72,24,55},{57,26,72},{52,54,69},{55,57,72},{15,43,23},{14,
    22,50},{14,40,31},{34,37,15},{34,22,37},{40,23,31},{15,37,43},{40,14,
    50},{31,23,43},{50,22,34},{14,66,22},{67,15,23},{37,22,66},{23,40,67},{66,
    14,31},{34,15,67},{66,43,37},{50,67,40},{43,66,31},{34,67,50},{4,35,
    39},{42,38,5},{8,39,35},{9,38,42},{59,9,8},{5,4,60},{5,70,4},{9,71,8},{35,
    59,8},{9,59,38},{60,4,39},{5,60,42},{4,70,35},{5,38,70},{71,39,8},{71,9,
    42},{59,35,70},{70,38,59},{39,71,60},{42,60,71},{25,46,18},{20,11,47},{11,
    32,41},{36,33,18},{36,20,33},{32,25,41},{18,46,36},{41,47,11},{25,32,
    46},{33,20,47},{20,65,11},{68,25,18},{36,65,20},{25,68,41},{32,11,65},{18,
    33,68},{36,46,65},{68,47,41},{65,46,32},{47,68,33},{53,10,17},{12,3,
    56},{10,45,30},{3,27,48},{30,45,12},{17,48,27},{27,53,17},{12,56,30},{10,
    53,45},{56,3,48},{12,61,3},{10,64,17},{3,61,27},{64,10,30},{12,45,61},{64,
    48,17},{53,27,61},{30,56,64},{61,45,53},{48,64,56},{7,51,16},{6,13,
    58},{29,44,7},{6,49,28},{29,13,44},{49,16,28},{28,16,51},{58,13,29},{7,44,
    51},{49,6,58},{6,62,13},{63,7,16},{62,6,28},{29,7,63},{44,13,62},{16,49,
    63},{51,62,28},{29,63,58},{62,51,44},{58,63,49}}
PolyhedronVertices[Icosahedron6Compound]:=
{{0,0,-Sqrt[5/4+Sqrt[5]/2]},{0,0,Sqrt[5/4+Sqrt[5]/2]},
 {-Sqrt[1/40-1/(40Sqrt[5])],(-15-7Sqrt[5])/20,-Sqrt[1/20-1/(10Sqrt[5])]},
 {-Sqrt[1/40-1/(40Sqrt[5])],(-1-Sqrt[5])/4,Sqrt[17/20+19/(10Sqrt[5])]},
 {-Sqrt[1/40-1/(40Sqrt[5])],(1+Sqrt[5])/4,Sqrt[17/20+19/(10Sqrt[5])]},
 {-Sqrt[1/40-1/(40Sqrt[5])],(15+7Sqrt[5])/20,-Sqrt[1/20-1/(10Sqrt[5])]},
 {Sqrt[1/40-1/(40Sqrt[5])],(-15-7Sqrt[5])/20,Sqrt[1/20-1/(10Sqrt[5])]},
 {Sqrt[1/40-1/(40Sqrt[5])],(-1-Sqrt[5])/4,-Sqrt[17/20+19/(10Sqrt[5])]},
 {Sqrt[1/40-1/(40Sqrt[5])],(1+Sqrt[5])/4,-Sqrt[17/20+19/(10Sqrt[5])]},
 {Sqrt[1/40-1/(40Sqrt[5])],(15+7Sqrt[5])/20,Sqrt[1/20-1/(10Sqrt[5])]},
 {-Sqrt[1/10+1/(10Sqrt[5])],(-5-3Sqrt[5])/10,-Sqrt[9/20+9/(10Sqrt[5])]},
 {-Sqrt[1/10+1/(10Sqrt[5])],(-5-Sqrt[5])/10,Sqrt[17/20+19/(10Sqrt[5])]},
 {-Sqrt[1/10+1/(10Sqrt[5])],(5+Sqrt[5])/10,Sqrt[17/20+19/(10Sqrt[5])]},
 {-Sqrt[1/10+1/(10Sqrt[5])],(5+3Sqrt[5])/10,-Sqrt[9/20+9/(10Sqrt[5])]},
 {Sqrt[1/10+1/(10Sqrt[5])],(-5-3Sqrt[5])/10,Sqrt[9/20+9/(10Sqrt[5])]},
 {Sqrt[1/10+1/(10Sqrt[5])],(-5-Sqrt[5])/10,-Sqrt[17/20+19/(10Sqrt[5])]},
 {Sqrt[1/10+1/(10Sqrt[5])],(5+Sqrt[5])/10,-Sqrt[17/20+19/(10Sqrt[5])]},
 {Sqrt[1/10+1/(10Sqrt[5])],(5+3Sqrt[5])/10,Sqrt[9/20+9/(10Sqrt[5])]},
 {-Sqrt[1/8+1/(8Sqrt[5])],(-3-Sqrt[5])/4,-Sqrt[1/4+1/(2Sqrt[5])]},
 {-Sqrt[1/8+1/(8Sqrt[5])],(-3-Sqrt[5])/4,Sqrt[1/4+1/(2Sqrt[5])]},
 {-Sqrt[1/8+1/(8Sqrt[5])],(3+Sqrt[5])/4,-Sqrt[1/4+1/(2Sqrt[5])]},
 {-Sqrt[1/8+1/(8Sqrt[5])],(3+Sqrt[5])/4,Sqrt[1/4+1/(2Sqrt[5])]},
 {Sqrt[1/8+1/(8Sqrt[5])],(-3-Sqrt[5])/4,-Sqrt[1/4+1/(2Sqrt[5])]},
 {Sqrt[1/8+1/(8Sqrt[5])],(-3-Sqrt[5])/4,Sqrt[1/4+1/(2Sqrt[5])]},
 {Sqrt[1/8+1/(8Sqrt[5])],(3+Sqrt[5])/4,-Sqrt[1/4+1/(2Sqrt[5])]},
 {Sqrt[1/8+1/(8Sqrt[5])],(3+Sqrt[5])/4,Sqrt[1/4+1/(2Sqrt[5])]},
 {-Sqrt[9/40+9/(40Sqrt[5])],(-5-3Sqrt[5])/20,-Sqrt[17/20+19/(10Sqrt[5])]},
 {-Sqrt[9/40+9/(40Sqrt[5])],(5+3Sqrt[5])/20,-Sqrt[17/20+19/(10Sqrt[5])]},
 {Sqrt[9/40+9/(40Sqrt[5])],(-5-3Sqrt[5])/20,Sqrt[17/20+19/(10Sqrt[5])]},
 {Sqrt[9/40+9/(40Sqrt[5])],(5+3Sqrt[5])/20,Sqrt[17/20+19/(10Sqrt[5])]},
 {-Sqrt[13/40+19/(40Sqrt[5])],(-5-Sqrt[5])/20,-Sqrt[17/20+19/(10Sqrt[5])]},
 {-Sqrt[13/40+19/(40Sqrt[5])],(5+Sqrt[5])/20,-Sqrt[17/20+19/(10Sqrt[5])]},
 {Sqrt[13/40+19/(40Sqrt[5])],(-5-Sqrt[5])/20,Sqrt[17/20+19/(10Sqrt[5])]},
 {Sqrt[13/40+19/(40Sqrt[5])],(5+Sqrt[5])/20,Sqrt[17/20+19/(10Sqrt[5])]},
 {-Sqrt[13/40+29/(40Sqrt[5])],(-3-Sqrt[5])/4,-Sqrt[1/20-1/(10Sqrt[5])]},
 {-Sqrt[13/40+29/(40Sqrt[5])],(5-Sqrt[5])/20,Sqrt[17/20+19/(10Sqrt[5])]},
 {-Sqrt[13/40+29/(40Sqrt[5])],(-5+Sqrt[5])/20,Sqrt[17/20+19/(10Sqrt[5])]},
 {-Sqrt[13/40+29/(40Sqrt[5])],(3+Sqrt[5])/4,-Sqrt[1/20-1/(10Sqrt[5])]},
 {Sqrt[13/40+29/(40Sqrt[5])],(-3-Sqrt[5])/4,Sqrt[1/20-1/(10Sqrt[5])]},
 {Sqrt[13/40+29/(40Sqrt[5])],(5-Sqrt[5])/20,-Sqrt[17/20+19/(10Sqrt[5])]},
 {Sqrt[13/40+29/(40Sqrt[5])],(-5+Sqrt[5])/20,-Sqrt[17/20+19/(10Sqrt[5])]},
 {Sqrt[13/40+29/(40Sqrt[5])],(3+Sqrt[5])/4,Sqrt[1/20-1/(10Sqrt[5])]},
 {-Sqrt[1/2+11/(10Sqrt[5])],(-5-3Sqrt[5])/10,Sqrt[1/20-1/(10Sqrt[5])]},
 {-Sqrt[1/2+11/(10Sqrt[5])],(-5-Sqrt[5])/10,Sqrt[9/20+9/(10Sqrt[5])]},
 {-Sqrt[1/2+11/(10Sqrt[5])],(5+Sqrt[5])/10,Sqrt[9/20+9/(10Sqrt[5])]},
 {-Sqrt[1/2+11/(10Sqrt[5])],(5+3Sqrt[5])/10,Sqrt[1/20-1/(10Sqrt[5])]},
 {Sqrt[1/2+11/(10Sqrt[5])],(-5-3Sqrt[5])/10,-Sqrt[1/20-1/(10Sqrt[5])]},
 {Sqrt[1/2+11/(10Sqrt[5])],(-5-Sqrt[5])/10,-Sqrt[9/20+9/(10Sqrt[5])]},
 {Sqrt[1/2+11/(10Sqrt[5])],(5+Sqrt[5])/10,-Sqrt[9/20+9/(10Sqrt[5])]},
 {Sqrt[1/2+11/(10Sqrt[5])],(5+3Sqrt[5])/10,-Sqrt[1/20-1/(10Sqrt[5])]},
 {-Sqrt[5/8+11/(8Sqrt[5])],(-1-Sqrt[5])/4,-Sqrt[1/4+1/(2Sqrt[5])]},
 {-Sqrt[5/8+11/(8Sqrt[5])],(-1-Sqrt[5])/4,Sqrt[1/4+1/(2Sqrt[5])]},
 {-Sqrt[5/8+11/(8Sqrt[5])],(1+Sqrt[5])/4,-Sqrt[1/4+1/(2Sqrt[5])]},
 {-Sqrt[5/8+11/(8Sqrt[5])],(1+Sqrt[5])/4,Sqrt[1/4+1/(2Sqrt[5])]},
 {Sqrt[5/8+11/(8Sqrt[5])],(-1-Sqrt[5])/4,-Sqrt[1/4+1/(2Sqrt[5])]},
 {Sqrt[5/8+11/(8Sqrt[5])],(-1-Sqrt[5])/4,Sqrt[1/4+1/(2Sqrt[5])]},
 {Sqrt[5/8+11/(8Sqrt[5])],(1+Sqrt[5])/4,-Sqrt[1/4+1/(2Sqrt[5])]},
 {Sqrt[5/8+11/(8Sqrt[5])],(1+Sqrt[5])/4,Sqrt[1/4+1/(2Sqrt[5])]},
 {-Sqrt[4/5+8/(5Sqrt[5])],0,-Sqrt[9/20+9/(10Sqrt[5])]},
 {Sqrt[4/5+8/(5Sqrt[5])],0,Sqrt[9/20+9/(10Sqrt[5])]},
 {-Sqrt[41/40+89/(40Sqrt[5])],(-5-3Sqrt[5])/20,Sqrt[1/20-1/(10Sqrt[5])]},
 {-Sqrt[41/40+89/(40Sqrt[5])],(5+3Sqrt[5])/20,Sqrt[1/20-1/(10Sqrt[5])]},
 {Sqrt[41/40+89/(40Sqrt[5])],(-5-3Sqrt[5])/20,-Sqrt[1/20-1/(10Sqrt[5])]},
 {Sqrt[41/40+89/(40Sqrt[5])],(5+3Sqrt[5])/20,-Sqrt[1/20-1/(10Sqrt[5])]},
 {-Sqrt[9/8+99/(40Sqrt[5])],(-5-Sqrt[5])/20,-Sqrt[1/20-1/(10Sqrt[5])]},
 {-Sqrt[9/8+99/(40Sqrt[5])],(5+Sqrt[5])/20,-Sqrt[1/20-1/(10Sqrt[5])]},
 {Sqrt[9/8+99/(40Sqrt[5])],(-5-Sqrt[5])/20,Sqrt[1/20-1/(10Sqrt[5])]},
 {Sqrt[9/8+99/(40Sqrt[5])],(5+Sqrt[5])/20,Sqrt[1/20-1/(10Sqrt[5])]},
 {-Sqrt[(5+2Sqrt[5])/5],0,-Sqrt[1/4+1/(2Sqrt[5])]},
 {-Sqrt[(5+2Sqrt[5])/5],0,Sqrt[1/4+1/(2Sqrt[5])]},
 {Sqrt[(5+2Sqrt[5])/5],0,-Sqrt[1/4+1/(2Sqrt[5])]},
 {Sqrt[(5+2Sqrt[5])/5],0,Sqrt[1/4+1/(2Sqrt[5])]}}

(* Kepler-Poinsot Solids *)

KeplerPoinsotSolids:={GreatDodecahedron,GreatIcosahedron,
    GreatStellatedDodecahedron,SmallStellatedDodecahedron}

(* Octahedron *)

PolyhedronName[Octahedron]:="octahedron"
PolyhedronDual[Octahedron]:=Cube
NetFaces[Octahedron]:={{2,6,7},{2,7,3},{3,7,8},{7,10,8},{1,3,4},{3,8,4},{4,8,9},{4,9,5}}
NetVertices[Octahedron]:={{-5/(2Sqrt[3]),3/2},{-(1/Sqrt[3]),0},{-(1/Sqrt[3]),1},{-(1/Sqrt[3]),2},{-(1/Sqrt[3]),3},{1/(2Sqrt[3]),-1/2},{1/(2Sqrt[3]),1/2},{1/(2Sqrt[3]),3/2},{1/(2Sqrt[3]),5/2},{2/Sqrt[3],1}}
PolyhedronPolyhedra[Octahedron]:={Range[8]}
PolyhedronFaces[Octahedron]:=PolyhedronFaces[Octahedron,"C4Square"]
PolyhedronVertices[Octahedron]:=PolyhedronVertices[Octahedron,"C4Square"]

(* Octahedron C3 *)

PolyhedronName[Octahedron,"C3"]:="octahedron oriented with z-axis along a C3 \
symmetry axis"
PolyhedronDual[Octahedron,"C3"]:=Sequence[Cube,"C3"]
PolyhedronPolyhedra[Octahedron,"C3"]:={Range[8]}
PolyhedronFaces[Octahedron,"C3"]:={{4,1,5},{6,3,2},{5,3,6},{1,2,3},{4,6,2},{5,1,3},{1,4,2},{4,5,6}}
PolyhedronVertices[Octahedron,"C3"]:={{-(1/Sqrt[3]),0,-(1/Sqrt[6])},{-1/(2Sqrt[3]),-1/2,
  1/Sqrt[6]},{-1/(2Sqrt[3]),1/2,1/Sqrt[6]},
 {1/(2Sqrt[3]),-1/2,-(1/Sqrt[6])},{1/(2Sqrt[3]),1/2,-(1/Sqrt[6])},
 {1/Sqrt[3],0,1/Sqrt[6]}}

(* Octahedron C4 Diamond *)

PolyhedronName[Octahedron,"C4Diamond"]:="octahedron oriented with z-axis along a C4 \
symmetry axis and midplane projected onto the xy-plane a diamond"
(*
PolyhedronDual[Octahedron,"C4Diamond"]:=Sequence[Cube,"C4"]
*)
PolyhedronPolyhedra[Octahedron,"C4Diamond"]:={Range[8]}
PolyhedronFaces[Octahedron,"C4Diamond"]:=
{{2,6,4},{2,4,5},{2,5,3},{2,3,6},{6,3,1},{6,1,4},{1,3,5},{4,1,5}}
PolyhedronVertices[Octahedron,"C4Diamond"]:=
{{0,0,-1},{0,0,1},{0,-1,0},{0,1,0},{-1,0,0},{1,0,0}}/Sqrt[2]
 
(* Octahedron C4 Square *)

PolyhedronName[Octahedron,"C4Square"]:="octahedron oriented with z-axis along a C4 \
symmetry axis and midplane projected onto the xy-plane a square"
(*
PolyhedronDual[Octahedron,"C4Square"]:=Sequence[Cube,"C4"]
*)
PolyhedronPolyhedra[Octahedron,"C4Square"]:={Range[8]}
PolyhedronFaces[Octahedron,"C4Square"]:=
{{4,5,6},{4,6,2},{4,2,1},{4,1,5},{5,1,3},{5,3,6},{3,1,2},{6,3,2}}
PolyhedronVertices[Octahedron,"C4Square"]:=
{{-1/2,-1/2,0},{-1/2,1/2,0},{0,0,-(1/Sqrt[2])},
 {0,0,1/Sqrt[2]},{1/2,-1/2,0},{1/2,1/2,0}}

(* Octahedron 2-Compound C3 *)

PolyhedronName[Octahedron2Compound,"C3"]:="octahedron 2-compound"
PolyhedronDual[Octahedron2Compound,"C3"]:=Sequence[Cube2Compound,"C3"]
PolyhedronPolyhedra[Octahedron2Compound,"C3"]:={{1,2,3,4,5,6,7,8},{9,10,11,12,13,14,15,16}}
PolyhedronFaces[Octahedron2Compound,"C3"]:=
{{5,3,9},{8,2,12},{2,5,9},{3,8,12},{2,8,5},{3,5,8},{12,2,9},{9,3,12},{7,11,
    1},{4,6,10},{10,6,1},{11,7,4},{6,7,1},{6,4,7},{1,11,10},{11,4,10}}
PolyhedronVertices[Octahedron2Compound,"C3"]:=
{{0,-2Sqrt[2/3],-2/Sqrt[3]},{0,-2Sqrt[2/3],2/Sqrt[3]},{0,2Sqrt[2/3],-2/Sqrt[3]},
 {0,2Sqrt[2/3],2/Sqrt[3]},{-Sqrt[2],-Sqrt[2/3],-2/Sqrt[3]},{-Sqrt[2],-Sqrt[2/3],2/Sqrt[3]},
 {-Sqrt[2],Sqrt[2/3],-2/Sqrt[3]},{-Sqrt[2],Sqrt[2/3],2/Sqrt[3]},{Sqrt[2],-Sqrt[2/3],-2/Sqrt[3]},
 {Sqrt[2],-Sqrt[2/3],2/Sqrt[3]},{Sqrt[2],Sqrt[2/3],-2/Sqrt[3]},
 {Sqrt[2],Sqrt[2/3],2/Sqrt[3]}}/(2Sqrt[2])

(* Octahedron 2-Compound C4 *)

PolyhedronName[Octahedron2Compound,"C4"]:="octahedron 2-compound"
PolyhedronDual[Octahedron2Compound,"C4"]:=Sequence[Cube2Compound,"C4"]
NetPieces[Octahedron2Compound,"C4"]:={{15,{1,2,3}}}
NetFaces[Octahedron2Compound,"C4"]:={{1,3,4},{1,2,4},{4,5,3},{4,6,2}}
NetVertices[Octahedron2Compound,"C4"]:=
 {{0,0},{1/2-1/Sqrt[2],-Sqrt[3]/2},{1/2-1/Sqrt[2],Sqrt[3]/2},
 {1-1/Sqrt[2],0},{(-3(-2+Sqrt[2]))/4,-(Sqrt[3](-2+Sqrt[2]))/4},
 {(-3(-2+Sqrt[2]))/4,(Sqrt[3](-2+Sqrt[2]))/4}}
PolyhedronPolyhedra[Octahedron2Compound,"C4"]:={{1,2,3,4,5,6,7,8},{9,10,11,12,13,14,15,16}}
PolyhedronFaces[Octahedron2Compound,"C4"]:=
{{4,10,6},{4,6,9},{4,9,5},{4,5,10},{10,5,3},{10,3,6},{3,5,9},{6,3,9},{4,7,
    8},{4,8,2},{4,2,1},{4,1,7},{7,1,3},{7,3,8},{3,1,2},{8,3,2}}
PolyhedronVertices[Octahedron2Compound,"C4"]:=
{{-1/2,-1/2,0},{-1/2,1/2,0},{0,0,-(1/Sqrt[2])},{0,0,1/Sqrt[2]},{0,-(1/Sqrt[2]),0},
 {0,1/Sqrt[2],0},{1/2,-1/2,0},{1/2,1/2,0},{-(1/Sqrt[2]),0,0},{1/Sqrt[2],0,0}}
 
(* Octahedron 3-Compound *)

PolyhedronName[Octahedron3Compound]:="octahedron 3-compound"
PolyhedronDual[Octahedron3Compound]:=Cube3Compound
NetPieces[Octahedron3Compound]:={{6,{1,2,3,4}},{12,{5,6,7,8}}}
NetFaces[Octahedron3Compound]:=
{{9,1,13,3,9},{13,1,14,11,13},{14,1,12,10,14},{12,1,8,2,12},{5,15,4},{5,16,
    4},{5,16,7},{5,15,6}}
NetVertices[Octahedron3Compound]:=
{{0,0},{0,-Sqrt[9/4-3/Sqrt[2]]},{0,Sqrt[9/4-3/Sqrt[2]]},{1,0},{3/2,0},{7/4,-Sqrt[3]/4},
 {7/4,Sqrt[3]/4},{1-Sqrt[2],-Sqrt[3(3-2Sqrt[2])]},{1-Sqrt[2],Sqrt[3(3-2Sqrt[2])]},
 {(3(-1+Sqrt[2]))/4,-Sqrt[9/16-3/(4Sqrt[2])]},
 {(3(-1+Sqrt[2]))/4,Sqrt[9/16-3/(4Sqrt[2])]},{-1+Sqrt[2],-Sqrt[3(3-2Sqrt[2])]},
 {-1+Sqrt[2],Sqrt[3(3-2Sqrt[2])]},{2(-1+Sqrt[2]),0},
 {(1+Sqrt[2])/2,-Sqrt[(3(3-2Sqrt[2]))/2]},{(1+Sqrt[2])/2,Sqrt[(3(3-2Sqrt[2]))/2]}}/Sqrt[2]
PolyhedronPolyhedra[Octahedron3Compound]:=
{{1,2,3,4,5,6,7,8},{9,10,11,12,13,14,15,16},{17,18,19,20,21,22,23,24}}
PolyhedronFaces[Octahedron3Compound]:=
{{6,1,8},{9,1,7},{1,6,7},{9,8,1},{6,8,10},{7,10,9},{7,6,10},{9,10,8},{11,15,
    2},{2,16,12},{15,11,5},{12,16,5},{12,11,2},{12,5,11},{2,15,16},{15,5,
    16},{13,3,17},{13,17,4},{3,14,18},{4,18,14},{14,3,13},{4,14,13},{3,18,
    17},{4,17,18}}
PolyhedronVertices[Octahedron3Compound]:=
{{-1,0,0},{0,-1,0},{0,0,-1},{0,0,1},{0,1,0},
 {0,-(1/Sqrt[2]),-(1/Sqrt[2])},{0,-(1/Sqrt[2]),
  1/Sqrt[2]},{0,1/Sqrt[2],-(1/Sqrt[2])},
 {0,1/Sqrt[2],1/Sqrt[2]},{1,0,0},
 {-(1/Sqrt[2]),0,-(1/Sqrt[2])},{-(1/Sqrt[2]),0,
  1/Sqrt[2]},{-(1/Sqrt[2]),-(1/Sqrt[2]),0},
 {-(1/Sqrt[2]),1/Sqrt[2],0},{1/Sqrt[2],0,
  -(1/Sqrt[2])},{1/Sqrt[2],0,1/Sqrt[2]},
 {1/Sqrt[2],-(1/Sqrt[2]),0},{1/Sqrt[2],1/Sqrt[2],0}}/Sqrt[2]

(* Octahedron 4-Compound *)

PolyhedronName[Octahedron4Compound]:="octahedron 4-compound"
PolyhedronDual[Octahedron4Compound]:=Cube4Compound
NetPieces[Octahedron4Compound]:={{24,{1,2}},{24,{3,4,5,6}}}
NetFaces[Octahedron4Compound]:=
{{1,4,5},{1,3,5},{2,8,9},{2,8,7,6},{9,8,11,12},{8,10,13,11}}
NetVertices[Octahedron4Compound]:=
{{0,0},{3/4,-1/4},{1/(3Sqrt[2]),-Sqrt[3/2]/5},{1/(3Sqrt[2]),1/(5Sqrt[6])},{Sqrt[2]/3,0},
 {(45-8Sqrt[2])/60,(-15+4Sqrt[6])/60},{(45-2Sqrt[2])/60,(-3+2Sqrt[6])/12},
 {(9+2Sqrt[2])/12,(-3+2Sqrt[6])/12},{(9+4Sqrt[2])/12,-1/4},
 {(45+16Sqrt[2])/60,(-15+16Sqrt[6])/60},{(45+22Sqrt[2])/60,(-3+2Sqrt[6])/12},
 {(45+28Sqrt[2])/60,(-15+4Sqrt[6])/60},{(63+32Sqrt[2])/84,(-21+20Sqrt[6])/84}}/Sqrt[2]
PolyhedronPolyhedra[Octahedron4Compound]:=
{{1,2,3,4,5,6,7,8},{9,10,11,12,13,14,15,16},{17,18,19,20,21,22,23,24},{25,26,
    27,28,29,30,31,32}}
PolyhedronFaces[Octahedron4Compound]:=
{{12,5,2},{5,13,2},{2,20,12},{12,23,5},{2,13,20},{13,5,23},{12,20,23},{13,23,
    20},{8,3,10},{15,3,8},{3,17,10},{8,10,22},{3,15,17},{22,15,8},{22,10,
    17},{22,17,15},{4,7,9},{4,16,7},{4,9,18},{21,9,7},{18,16,4},{21,7,16},{9,
    21,18},{18,21,16},{11,1,6},{14,6,1},{11,19,1},{24,11,6},{14,1,19},{24,6,
    14},{19,11,24},{14,19,24}}
PolyhedronVertices[Octahedron4Compound]:=
{{-2/3,-2/3,-1/3},{-2/3,-2/3,1/3},
 {-2/3,-1/3,-2/3},{-2/3,-1/3,2/3},{-2/3,1/3,-2/3},
 {-2/3,1/3,2/3},{-2/3,2/3,-1/3},{-2/3,2/3,1/3},
 {-1/3,-2/3,-2/3},{-1/3,-2/3,2/3},{-1/3,2/3,-2/3},
 {-1/3,2/3,2/3},{1/3,-2/3,-2/3},{1/3,-2/3,2/3},
 {1/3,2/3,-2/3},{1/3,2/3,2/3},{2/3,-2/3,-1/3},
 {2/3,-2/3,1/3},{2/3,-1/3,-2/3},{2/3,-1/3,2/3},
 {2/3,1/3,-2/3},{2/3,1/3,2/3},{2/3,2/3,-1/3},
 {2/3,2/3,1/3}}

(* Octahedron 5-Compound *)

PolyhedronName[Octahedron5Compound]:="octahedron 5-compound"
PolyhedronDual[Octahedron5Compound]:=Cube5Compound
NetPieces[Octahedron5Compound]:={{30,{1,2,3,4}}}
NetFaces[Octahedron5Compound]:={{2,6,1},{2,5,1},{4,6,1},{3,5,1}}
NetVertices[Octahedron5Compound]:=
{{0,0},{0,Sqrt[2/5]},{-Sqrt[3/10],-(1/Sqrt[10])},{Sqrt[3/10],-(1/Sqrt[10])},
 {-Sqrt[9/4-(3Sqrt[5])/4],Sqrt[3/4-Sqrt[5]/4]},
 {Sqrt[9/4-(3Sqrt[5])/4],Sqrt[3/4-Sqrt[5]/4]}}/Sqrt[3+Sqrt[5]]
PolyhedronPolyhedra[Octahedron5Compound]:=
{{1,2,3,4,5,6,7,8},{9,10,11,12,13,14,15,16},{17,18,19,20,21,22,23,24},{25,26,
    27,28,29,30,31,32},{33,34,35,36,37,38,39,40}}
PolyhedronFaces[Octahedron5Compound]:=
{{1,11,17},{24,11,1},{17,14,1},{1,14,24},{17,11,2},{2,11,24},{2,14,17},{24,14,
    2},{3,5,20},{23,5,3},{3,20,28},{28,23,3},{21,6,4},{4,6,22},{27,21,4},{4,
    22,27},{30,20,5},{5,23,30},{6,21,29},{29,22,6},{7,9,16},{18,9,7},{7,16,
    26},{26,18,7},{15,10,8},{8,10,19},{25,15,8},{8,19,25},{9,13,16},{18,13,
    9},{15,12,10},{10,12,19},{12,15,25},{25,19,12},{26,16,13},{13,18,26},{30,
    28,20},{21,27,29},{29,27,22},{23,28,30}}
PolyhedronVertices[Octahedron5Compound]:=
	{{0,(-1-Sqrt[5])/2,0},{0,(1+Sqrt[5])/2,0},
 {Sqrt[1/8-1/(8Sqrt[5])],(-1-Sqrt[5])/4,
  -Sqrt[1+2/Sqrt[5]]},{Sqrt[1/8-1/(8Sqrt[5])],
  (1+Sqrt[5])/4,-Sqrt[1+2/Sqrt[5]]},
 {Sqrt[1/8+1/(8Sqrt[5])],(-3-Sqrt[5])/4,
  Sqrt[(5+Sqrt[5])/10]},{Sqrt[1/8+1/(8Sqrt[5])],
  (3+Sqrt[5])/4,Sqrt[(5+Sqrt[5])/10]},
 {Sqrt[1/4+1/(2Sqrt[5])],-1/2,Sqrt[1+2/Sqrt[5]]},
 {Sqrt[1/4+1/(2Sqrt[5])],1/2,Sqrt[1+2/Sqrt[5]]},
 {Sqrt[5/8+11/(8Sqrt[5])],(-1-Sqrt[5])/4,
  Root[1-5#1^2+5#1^4&,1,0]},
 {Sqrt[5/8+11/(8Sqrt[5])],(1+Sqrt[5])/4,
  Root[1-5#1^2+5#1^4&,1,0]},
 {-Sqrt[1+2/Sqrt[5]],0,Root[1-5#1^2+5#1^4&,1,
   0]},{-Sqrt[1+2/Sqrt[5]]/2,-1/2,
  -Sqrt[1+2/Sqrt[5]]},{-Sqrt[1+2/Sqrt[5]]/2,1/2,
  -Sqrt[1+2/Sqrt[5]]},{Sqrt[1+2/Sqrt[5]],0,
  Sqrt[(5+Sqrt[5])/10]},{Sqrt[5/8+Sqrt[5]/8],
  -(1+Sqrt[5])^2/8,0},{Sqrt[5/8+Sqrt[5]/8],
  (3+Sqrt[5])/4,0},{Sqrt[(5+Sqrt[5])/10],0,
  -Sqrt[1+2/Sqrt[5]]},{-Sqrt[(5+Sqrt[5])/2]/2,
  -(1+Sqrt[5])^2/8,0},{-Sqrt[(5+Sqrt[5])/2]/2,
  (3+Sqrt[5])/4,0},{-Sqrt[5+2Sqrt[5]]/2,-1/2,0},
 {-Sqrt[5+2Sqrt[5]]/2,1/2,0},{Sqrt[5+2Sqrt[5]]/2,
  -1/2,0},{Sqrt[5+2Sqrt[5]]/2,1/2,0},
 {Root[1-5#1^2+5#1^4&,1,0],0,
  Sqrt[1+2/Sqrt[5]]},{Root[1-100#1^2+80#1^4&,1,
   0],(-1-Sqrt[5])/4,Sqrt[(5+Sqrt[5])/10]},
 {Root[1-100#1^2+80#1^4&,1,0],(1+Sqrt[5])/4,
  Sqrt[(5+Sqrt[5])/10]},
 {Root[1-20#1^2+80#1^4&,1,0],(-3-Sqrt[5])/4,
  Root[1-5#1^2+5#1^4&,1,0]},
 {Root[1-20#1^2+80#1^4&,1,0],(3+Sqrt[5])/4,
  Root[1-5#1^2+5#1^4&,1,0]},
 {Root[1-20#1^2+80#1^4&,2,0],(-1-Sqrt[5])/4,
  Sqrt[1+2/Sqrt[5]]},{Root[1-20#1^2+80#1^4&,2,
   0],(1+Sqrt[5])/4,Sqrt[1+2/Sqrt[5]]}}/Sqrt[3+Sqrt[5]]

(* Octahedron 6-Compound *)

(* Octahedron 10-Compound *)

PolyhedronName[Octahedron10Compound]:="octahedron 10-compound"
PolyhedronDual[Octahedron10Compound]:="cube 10-compound"
PolyhedronPolyhedra[Octahedron10Compound]:=Partition[Range[8 10],8]
PolyhedronFaces[Octahedron10Compound]:=
{{15,33,22},{32,18,27},{32,27,15},{22,33,18},{22,18,32},{33,15,27},{32,15,
    22},{33,27,18},{48,46,38},{59,57,43},{46,48,43},{38,46,57},{59,48,38},{38,
    57,59},{46,43,57},{59,43,48},{54,53,41},{52,51,40},{41,51,52},{53,40,
    51},{52,40,54},{53,54,40},{41,53,51},{52,54,41},{42,55,56},{50,39,49},{50,
    49,42},{55,49,39},{56,39,50},{56,55,39},{55,42,49},{50,42,56},{34,16,
    21},{17,31,28},{16,28,31},{17,34,21},{21,31,17},{16,34,28},{21,16,31},{17,
    28,34},{47,37,45},{44,58,60},{44,47,45},{45,37,58},{37,47,60},{37,60,
    58},{45,58,44},{44,60,47},{12,4,8},{1,5,9},{4,9,5},{1,12,8},{8,5,1},{8,4,
    5},{1,9,12},{4,12,9},{35,13,24},{20,30,25},{25,30,13},{35,24,20},{30,20,
    24},{25,13,35},{30,24,13},{35,20,25},{3,11,7},{6,2,10},{6,10,3},{7,11,
    2},{7,2,6},{6,3,7},{11,10,2},{11,3,10},{23,14,36},{19,26,29},{29,26,
    14},{23,36,19},{23,19,29},{26,36,14},{29,14,23},{26,19,36}}
PolyhedronVertices[Octahedron10Compound]:=
{{0,Root[-1+6#1-7#1^2-6#1^3+9#1^4&,2,0],
  Root[-1+6#1-7#1^2-6#1^3+9#1^4&,1,0]},
 {0,Root[-1+6#1-7#1^2-6#1^3+9#1^4&,2,0],
  Root[-1-6#1-7#1^2+6#1^3+9#1^4&,4,0]},
 {0,Root[-1-6#1-7#1^2+6#1^3+9#1^4&,3,0],
  Root[-1+6#1-7#1^2-6#1^3+9#1^4&,1,0]},
 {0,Root[-1-6#1-7#1^2+6#1^3+9#1^4&,3,0],
  Root[-1-6#1-7#1^2+6#1^3+9#1^4&,4,0]},
 {-(1/Sqrt[2]),Root[-1+8#1-10#1^2-24#1^3+36#1^4&,4,0],
  Root[-1+8#1-10#1^2-24#1^3+36#1^4&,2,0]},
 {-(1/Sqrt[2]),Root[-1+8#1-10#1^2-24#1^3+36#1^4&,4,0],
  Root[-1-8#1-10#1^2+24#1^3+36#1^4&,3,0]},
 {-(1/Sqrt[2]),Root[-1-8#1-10#1^2+24#1^3+36#1^4&,1,0],
  Root[-1+8#1-10#1^2-24#1^3+36#1^4&,2,0]},
 {-(1/Sqrt[2]),Root[-1-8#1-10#1^2+24#1^3+36#1^4&,1,0],
  Root[-1-8#1-10#1^2+24#1^3+36#1^4&,3,0]},
 {1/Sqrt[2],Root[-1+8#1-10#1^2-24#1^3+36#1^4&,4,0],
  Root[-1+8#1-10#1^2-24#1^3+36#1^4&,2,0]},
 {1/Sqrt[2],Root[-1+8#1-10#1^2-24#1^3+36#1^4&,4,0],
  Root[-1-8#1-10#1^2+24#1^3+36#1^4&,3,0]},
 {1/Sqrt[2],Root[-1-8#1-10#1^2+24#1^3+36#1^4&,1,0],
  Root[-1+8#1-10#1^2-24#1^3+36#1^4&,2,0]},
 {1/Sqrt[2],Root[-1-8#1-10#1^2+24#1^3+36#1^4&,1,0],
  Root[-1-8#1-10#1^2+24#1^3+36#1^4&,3,0]},
 {(-2-Sqrt[10])/6,(2-Sqrt[7+3Sqrt[5]])/6,(-2-Sqrt[7-3Sqrt[5]])/6},
 {(-2-Sqrt[10])/6,(2-Sqrt[7+3Sqrt[5]])/6,(2+Sqrt[7-3Sqrt[5]])/6},
 {(-2-Sqrt[10])/6,(-2+Sqrt[7+3Sqrt[5]])/6,(-2-Sqrt[7-3Sqrt[5]])/6},
 {(-2-Sqrt[10])/6,(-2+Sqrt[7+3Sqrt[5]])/6,(2+Sqrt[7-3Sqrt[5]])/6},
 {(2+Sqrt[10])/6,(2-Sqrt[7+3Sqrt[5]])/6,(-2-Sqrt[7-3Sqrt[5]])/6},
 {(2+Sqrt[10])/6,(2-Sqrt[7+3Sqrt[5]])/6,(2+Sqrt[7-3Sqrt[5]])/6},
 {(2+Sqrt[10])/6,(-2+Sqrt[7+3Sqrt[5]])/6,(-2-Sqrt[7-3Sqrt[5]])/6},
 {(2+Sqrt[10])/6,(-2+Sqrt[7+3Sqrt[5]])/6,(2+Sqrt[7-3Sqrt[5]])/6},
 {(-2-Sqrt[7-3Sqrt[5]])/6,(-2-Sqrt[10])/6,(2-Sqrt[7+3Sqrt[5]])/6},
 {(-2-Sqrt[7-3Sqrt[5]])/6,(-2-Sqrt[10])/6,(-2+Sqrt[7+3Sqrt[5]])/6},
 {(-2-Sqrt[7-3Sqrt[5]])/6,(2+Sqrt[10])/6,(2-Sqrt[7+3Sqrt[5]])/6},
 {(-2-Sqrt[7-3Sqrt[5]])/6,(2+Sqrt[10])/6,(-2+Sqrt[7+3Sqrt[5]])/6},
 {(2+Sqrt[7-3Sqrt[5]])/6,(-2-Sqrt[10])/6,(2-Sqrt[7+3Sqrt[5]])/6},
 {(2+Sqrt[7-3Sqrt[5]])/6,(-2-Sqrt[10])/6,(-2+Sqrt[7+3Sqrt[5]])/6},
 {(2+Sqrt[7-3Sqrt[5]])/6,(2+Sqrt[10])/6,(2-Sqrt[7+3Sqrt[5]])/6},
 {(2+Sqrt[7-3Sqrt[5]])/6,(2+Sqrt[10])/6,(-2+Sqrt[7+3Sqrt[5]])/6},
 {(2-Sqrt[7+3Sqrt[5]])/6,(-2-Sqrt[7-3Sqrt[5]])/6,(-2-Sqrt[10])/6},
 {(2-Sqrt[7+3Sqrt[5]])/6,(-2-Sqrt[7-3Sqrt[5]])/6,(2+Sqrt[10])/6},
 {(2-Sqrt[7+3Sqrt[5]])/6,(2+Sqrt[7-3Sqrt[5]])/6,(-2-Sqrt[10])/6},
 {(2-Sqrt[7+3Sqrt[5]])/6,(2+Sqrt[7-3Sqrt[5]])/6,(2+Sqrt[10])/6},
 {(-2+Sqrt[7+3Sqrt[5]])/6,(-2-Sqrt[7-3Sqrt[5]])/6,(-2-Sqrt[10])/6},
 {(-2+Sqrt[7+3Sqrt[5]])/6,(-2-Sqrt[7-3Sqrt[5]])/6,(2+Sqrt[10])/6},
 {(-2+Sqrt[7+3Sqrt[5]])/6,(2+Sqrt[7-3Sqrt[5]])/6,(-2-Sqrt[10])/6},
 {(-2+Sqrt[7+3Sqrt[5]])/6,(2+Sqrt[7-3Sqrt[5]])/6,(2+Sqrt[10])/6},
 {Root[-1+6#1-7#1^2-6#1^3+9#1^4&,1,0],0,
  Root[-1+6#1-7#1^2-6#1^3+9#1^4&,2,0]},
 {Root[-1+6#1-7#1^2-6#1^3+9#1^4&,1,0],0,
  Root[-1-6#1-7#1^2+6#1^3+9#1^4&,3,0]},
 {Root[-1+6#1-7#1^2-6#1^3+9#1^4&,2,0],
  Root[-1+6#1-7#1^2-6#1^3+9#1^4&,1,0],0},
 {Root[-1+6#1-7#1^2-6#1^3+9#1^4&,2,0],
  Root[-1-6#1-7#1^2+6#1^3+9#1^4&,4,0],0},
 {Root[-1-6#1-7#1^2+6#1^3+9#1^4&,3,0],
  Root[-1+6#1-7#1^2-6#1^3+9#1^4&,1,0],0},
 {Root[-1-6#1-7#1^2+6#1^3+9#1^4&,3,0],
  Root[-1-6#1-7#1^2+6#1^3+9#1^4&,4,0],0},
 {Root[-1-6#1-7#1^2+6#1^3+9#1^4&,4,0],0,
  Root[-1+6#1-7#1^2-6#1^3+9#1^4&,2,0]},
 {Root[-1-6#1-7#1^2+6#1^3+9#1^4&,4,0],0,
  Root[-1-6#1-7#1^2+6#1^3+9#1^4&,3,0]},
 {Root[-1+8#1-10#1^2-24#1^3+36#1^4&,2,0],-(1/Sqrt[2]),
  Root[-1+8#1-10#1^2-24#1^3+36#1^4&,4,0]},
 {Root[-1+8#1-10#1^2-24#1^3+36#1^4&,2,0],-(1/Sqrt[2]),
  Root[-1-8#1-10#1^2+24#1^3+36#1^4&,1,0]},
 {Root[-1+8#1-10#1^2-24#1^3+36#1^4&,2,0],1/Sqrt[2],
  Root[-1+8#1-10#1^2-24#1^3+36#1^4&,4,0]},
 {Root[-1+8#1-10#1^2-24#1^3+36#1^4&,2,0],1/Sqrt[2],
  Root[-1-8#1-10#1^2+24#1^3+36#1^4&,1,0]},
 {Root[-1+8#1-10#1^2-24#1^3+36#1^4&,4,0],
  Root[-1+8#1-10#1^2-24#1^3+36#1^4&,2,0],-(1/Sqrt[2])},
 {Root[-1+8#1-10#1^2-24#1^3+36#1^4&,4,0],
  Root[-1+8#1-10#1^2-24#1^3+36#1^4&,2,0],1/Sqrt[2]},
 {Root[-1+8#1-10#1^2-24#1^3+36#1^4&,4,0],
  Root[-1-8#1-10#1^2+24#1^3+36#1^4&,3,0],-(1/Sqrt[2])},
 {Root[-1+8#1-10#1^2-24#1^3+36#1^4&,4,0],
  Root[-1-8#1-10#1^2+24#1^3+36#1^4&,3,0],1/Sqrt[2]},
 {Root[-1-8#1-10#1^2+24#1^3+36#1^4&,1,0],
  Root[-1+8#1-10#1^2-24#1^3+36#1^4&,2,0],-(1/Sqrt[2])},
 {Root[-1-8#1-10#1^2+24#1^3+36#1^4&,1,0],
  Root[-1+8#1-10#1^2-24#1^3+36#1^4&,2,0],1/Sqrt[2]},
 {Root[-1-8#1-10#1^2+24#1^3+36#1^4&,1,0],
  Root[-1-8#1-10#1^2+24#1^3+36#1^4&,3,0],-(1/Sqrt[2])},
 {Root[-1-8#1-10#1^2+24#1^3+36#1^4&,1,0],
  Root[-1-8#1-10#1^2+24#1^3+36#1^4&,3,0],1/Sqrt[2]},
 {Root[-1-8#1-10#1^2+24#1^3+36#1^4&,3,0],-(1/Sqrt[2]),
  Root[-1+8#1-10#1^2-24#1^3+36#1^4&,4,0]},
 {Root[-1-8#1-10#1^2+24#1^3+36#1^4&,3,0],-(1/Sqrt[2]),
  Root[-1-8#1-10#1^2+24#1^3+36#1^4&,1,0]},
 {Root[-1-8#1-10#1^2+24#1^3+36#1^4&,3,0],1/Sqrt[2],
  Root[-1+8#1-10#1^2-24#1^3+36#1^4&,4,0]},
 {Root[-1-8#1-10#1^2+24#1^3+36#1^4&,3,0],1/Sqrt[2],
  Root[-1-8#1-10#1^2+24#1^3+36#1^4&,1,0]}}

(* Platonic *)

Platonic[1]:=Cube
Platonic[2]:=Dodecahedron
Platonic[3]:=Icosahedron
Platonic[4]:=Octahedron
Platonic[5]:=Tetrahedron

Platonics:=Platonic/@Range[5]

Inradius[Cube]:=1/2
Midradius[Cube]:=Sqrt[2]/2
Circumradius[Cube]:=Sqrt[3]/2

Inradius[Dodecahedron]:=Sqrt[250+110Sqrt[5]]/20
Midradius[Dodecahedron]:=(3+Sqrt[5])/4
Circumradius[Dodecahedron]:=(Sqrt[15]+Sqrt[3])/4

Inradius[Icosahedron]:=(3Sqrt[3]+Sqrt[15])/12
Midradius[Icosahedron]:=(1+Sqrt[5])/4
Circumradius[Icosahedron]:=Sqrt[10+2Sqrt[5]]/4

Inradius[Octahedron]:=Sqrt[6]/6
Midradius[Octahedron]:=1/2
Circumradius[Octahedron]:=Sqrt[2]/2

Inradius[Tetrahedron]:=Sqrt[6]/12
Midradius[Tetrahedron]:=Sqrt[2]/4
Circumradius[Tetrahedron]:=Sqrt[6]/4

(* Small Stellated Dodecahedron *)

PolyhedronName[SmallStellatedDodecahedron]:="small stellated dodecahedron"
PolyhedronDual[SmallStellatedDodecahedron]:=GreatDodecahedron

PolyhedronFaces[SmallStellatedDodecahedron]:=
{{11,3,7,1,9},{12,10,1,8,3},{4,6,11,8,1},{4,1,7,12,5},{6,9,1,10,5},{8,3,2,6,
    9},{7,10,5,2,3},{4,9,11,2,5},{4,6,2,12,10},{8,7,12,2,11},{11,6,5,12,3},{4,
    10,7,8,9}}
PolyhedronVertices[SmallStellatedDodecahedron]:=
{{0,0,-5/Sqrt[50-10Sqrt[5]]},{0,0,5/Sqrt[50-10Sqrt[5]]},
 {-Sqrt[2/(5-Sqrt[5])],0,-(1/Sqrt[10-2Sqrt[5]])},
 {Sqrt[2/(5-Sqrt[5])],0,1/Sqrt[10-2Sqrt[5]]},
 {-(-1+Sqrt[5])/(2Sqrt[10-2Sqrt[5]]),-Sqrt[(5+Sqrt[5])/(5-Sqrt[5])]/2,
  -(1/Sqrt[10-2Sqrt[5]])},{-(-1+Sqrt[5])/(2Sqrt[10-2Sqrt[5]]),
  Sqrt[(5+Sqrt[5])/(5-Sqrt[5])]/2,-(1/Sqrt[10-2Sqrt[5]])},
 {(-1+Sqrt[5])/(2Sqrt[10-2Sqrt[5]]),-Sqrt[(5+Sqrt[5])/(5-Sqrt[5])]/2,
  1/Sqrt[10-2Sqrt[5]]},{(-1+Sqrt[5])/(2Sqrt[10-2Sqrt[5]]),
  Sqrt[(5+Sqrt[5])/(5-Sqrt[5])]/2,1/Sqrt[10-2Sqrt[5]]},
 {-(1+Sqrt[5])/(2Sqrt[10-2Sqrt[5]]),-1/2,1/Sqrt[10-2Sqrt[5]]},
 {-(1+Sqrt[5])/(2Sqrt[10-2Sqrt[5]]),1/2,1/Sqrt[10-2Sqrt[5]]},
 {(1+Sqrt[5])/(2Sqrt[10-2Sqrt[5]]),-1/2,-(1/Sqrt[10-2Sqrt[5]])},
 {(1+Sqrt[5])/(2Sqrt[10-2Sqrt[5]]),1/2,-(1/Sqrt[10-2Sqrt[5]])}}

PolyhedronFaces[SmallStellatedDodecahedron,Split->True]:=
{{20,19,10},{20,10,9},{20,9,18},{20,18,4},{20,4,19},{17,3,16},{17,16,12},{17,
    12,11},{17,11,15},{17,15,3},{28,15,11},{28,11,7},{28,7,25},{28,25,13},{28,
    13,15},{23,11,12},{23,12,8},{23,8,2},{23,2,7},{23,7,11},{27,12,16},{27,16,
    14},{27,14,26},{27,26,8},{27,8,12},{24,16,3},{24,3,1},{24,1,6},{24,6,
    14},{24,14,16},{21,3,15},{21,15,13},{21,13,5},{21,5,1},{21,1,3},{32,26,
    14},{32,14,6},{32,6,10},{32,10,19},{32,19,26},{22,6,1},{22,1,5},{22,5,
    9},{22,9,10},{22,10,6},{30,5,13},{30,13,25},{30,25,18},{30,18,9},{30,9,
    5},{31,25,7},{31,7,2},{31,2,4},{31,4,18},{31,18,25},{29,2,8},{29,8,
    26},{29,26,19},{29,19,4},{29,4,2}}
PolyhedronVertices[SmallStellatedDodecahedron,Split->True]:=
{{-Sqrt[1+2/Sqrt[5]],0,Root[1-20#1^2+80#1^4&,2,0]},{Sqrt[1+2/Sqrt[5]],0,Sqrt[1/8-1/(8Sqrt[5])]},
 {Root[1-5#1^2+5#1^4&,1,0],0,Root[1-100#1^2+80#1^4&,1,0]},
 {Sqrt[(5+Sqrt[5])/10],0,Sqrt[5/8+11/(8Sqrt[5])]},{Root[1-100#1^2+80#1^4&,1,0],(-1-Sqrt[5])/4,
  Sqrt[1/8-1/(8Sqrt[5])]},{Root[1-100#1^2+80#1^4&,1,0],(1+Sqrt[5])/4,Sqrt[1/8-1/(8Sqrt[5])]},
 {Sqrt[5/8+11/(8Sqrt[5])],(-1-Sqrt[5])/4,Root[1-20#1^2+80#1^4&,2,0]},
 {Sqrt[5/8+11/(8Sqrt[5])],(1+Sqrt[5])/4,Root[1-20#1^2+80#1^4&,2,0]},
 {-Sqrt[1+2/Sqrt[5]]/2,-1/2,Sqrt[5/8+11/(8Sqrt[5])]},{-Sqrt[1+2/Sqrt[5]]/2,1/2,Sqrt[5/8+11/(8Sqrt[5])]},
 {Sqrt[1/4+1/(2Sqrt[5])],-1/2,Root[1-100#1^2+80#1^4&,1,0]},
 {Sqrt[1/4+1/(2Sqrt[5])],1/2,Root[1-100#1^2+80#1^4&,1,0]},
 {Root[1-20#1^2+80#1^4&,1,0],(-3-Sqrt[5])/4,Root[1-20#1^2+80#1^4&,2,0]},
 {Root[1-20#1^2+80#1^4&,1,0],(3+Sqrt[5])/4,Root[1-20#1^2+80#1^4&,2,0]},
 {Root[1-20#1^2+80#1^4&,2,0],(-1-Sqrt[5])/4,Root[1-100#1^2+80#1^4&,1,0]},
 {Root[1-20#1^2+80#1^4&,2,0],(1+Sqrt[5])/4,Root[1-100#1^2+80#1^4&,1,0]},
 {0,0,Root[5-100#1^2+16#1^4&,1,0]},{Sqrt[1/8-1/(8Sqrt[5])],(-1-Sqrt[5])/4,
  Sqrt[5/8+11/(8Sqrt[5])]},{Sqrt[1/8-1/(8Sqrt[5])],(1+Sqrt[5])/4,Sqrt[5/8+11/(8Sqrt[5])]},
 {0,0,Sqrt[25/8+(11Sqrt[5])/8]},{Root[1-260#1^2+80#1^4&,1,0],(-3-Sqrt[5])/4,
  Root[1-100#1^2+80#1^4&,1,0]},{Root[1-25#1^2+5#1^4&,1,0],0,Sqrt[5/8+11/(8Sqrt[5])]},
 {Sqrt[5/2+11/(2Sqrt[5])],0,Root[1-100#1^2+80#1^4&,1,0]},
 {Root[1-260#1^2+80#1^4&,1,0],(3+Sqrt[5])/4,Root[1-100#1^2+80#1^4&,1,0]},
 {Sqrt[1/8+1/(8Sqrt[5])],(-3-Sqrt[5])/4,Sqrt[1/8-1/(8Sqrt[5])]},
 {Sqrt[1/8+1/(8Sqrt[5])],(3+Sqrt[5])/4,Sqrt[1/8-1/(8Sqrt[5])]},
 {Sqrt[1+2/Sqrt[5]]/2,(2+Sqrt[5])/2,Root[1-100#1^2+80#1^4&,1,0]},
 {Sqrt[1+2/Sqrt[5]]/2,-1-Sqrt[5]/2,Root[1-100#1^2+80#1^4&,1,0]},
 {Sqrt[13/8+29/(8Sqrt[5])],(3+Sqrt[5])/4,Sqrt[5/8+11/(8Sqrt[5])]},
 {-Sqrt[1+2/Sqrt[5]]/2,-1-Sqrt[5]/2,Sqrt[5/8+11/(8Sqrt[5])]},{Sqrt[13/8+29/(8Sqrt[5])],(-3-Sqrt[5])/4,
  Sqrt[5/8+11/(8Sqrt[5])]},{-Sqrt[1+2/Sqrt[5]]/2,(2+Sqrt[5])/2,Sqrt[5/8+11/(8Sqrt[5])]}}

(* Stella Octangula *)

PolyhedronName[StellaOctangula]:="stella octangula"

NetPieces[StellaOctangula]:={{8,{1,2,3}}}
NetFaces[StellaOctangula]:={{2,3,1},{2,4,3},{4,3,5}}
NetVertices[
    StellaOctangula]:={{-1,-(1/Sqrt[3])},{-1/2,
      1/(2Sqrt[3])},{0,-(1/Sqrt[3])},{1/2,1/(2Sqrt[3])},{1,-(1/Sqrt[3])}}

PolyhedronPolyhedra[StellaOctangula]:={{1,2,3,4},{5,6,7,8}}
PolyhedronFaces[StellaOctangula]:=
{{4,5,8},{5,4,2},{8,2,4},{2,8,5},{6,7,3},{7,6,1},{3,1,6},{1,3,7}}
PolyhedronVertices[StellaOctangula]:=
{{0,0,-Sqrt[3/2]/2},{0,0,Sqrt[3/2]/2},{-(1/Sqrt[3]),0,1/(2Sqrt[6])},
 {-1/(2Sqrt[3]),-1/2,-1/(2Sqrt[6])},{-1/(2Sqrt[3]),1/2,-1/(2Sqrt[6])},
 {1/(2Sqrt[3]),-1/2,1/(2Sqrt[6])},{1/(2Sqrt[3]),1/2,1/(2Sqrt[6])},
 {1/Sqrt[3],0,-1/(2Sqrt[6])}}

PolyhedronFaces[StellaOctangula,Split->True]:=
{{4,5,9},{6,10,5},{14,9,10},{5,10,1},{10,9,1},{9,5,1},{6,5,8},{4,7,5},{2,8,
    7},{5,7,3},{7,8,3},{8,5,3},{14,12,9},{2,7,12},{4,9,7},{12,7,11},{7,9,
    11},{9,12,11},{2,12,8},{14,10,12},{6,8,10},{12,10,13},{10,8,13},{8,12,13}}
PolyhedronVertices[StellaOctangula,Split->True]:=
{{0,0,-Sqrt[3/2]/2},{0,0,Sqrt[3/2]/2},{-(1/Sqrt[3]),0,1/(2Sqrt[6])},
 {-1/(2Sqrt[3]),-1/2,-1/(2Sqrt[6])},{-1/(2Sqrt[3]),0,-1/(2Sqrt[6])},
 {-1/(2Sqrt[3]),1/2,-1/(2Sqrt[6])},{-1/(4Sqrt[3]),-1/4,1/(2Sqrt[6])},
 {-1/(4Sqrt[3]),1/4,1/(2Sqrt[6])},{1/(4Sqrt[3]),-1/4,-1/(2Sqrt[6])},
 {1/(4Sqrt[3]),1/4,-1/(2Sqrt[6])},{1/(2Sqrt[3]),-1/2,1/(2Sqrt[6])},
 {1/(2Sqrt[3]),0,1/(2Sqrt[6])},{1/(2Sqrt[3]),1/2,1/(2Sqrt[6])},
 {1/Sqrt[3],0,-1/(2Sqrt[6])}}

(* Tetrahedron *)

PolyhedronName[Tetrahedron]:="tetrahedron"
PolyhedronDual[Tetrahedron]:=Tetrahedron
NetFaces[Tetrahedron]:={{2,3,1},{4,2,5},{2,5,3},{5,3,6}}
NetVertices[Tetrahedron]:={{-1,-(1/Sqrt[3])},{-1/2,1/(2Sqrt[3])},{0,-(1/Sqrt[3])},{0,2/Sqrt[3]},{1/2,1/(2Sqrt[3])},{1,-(1/Sqrt[3])}}
PolyhedronPolyhedra[Tetrahedron]:={Range[4]}
PolyhedronFaces[Tetrahedron]:=PolyhedronFaces[Tetrahedron,"C3"]
PolyhedronVertices[Tetrahedron]:=PolyhedronVertices[Tetrahedron,"C3"]

(* Tetrahedron C2 *)

PolyhedronName[Tetrahedron,"C2"]:="tetrahedron oriented with the z-axis along a \
C2 symmetry axis and top edge having slope -1 in the xy-plane."
PolyhedronDual[Tetrahedron,"C2"]:=Tetrahedron
PolyhedronPolyhedra[Tetrahedron,"C2"]:={Range[4]}
PolyhedronVertices[Tetrahedron,"C2"]:=
{{-1,-1,-1},{-1,1,1},{1,-1,1},{1,1,-1}}/(2Sqrt[2])
PolyhedronFaces[Tetrahedron,"C2"]:={{3,2,1},{1,2,4},{4,3,1},{2,3,4}}

(* Tetrahedron C3 *)

PolyhedronName[Tetrahedron,"C3"]:="tetrahedron oriented with the z-axis along a \
C3 symmetry axis and vertex pointing in the +z direction."
PolyhedronPolyhedra[Tetrahedron,"C3"]:={Range[4]}
PolyhedronVertices[Tetrahedron,"C3"]:=
{{0,0,Sqrt[2/3]-1/(2Sqrt[6])},{-1/(2Sqrt[3]),-1/2,
  -1/(2Sqrt[6])},{-1/(2Sqrt[3]),1/2,-1/(2Sqrt[6])},{1/Sqrt[3],0,-1/(2Sqrt[6])}}
PolyhedronFaces[Tetrahedron,"C3"]:=
{{2,3,4},{3,2,1},{4,1,2},{1,4,3}}

(* Tetrahedron 2-Compound *)

PolyhedronName[Tetrahedron2Compound,chiral_:1]:=
	"tetrahedron 2-compound (dextro)"
PolyhedronFaces[Tetrahedron2Compound,chiral_:1]=
{{3,4,6},{4,3,2},{6,2,3},{2,6,4},{8,7,5},{1,5,7},{5,1,8},{7,8,1}}
PolyhedronVertices[Tetrahedron2Compound,chiral_:1]=
{{0,0,-Sqrt[3/2]/2},{0,0,Sqrt[3/2]/2},{-1/(2Sqrt[3]),-1/2,-1/(2Sqrt[6])},
 {-1/(2Sqrt[3]),1/2,-1/(2Sqrt[6])},{-1/(4Sqrt[3]),-Sqrt[5]/4,1/(2Sqrt[6])},{1/Sqrt[3],0,-1/(2Sqrt[6])},
 {-Sqrt[(23-3Sqrt[5])/6]/4,(1+Sqrt[5])/8,1/(2Sqrt[6])},{Sqrt[23/96+Sqrt[5]/32],(-1+Sqrt[5])/8,
  1/(2Sqrt[6])}}

(* Tetrahedron 3-Compound *)

PolyhedronName[Tetrahedron3Compound]:="tetrahedron 3-compound"
PolyhedronPolyhedra[Tetrahedron3Compound]:=Partition[Range[3 4],4]
PolyhedronFaces[Tetrahedron3Compound]:={{3,2,1},{1,2,4},{4,3,1},{2,3,4},{9,12,6},{6,12,7},{7,9,6},{12,9,7},{8,5,
    11},{11,5,10},{10,8,11},{5,8,10}}
PolyhedronVertices[Tetrahedron3Compound]:=
{{-1/(2Sqrt[2]),-1/(2Sqrt[2]),-1/(2Sqrt[2])},
 {-1/(2Sqrt[2]),1/(2Sqrt[2]),1/(2Sqrt[2])},
 {1/(2Sqrt[2]),-1/(2Sqrt[2]),1/(2Sqrt[2])},
 {1/(2Sqrt[2]),1/(2Sqrt[2]),-1/(2Sqrt[2])},
 {-Sqrt[2-Sqrt[3]]/4,-Sqrt[1/8+Sqrt[3]/16],1/(2Sqrt[2])},
 {-Sqrt[2-Sqrt[3]]/4,Sqrt[1/8+Sqrt[3]/16],-1/(2Sqrt[2])},
 {Sqrt[2-Sqrt[3]]/4,-Sqrt[1/8+Sqrt[3]/16],-1/(2Sqrt[2])},
 {Sqrt[2-Sqrt[3]]/4,Sqrt[1/8+Sqrt[3]/16],1/(2Sqrt[2])},
 {-Sqrt[1/8+Sqrt[3]/16],-Sqrt[2-Sqrt[3]]/4,1/(2Sqrt[2])},
 {-Sqrt[1/8+Sqrt[3]/16],Sqrt[2-Sqrt[3]]/4,-1/(2Sqrt[2])},
 {Sqrt[1/8+Sqrt[3]/16],-Sqrt[2-Sqrt[3]]/4,-1/(2Sqrt[2])},
 {Sqrt[1/8+Sqrt[3]/16],Sqrt[2-Sqrt[3]]/4,1/(2Sqrt[2])}}

(* Tetrahedron 4-Compound *)

PolyhedronName[Tetrahedron4Compound,"C3"]:="tetrahedron rotation compound of \
4 tetrahedra by angle Pi/3."
NetPieces[Tetrahedron4Compound,"C3"]:=
 {{4,{1,2,3}},{12,{4,5,6}},{12,{7,8}},{12,{9,10}}}
NetFaces[Tetrahedron4Compound,"C3"]:=
 {{10,1,2,6,5,7,4,3},{10,3,8,9,11,12,13,18},{10,18,17,14,16,15,19,20},{24,27,
    26,25},{24,21,22,23},{23,24,25},{28,29,31},{28,30,31},{35,34,32},{35,33,
    32}}
NetVertices[Tetrahedron4Compound,"C3"]:=
{{-3/5,0},{-5/14,Sqrt[3]/14},{-3/10,(3Sqrt[3])/10},{-2/7,Sqrt[3]/7},{-9/32,(3Sqrt[3])/32},
 {-5/18,1/(6Sqrt[3])},{-2/9,1/(3Sqrt[3])},{-1/14,(3Sqrt[3])/14},{-1/18,1/(2Sqrt[3])},{0,0},
 {0,(3Sqrt[3])/16},{1/18,1/(2Sqrt[3])},{1/14,(3Sqrt[3])/14},{2/9,1/(3Sqrt[3])},{5/18,1/(6Sqrt[3])},
 {9/32,(3Sqrt[3])/32},{2/7,Sqrt[3]/7},{3/10,(3Sqrt[3])/10},{5/14,Sqrt[3]/14},{3/5,0},{23/30,1/(2Sqrt[3])},
 {37/42,1/(14Sqrt[3])},{1,0},{7/6,1/(2Sqrt[3])},{4/3,0},{61/42,1/(14Sqrt[3])},{47/30,1/(2Sqrt[3])},
 {15/8,1/8},{187/96,(4+Sqrt[3])/32},{331/168,(21-8Sqrt[3])/168},{143/72,1/8},{181/72,1/8},
 {425/168,(21-8Sqrt[3])/168},{245/96,(4+Sqrt[3])/32},{21/8,1/8}}

PolyhedronPolyhedra[Tetrahedron4Compound,"C3"]:=Partition[Range[4 4],4]
PolyhedronFaces[Tetrahedron4Compound,"C3"]:=
{{12,2,13},{2,12,1},{13,1,12},{1,13,2},{4,9,16},{9,4,8},{16,8,4},{8,16,9},{3,
    7,10},{7,3,15},{10,15,3},{15,10,7},{6,11,14},{11,6,5},{14,5,6},{5,14,11}}
PolyhedronVertices[Tetrahedron4Compound,"C3"]:=
{{0,0,Sqrt[3/2]/2},{-(1/Sqrt[3]),0,-1/(2Sqrt[6])},
 {-2/(3Sqrt[3]),-1/3,5/(6Sqrt[6])},{-2/(3Sqrt[3]),0,-7/(6Sqrt[6])},
 {-2/(3Sqrt[3]),1/3,5/(6Sqrt[6])},{-1/(2Sqrt[3]),-1/2,-1/(2Sqrt[6])},
 {-1/(2Sqrt[3]),1/2,-1/(2Sqrt[6])},{-1/(6Sqrt[3]),-1/2,5/(6Sqrt[6])},
 {-1/(6Sqrt[3]),1/2,5/(6Sqrt[6])},{1/(3Sqrt[3]),-1/3,-7/(6Sqrt[6])},
 {1/(3Sqrt[3]),1/3,-7/(6Sqrt[6])},{1/(2Sqrt[3]),-1/2,-1/(2Sqrt[6])},
 {1/(2Sqrt[3]),1/2,-1/(2Sqrt[6])},{5/(6Sqrt[3]),-1/6,5/(6Sqrt[6])},
 {5/(6Sqrt[3]),1/6,5/(6Sqrt[6])},{1/Sqrt[3],0,-1/(2Sqrt[6])}}

(* Tetrahedron 5-Compound *)

PolyhedronName[Tetrahedron5Compound,1]:=
	"tetrahedron 5-compound (dextro)"
NetPieces[Tetrahedron5Compound,1]:={{20,{1,2,3}}}
NetFaces[Tetrahedron5Compound,1]:=
{{1,7,9,10,4},{1,6,11,8,7},{1,3,5,2,6}}
NetVertices[Tetrahedron5Compound,1]:=
{{0,0},{-Sqrt[5/2]/2,Sqrt[3/2]/2},{-Sqrt[3-Sqrt[5]],0},{Sqrt[3-Sqrt[5]],0},
 {-Sqrt[15/16-(5Sqrt[5])/16],Sqrt[9/16-(3Sqrt[5])/16]},
 {-Sqrt[3/4-Sqrt[5]/4],Sqrt[9/4-(3Sqrt[5])/4]},
 {Sqrt[3/4-Sqrt[5]/4],Sqrt[9/4-(3Sqrt[5])/4]},
 {Sqrt[7/16-(3Sqrt[5])/16],Sqrt[9/16+(3Sqrt[5])/16]},
 {Sqrt[3/16+Sqrt[5]/16],Sqrt[21/16-(9Sqrt[5])/16]},
 {Sqrt[7/16+(3Sqrt[5])/16],Sqrt[9/16-(3Sqrt[5])/16]},
 {1/(2Sqrt[2(9+4Sqrt[5])]),Sqrt[3/2]/2}}
PolyhedronFaces[Tetrahedron5Compound,1]:=
{{17,11,1},{1,11,20},{20,17,1},{11,17,20},{14,9,2},{2,9,15},{2,14,15},{9,14,
    15},{3,8,10},{19,8,3},{3,10,19},{8,10,19},{12,6,4},{4,6,13},{4,12,13},{6,
    12,13},{16,7,5},{5,7,18},{18,16,5},{7,16,18}}
PolyhedronVertices[Tetrahedron5Compound,1]:=PolyhedronVertices[Dodecahedron]

PolyhedronName[Tetrahedron5Compound,-1]:=
	"tetrahedron 5-compound (laevo)"
PolyhedronFaces[Tetrahedron5Compound,-1]:=
{{1,12,18},{19,12,1},{1,18,19},{19,18,12},{13,10,2},{2,10,16},{2,13,16},{16,
    13,10},{3,7,9},{3,7,20},{3,9,20},{20,9,7},{11,5,4},{14,5,4},{14,11,4},{5,
    11,14},{15,8,6},{6,8,17},{17,15,6},{8,15,17}}
PolyhedronVertices[Tetrahedron5Compound,-1]:=PolyhedronVertices[Dodecahedron]
PolyhedronPolyhedra[Tetrahedron5Compound]:=
	{{1,2,3,4},{5,6,7,8},{9,10,11,12},{13,14,15,16},{17,18,19,20}}

(* Tetrahedron 6-Compound *)

PolyhedronName[Tetrahedron6Compound]:="tetrahedron 6-compound"
NetPieces[Tetrahedron6Compound]:={{24,{1,2,3}},{24,{4,5}}}
NetFaces[Tetrahedron6Compound]:=
{{9,10,8,2,6,7},{10,8,11,1},{2,8,3,12},{4,14,5},{4,13,5}}
NetVertices[Tetrahedron6Compound]:=
{{-1/2,0},{0,0},{0,Sqrt[9/8-3/(2Sqrt[2])]},{1/2,1/8},{1/2,(-3+2Sqrt[2])/8},
 {(-2+Sqrt[2])/6,-Sqrt[1/4-1/(3Sqrt[2])]},{(-2+Sqrt[2])/4,0},
 {(-2+Sqrt[2])/4,Sqrt[9/8-3/(2Sqrt[2])]},{(-2+Sqrt[2])/3,-Sqrt[1/4-1/(3Sqrt[2])]},
 {(-2+Sqrt[2])/2,0},{(-2+Sqrt[2])/2,Sqrt[9/8-3/(2Sqrt[2])]},{(-1+Sqrt[2])/2,0},
 {(6-Sqrt[6(3-2Sqrt[2])])/12,(-5+4Sqrt[2])/24},{(6+Sqrt[6(3-2Sqrt[2])])/12,
  (-5+4Sqrt[2])/24}}

PolyhedronPolyhedra[Tetrahedron6Compound]:=Partition[Range[6 4],4]
PolyhedronFaces[Tetrahedron6Compound]:=
{{6,8,1},{1,8,13},{13,6,1},{8,6,13},{2,14,7},{7,14,5},{5,2,7},{14,2,5},{4,16,
    9},{9,16,10},{10,4,9},{16,4,10},{11,12,15},{15,12,3},{3,11,15},{12,11,
    3},{22,23,17},{17,23,20},{20,22,17},{23,22,20},{21,24,19},{19,24,18},{18,
    21,19},{24,21,18}}
PolyhedronVertices[Tetrahedron6Compound]:=
{{-1/2,0,-1/(2Sqrt[2])},{-1/2,0,1/(2Sqrt[2])},
 {-1/2,-1/(2Sqrt[2]),0},{-1/2,1/(2Sqrt[2]),0},
 {0,-1/2,-1/(2Sqrt[2])},{0,-1/2,1/(2Sqrt[2])},
 {0,1/2,-1/(2Sqrt[2])},{0,1/2,1/(2Sqrt[2])},
 {0,-1/(2Sqrt[2]),-1/2},{0,-1/(2Sqrt[2]),1/2},
 {0,1/(2Sqrt[2]),-1/2},{0,1/(2Sqrt[2]),1/2},
 {1/2,0,-1/(2Sqrt[2])},{1/2,0,1/(2Sqrt[2])},
 {1/2,-1/(2Sqrt[2]),0},{1/2,1/(2Sqrt[2]),0},
 {-1/(2Sqrt[2]),-1/2,0},{-1/(2Sqrt[2]),0,-1/2},
 {-1/(2Sqrt[2]),0,1/2},{-1/(2Sqrt[2]),1/2,0},
 {1/(2Sqrt[2]),-1/2,0},{1/(2Sqrt[2]),0,-1/2},
 {1/(2Sqrt[2]),0,1/2},{1/(2Sqrt[2]),1/2,0}}

(* Tetrahedron 10-Compound *)

PolyhedronName[Tetrahedron10Compound]:="tetrahedron 10-compound"
PolyhedronFaces[Tetrahedron10Compound]:=
{{17,11,1},{1,11,20},{20,17,1},{11,17,20},{1,12,18},{19,12,1},{1,18,19},{19,
    18,12},{14,9,2},{2,9,15},{2,14,15},{9,14,15},{13,10,2},{2,10,16},{2,13,
    16},{16,13,10},{3,7,9},{3,7,20},{3,9,20},{20,9,7},{3,8,10},{19,8,3},{3,10,
    19},{8,10,19},{11,5,4},{14,5,4},{14,11,4},{5,11,14},{12,6,4},{4,6,13},{4,
    12,13},{6,12,13},{16,7,5},{5,7,18},{18,16,5},{7,16,18},{15,8,6},{6,8,
    17},{17,15,6},{8,15,17}}
PolyhedronVertices[Tetrahedron10Compound]:=PolyhedronVertices[Dodecahedron]


End[]

(* Protect[  ] *)

EndPackage[]

Archimideans

(*:Mathematica Version: 5.2 *)

(*:Package Version: 1.47 *)

(*:Name: MathWorld`Archimedean` *)

(*:Author: Eric W. Weisstein *)

(*:URL:
  http://mathworld.wolfram.com/packages/Archimedean.m
*)

(*:Summary:
*)

(*:History:
  v1.00 (1999-07-25): Put in package format
  v1.02 (1999-07-26): Added factor of 1/2 to TruncatedCube,
                      1/(4Sqrt[3]) to TruncatedOctahedron, and
                      9/5 to TruncatedTetrahedron
                      to get unit side lengths
  v1.03 (1999-07-28): Multiplied cuboctahedron by 1/2,
                      TruncatedCube by 2
  v1.04 (1999-07-29): Multiplied TruncatedDodecahedronExact by (3 + Sqrt[5])/4
                      Corrected TruncatedOctahedronExact.
                      Multiplied TruncatedTetrahedron by 9/5
  v1.05 (1999-07-30): Archimdeanr, ArchimdeanR, ArchimdeanRho.
                      Scaled GreatRhombicuboctahedronExact and
                      SmallRhombicuboctahedronExact.
  v1.06 (1999-07-31): Moved Netlist Archimedean solids
                      here
  v1.07 (1999-08-02): Renamed ArchimedeanExact` to Archimedean`
  v1.08 (1999-08-03): Implemented OrientCentered instead of Orient
  v1.09 (1999-08-14): GreatRhombicosidodecahedronExact, IcosidodecahedronExact,
                      SnubCubeExact, SnubDodecahedronExact
  v1.10 (1999-08-18): Simplified DodecahedronExact.
  v1.11 (1999-08-27): CuboctahedronDualExact
  v1.12 (1999-08-29): DeltoidalIcositetrahedronExact,
                      IcosidodecahedronDualExact,
                      PentakisDodecahedronExact,
                      RhombicTriacontahedronExact,
                      SmallRhombicuboctahedronDualExact
                      TetrakisHexahedronExact, TruncatedIcosahedronDualExact,
                      TruncatedOctahedronDualExact
  v1.13 (1999-08-30): Corrected TruncatedDodecahedronExact
  v1.14 (1999-08-30): PentagonalIcositetrahedronExact, TruncatedDodecahedronDualExact
  v1.15 (1999-08-31): SnubCubeDualExact
  v1.16 (1999-10-01): Added  context for SnubDodecahedronExact
  v1.17 (1999-10-07): Added  contexts
  v1.18 (2000-01-01): Updated URL
  v1.19 (2000-01-04): rhombicDodecahedron moved from  to here
  v1.20 (2000-02-15): fixed UniformPolyhedron non-exact entries.
                      Fixed ToExactN for SnubDodecahedronExact
  v1.21 (2000-02-25): Added names for some duals.  ArchimedeanName,
                      ArchimedeanDualName, ArchimedeanCompound,
                      PentagonalHexecontahedron.  Archimedean, ArchimedeanDual.
  v1.22 (2000-11-13): PolyhedronOperations now added to BeginPackage
  v1.23 (2000-12-29): Added OrientCentered to TriakisTetrahedronExact
  v1.24 (2002-10-11): Added missing non-exact definitions,
  v1.25 (2002-10-14): Added nice embedding for the SnubCube.
                      GreatRhombicosidodecahedronExact
  v1.26 (2002-10-27): Added remaining missing inexact versions
  v1.30 (2002-11-09): Switched to new syntax Polyhedron[poly].  Completely cleaned
                      up and uniformized syntax and documentation, moving
                      incomplete code into individual notebooks.  This package
                      is now free-standing, requiring only PolyhedronOperations.m
                      to define the Polyhedron[] commands.  Added nets.
  v1.31 (2002-11-15): Added numerical vertices for SmallRhombicosidodecahedron,
                      snub cube, snub dodecahedron
  v1.35 (2003-02-19): Corrected orientations as needed, cleaned up radii.
                      Added SnubCube, PentagonalIcositetrahedron, 
                      SmallRhombicosidodecahedron, and  DeltoidalHexecontahedron
  v1.36 (2003-08-10): GreatRhombicTriacontahedron, RhombicHexecontahedron
  v1.37 (2003-08-13): Fixed remaining occurrences of Exact
  v1.38 (2003-10-18): updated context
  v1.39 (2003-11-09): updated context of PolyhedronOperations in BeginPackage.
                      Fixed approximate zeros for NetVertices of Archimedean duals
  v1.40 (2003-11-21): Added Archimedeans and ArchimedeanDuals, fixed some PolyhedronDuals
  v1.41 (2003-12-07): Moved GreatRhombicTriacontahedron to MiscPolyhedra` where it
                      belongs
  v1.42 (2003-12-20): Added versions of SnubCube and SnubDodecahedron constructed by
                      snubification and that therefore have simpler and more natural
                      coordinates.  Also added both handednesses for these
  v1.43 (2004-05-16): Return ArchimedeanDuals as lists
  v1.44 (2004-09-08): Document Archimedeans to be in alphabetical order
  v1.45 (2004-11-10): Fixed RhombicDodecahedron and PentagonalHexecontahedron
  v1.46 (2004-12-01): Fixed PentagonalHexecontahedron "dextro" and "laevo"
  v1.47 (2005-09-16): Fixed orientation of RhombicDodecahedron
 
  (c) 1999-2005 Eric W. Weisstein
*)

(*:Keywords:
  
*)

(*:Requirements: None. *)

(*:Discussion: *)

(*:References: *)

(*:Limitations: 

Exact net vertices still missing for some solids

Need to add Archimedean compounds for Archimedeans #2,3,5,9

Definitions of laevo and dextro for snubs unceratain.

Currently, the duals of the laevo snubs are listed as laevo, etc., 
but this should perhaps be inverted...

Neeed to add ArchimedeanDual[7,8] dextro

*)

BeginPackage["MathWorld`Archimedean`",
	{
	"MathWorld`PolyhedronOperations`"
	}
]


Archimedean::usage =
"Archimedean[n] gives a representation of the nth Archimedean solid as \
a list of Polygons."

ArchimedeanCompound::usage =
"ArchimedeanCompound[n] gives a list of Polygons forming the \
compound of the nth Archimedean solid and its dual."

ArchimedeanDual::usage =
"ArchimedeanDual[n] gives an exact representation (or inexact if exact is not available) \
of the nth Archimedean dual."

ArchimedeanDuals::usage =
"ArchimedeanDuals gives a list of Archimedean duals."

Archimedeans::usage =
"Archimedeans gives a list of Archimedean solids in alphabetical order."

Cuboctahedron::usage =
"Polyhedron[Cuboctahedron] gives a list of Polygons with analytic vertices \
representing this Archimedean solid."

DeltoidalHexecontahedron::usage =
"Polyhedron[DeltoidalHexecontahedron] gives a list of Polygons with \
analytic vertices representing this Archimedean dual."

DeltoidalIcositetrahedron::usage =
"Polyhedron[DeltoidalIcositetrahedron] gives a list of Polygons with \
analytic vertices representing this Archimedean dual."

DisdyakisTriacontahedron::usage =
"Polyhedron[DisdyakisTriacontahedron] gives a list of Polygons with \
analytic vertices representing this Archimedean dual."

DisdyakisDodecahedron::usage =
"Polyhedron[DisdyakisDodecahedron] gives a list of Polygons with \
analytic vertices representing this Archimedean dual."

GreatRhombicosidodecahedron::usage =
"Polyhedron[GreatRhombicosidodecahedron] gives a list of Polygons with \
analytic vertices representing this Archimedean solid."

GreatRhombicuboctahedron::usage =
"Polyhedron[GreatRhombicuboctahedron] gives a list of Polygons with \
analytic vertices representing this Archimedean solid."
 
Icosidodecahedron::usage =
"Polyhedron[Icosidodecahedron] gives a list of Polygons with analytic vertices \
representing the icosidodecahedron with unit edge lengths."

PentagonalHexecontahedron::usage =
"Polyhedron[PentagonalHexecontahedron] gives a list of Polygons with \
analytic vertices representing the left-handed version of thie Archimedean dual.  \
Polyhedron[PentagonalHexecontahedron,(\"laevo\"|\"dextro\")] gives the pentagonal \
hexecontahedron with specified handedness."

PentagonalIcositetrahedron::usage =
"Polyhedron[PentagonalIcositetrahedron] gives a list of Polygons with \
analytic vertices representing the left-handed version of this Archimedean dual.  \
Polyhedron[PentagonalIcositetrahedron,(\"laevo\"|\"dextro\")] gives the pentagonal \
icositetrahedron with specified handedness."

PentakisDodecahedron::usage =
"Polyhedron[PentakisDodecahedron] gives a list of Polygons with \
analytic vertices representing this Archimedean dual."

RhombicDodecahedron::usage =
"Polyhedron[RhombicDodecahedron] gives a list of Polygons with analytic \
vertices representing the rhombic dodecahedron Archimedean dual."

RhombicHexecontahedron::usage =
"Polyhedron[RhombicHexecontahedron] gives a list of Polygons with analytic \
vertices representing the rhombic hexecontahedron."

RhombicTriacontahedron::usage =
"Polyhedron[RhombicTriacontahedron] gives a list of Polygons with analytic \
vertices representing the rhombic dodecahedron Archimedean dual."

SmallRhombicosidodecahedron::usage =
"Polyhedron[SmallRhombicosidodecahedron] gives a list of Polygons with exact \
vertices representing this Archimedean solid."

SmallRhombicuboctahedron::usage =
"Polyhedron[SmallRhombicuboctahedron] gives a list of Polygons with exact \
vertices representing this Archimedean solid."

SmallTriakisOctahedron::usage =
"Polyhedron[SmallTriakisOctahedron] gives a list of Polygons with analytic \
vertices representing the small triakis octahedron Archimedean dual."

SnubCube::usage =
"Polyhedron[SnubCube] gives a list of Polygons with analytic vertices \
representing the laevo-handed snub cube Archimedean solid \
constructed  by snubification of a cube in normal position.  \
Polyhedron[SnubCube,(\"laevo\"|\"dextro\")] gives the snub cube with specified \
handedness."

SnubDodecahedron::usage =
"Polyhedron[SnubDodecahedron] gives a list of Polygons with analytic \
vertices representing the laevo-handed snub dodecahedron Archimedean solid \
constructed  by snubification of a dodecahedron in normal position.  \
Polyhedron[SnubDodecahedron,(\"laevo\"|\"dextro\")] gives the snub dodecahedron \
with specified handedness."

TetrakisHexahedron::usage =
"Polyhedron[TetrakisHexahedron] gives a list of Polygons with analytic \
vertices representing the triakis hexahedron Archimedean dual."

TriakisIcosahedron::usage =
"Polyhedron[TriakisIcosahedron] gives a list of Polygons with analytic \
vertices representing thetriakis icosahedron Archimedean dual."

TriakisTetrahedron::usage =
"Polyhedron[TriakisTetrahedron] gives a list of Polygons with analytic \
vertices representing the tetrakis tetrahedron Archimedean dual."

TruncatedCube::usage =
"Polyhedron[TruncatedCube] gives Polygons with analytic vertices representing \
this Archimedean solid."

TruncatedDodecahedron::usage =
"Polyhedron[TruncatedDodecahedron] gives a list of Polygons with analytic \
vertices representing this Archimedean solid."

TruncatedIcosahedron::usage =
"Polyhedron[TruncatedIcosahedron] gives a list of Polygons with analytic \
vertices representing this Archimedean solid."

TruncatedOctahedron::usage =
"Polyhedorn[TruncatedOctahedron] gives a list of Polygons with analytic \
vertices representing this Archimedean solid."

TruncatedTetrahedron::usage =
"Polyhedron[TruncatedTetrahedron] gives a list of Polygons with analytic \
vertices representing this Archimedean solid."


Begin["`Private`"]

Archimedean[ 1]:=Cuboctahedron
Archimedean[ 2]:=GreatRhombicosidodecahedron
Archimedean[ 3]:=GreatRhombicuboctahedron
Archimedean[ 4]:=Icosidodecahedron
Archimedean[ 5]:=SmallRhombicosidodecahedron
Archimedean[ 6]:=SmallRhombicuboctahedron
Archimedean[ 7]:=SnubCube
Archimedean[ 8]:=SnubDodecahedron
Archimedean[ 9]:=TruncatedCube
Archimedean[10]:=TruncatedDodecahedron
Archimedean[11]:=TruncatedIcosahedron
Archimedean[12]:=TruncatedOctahedron
Archimedean[13]:=TruncatedTetrahedron

Inradius[Cuboctahedron]:=3/4
Midradius[Cuboctahedron]:=Sqrt[3]/2
Circumradius[Cuboctahedron]:=1

Inradius[GreatRhombicosidodecahedron]:=1/241(105+6Sqrt[5])Sqrt[31+12Sqrt[5]]
Midradius[GreatRhombicosidodecahedron]:=1/2Sqrt[30+12Sqrt[5]]
Circumradius[GreatRhombicosidodecahedron]:= 1/2Sqrt[31+12Sqrt[5]]

Inradius[GreatRhombicuboctahedron]:=3/97(14+Sqrt[2])Sqrt[13+6Sqrt[2]]
Midradius[GreatRhombicuboctahedron]:=1/2Sqrt[12+6Sqrt[2]]
Circumradius[GreatRhombicuboctahedron]:=1/2Sqrt[13+6Sqrt[2]]

Inradius[Icosidodecahedron]:=1/8(5+3Sqrt[5])
Midradius[Icosidodecahedron]:=1/2Sqrt[5+2Sqrt[5]]
Circumradius[Icosidodecahedron]:=1/2(1+Sqrt[5])

Inradius[SmallRhombicosidodecahedron]:=1/41(15+2Sqrt[5])Sqrt[11+4Sqrt[5]]
Midradius[SmallRhombicosidodecahedron]:=1/2Sqrt[10+4Sqrt[5]]
Circumradius[SmallRhombicosidodecahedron]:=1/2Sqrt[11+4Sqrt[5]]

Inradius[SmallRhombicuboctahedron]:=1/17(6+Sqrt[2])Sqrt[5+2Sqrt[2]]
Midradius[SmallRhombicuboctahedron]:=1/2Sqrt[4+2Sqrt[2]]
Circumradius[SmallRhombicuboctahedron]:=1/2Sqrt[5+2Sqrt[2]]

Inradius[SnubCube]:=Root[-1+64#1^2-1248#1^4+896#1^6&,2]
Midradius[SnubCube]:=Root[-1+20#1^2-112#1^4+64#1^6&,2]
Circumradius[SnubCube]:=Root[-7+44#1^2-80#1^4+32#1^6&,2]

Inradius[SnubDodecahedron]:=Root[1-128#1^2+6384#1^4-149376#1^6+1443072#1^8-3900416#1^10+856064#1^12&,8]
Midradius[SnubDodecahedron]:=Root[1-40#1^2+624#1^4-4672#1^6+16384#1^8-21504#1^10+4096#1^12&,8]
Circumradius[SnubDodecahedron]:=Root[209-2696#1^2+13872#1^4-35776#1^6+47104#1^8-27648#1^10+4096#1^12&,8]

Inradius[TruncatedCube]:=1/17(5+2Sqrt[2])Sqrt[7+4Sqrt[2]]
Midradius[TruncatedCube]:=1/2(2+Sqrt[2])
Circumradius[TruncatedCube]:=1/2Sqrt[7+4Sqrt[2]]

Inradius[TruncatedDodecahedron]:=5/488(17Sqrt[2]+3Sqrt[10])Sqrt[37+15Sqrt[5]]
Midradius[TruncatedDodecahedron]:=1/4(5+3Sqrt[5])
Circumradius[TruncatedDodecahedron]:=1/4Sqrt[74+30Sqrt[5]]

Inradius[TruncatedIcosahedron]:=9/872(21+Sqrt[5])Sqrt[58+18Sqrt[5]]
Midradius[TruncatedIcosahedron]:=3/4(1+Sqrt[5])
Circumradius[TruncatedIcosahedron]:=1/4Sqrt[58+18Sqrt[5]]

Inradius[TruncatedOctahedron]:=9/20Sqrt[10]
Midradius[TruncatedOctahedron]:=3/2
Circumradius[TruncatedOctahedron]:=1/2Sqrt[10]

Inradius[TruncatedTetrahedron]:=9/44Sqrt[22]
Midradius[TruncatedTetrahedron]:=3/4Sqrt[2]
Circumradius[TruncatedTetrahedron]:=1/4Sqrt[22]

(* Archimedean Compound *)

ArchimedeanCompound[1]:=MidpointCumulate[
	Polyhedron[Cuboctahedron],
	{{3,1/4/Sqrt[6]},{4,1/2/Sqrt[2]}}
]
	
ArchimedeanCompound[2]:={}

ArchimedeanCompound[3]:={}

ArchimedeanCompound[4]:=MidpointCumulate[
	Polyhedron[Icosidodecahedron],
	{{3,Sqrt[(7-3Sqrt[5])/6]/4},{5,Sqrt[(5+2Sqrt[5])/5]/4}}
]
 	
ArchimedeanCompound[5]:={}

ArchimedeanCompound[6]:=MidpointCumulate[
	Polyhedron[SmallRhombicuboctahedron],
	{{3,Sqrt[(11-6Sqrt[2])/3]/14},{4,(Sqrt[2]-1)/2}}
]

ArchimedeanCompound[7]:=MidpointCumulate[
	Polyhedron[SnubCube],
{{3,Root[-1+216#1^2-864#1^4+3456#1^6&,2]},
{4,Root[-1+16#1^2+96#1^4+128#1^6&,2]}}
]

ArchimedeanCompound[8]:=MidpointCumulate[
	Polyhedron[TruncatedCube],
	{{8,(1+Sqrt[2])/2},{3,Sqrt[(17-12Sqrt[2])/3]/2}}
]

ArchimedeanCompound[9]:={}

ArchimedeanCompound[10]:=MidpointCumulate[
	Polyhedron[TruncatedDodecahedron],
	{{3,Sqrt[(63+5Sqrt[5])/6]/62},{10,Sqrt[(5+Sqrt[5])/2]/2}}
]

ArchimedeanCompound[11]:=MidpointCumulate[
	Polyhedron[TruncatedIcosahedron],
	{{5,Sqrt[(305+131Sqrt[5])/10]/38},{6,Sqrt[3/2(7-3Sqrt[5])]/2}}
]

ArchimedeanCompound[12]:=MidpointCumulate[
	Polyhedron[TruncatedOctahedron],
	{{4,Sqrt[2]/8},{6,Sqrt[6]/4}}
]

ArchimedeanCompound[13]:=MidpointCumulate[
	Polyhedron[TruncatedTetrahedron],
	{{3,Sqrt[6]/30},{6,Sqrt[6]/2}}
]

(* Archimedean Duals *)

ArchimedeanDual[n_Integer]:={PolyhedronDual[Archimedean[n]]} /; 1<=n<=13

ArchimedeanDuals:=ArchimedeanDual/@Range[13]


(* Archimedeans *)

Archimedeans:=Archimedean/@Range[13]

(* Cuboctahedron *)

PolyhedronName[Cuboctahedron]:="cuboctahedron"
NetFaces[Cuboctahedron]:={{2,4,5},{5,4,8,9},{9,8,13},{4,7,8},{1,3,7,4},{8,7,11,12},{12,11,15},{6,10,11},{11,10,14,15},{15,14,17},{12,15,18,16},{16,18,19},{19,18,20,21},{21,20,22}}
NetVertices[Cuboctahedron]:={{-(1/Sqrt[3]),-1},{-(1/Sqrt[3]),0},{(3-2Sqrt[3])/6,(-2-Sqrt[3])/2},{1/(2Sqrt[3]),-1/2},{1/(2Sqrt[3]),1/2},{(3+Sqrt[3])/6,(-3-Sqrt[3])/2},{(3+Sqrt[3])/6,(-1-Sqrt[3])/2},{(6+Sqrt[3])/6,-1/2},{(6+Sqrt[3])/6,1/2},{(3+4Sqrt[3])/6,(-4-Sqrt[3])/2},{(3+4Sqrt[3])/6,(-2-Sqrt[3])/2},{(3+2Sqrt[3])/3,-1},{(3+2Sqrt[3])/3,0},{(9+4Sqrt[3])/6,(-4-Sqrt[3])/2},{(9+4Sqrt[3])/6,(-2-Sqrt[3])/2},{(6+7Sqrt[3])/6,-1/2},{(9+7Sqrt[3])/6,(-3-Sqrt[3])/2},{(9+7Sqrt[3])/6,(-1-Sqrt[3])/2},{(12+7Sqrt[3])/6,-1/2},{(9+10Sqrt[3])/6,(-2-Sqrt[3])/2},{(6+5Sqrt[3])/3,-1},{(5(3+2Sqrt[3]))/6,(-2-Sqrt[3])/2}}
PolyhedronDual[Cuboctahedron]:=RhombicDodecahedron
PolyhedronFaces[Cuboctahedron]:=
{{4,10,8,2},{3,9,11,5},{9,6,8,12},{3,1,2,6},{5,7,4,1},{11,12,10,7},{12,11,
    9},{3,5,1},{6,9,3},{5,11,7},{8,10,12},{1,4,2},{2,8,6},{7,10,4}}
PolyhedronVertices[Cuboctahedron]:=
{{-1,0,0},{-1/2,-1/2,-(1/Sqrt[2])},
{-1/2,-1/2,1/Sqrt[2]},{-1/2,1/2,-(1/Sqrt[2])},
{-1/2,1/2,1/Sqrt[2]},{0,-1,0},{0,1,0},
{1/2,-1/2,-(1/Sqrt[2])},{1/2,-1/2,1/Sqrt[2]},
{1/2,1/2,-(1/Sqrt[2])},{1/2,1/2,1/Sqrt[2]},{1,0,0}}

(* Deltoidal Hexecontahedron *)

PolyhedronName[DeltoidalHexecontahedron]:="deltoidal hexecontahedron"
NetFaces[DeltoidalHexecontahedron]:={{33,29,24,22},{33,22,20,26},{33,26,32,41},{33,41,45,44},
{33,44,43,36},{28,16,24,29},{28,29,38,40},{28,40,42,34},{28,34,25,19},{28,19,13,15},{9,11,20,22},
{9,22,23,14},{9,14,6,4},{9,4,1,2},{9,2,3,7},{21,31,32,26},{21,26,17,12},{21,12,5,8},{21,8,10,18},
{21,18,27,30},{47,51,45,41},{47,41,35,37},{47,37,39,46},{47,46,52,56},{47,56,58,55},{54,49,43,44},
{54,44,48,53},{54,53,59,62},{54,62,64,61},{54,61,57,50},{68,75,84,79},{68,79,73,66},{68,66,64,62},
{68,62,60,63},{68,63,65,71},{89,101,97,88},{89,88,83,78},{89,78,74,81},{89,81,87,95},{89,95,103,104},
{113,112,106,102},{113,102,98,105},{113,105,110,116},{113,116,121,122},{113,122,120,115},{109,96,94,100},
{109,100,108,114},{109,114,118,119},{109,119,117,111},{109,111,107,99},{82,72,80,86},{82,86,92,93},
{82,93,91,85},{82,85,76,70},{82,70,67,69},{90,79,84,88},{90,88,97,102},{90,102,106,100},{90,100,94,86},
{90,86,80,77}}
NetVertices[DeltoidalHexecontahedron]:={{-3.08833572923,0.641509221705},{-2.92700102126,0.0122346606165},
{-2.70022373872,-0.596524261577},{-2.61051993394,1.08163128091},{-2.14974158858,-1.37845515939},
{-2.08896677291,1.46892659646},{-2.05666433631,-0.685105618551},{-1.98840688061,-2.00772972048},
{-1.97279056825,0.311370769042},{-1.76162959806,-2.61648864267},{-1.69256484556,-0.648563366375},
{-1.67192579328,-0.938333100188},{-1.57323330852,2.27420016506},{-1.50085056964,1.19300143871},
{-1.41189860055,1.64492560397},{-1.24039771319,1.32167458542},{-1.15037263225,-0.551037784637},
{-1.11807019565,-2.70506999965},{-1.09541751322,2.71432222426},{-1.07944438301,-0.433857382216},
{-1.03419642759,-1.70859361205},{-0.978165739025,0.207826338554},{-0.945101407048,0.85661144256},
{-0.809801596111,0.835256714461},{-0.573864352191,3.10161753981},{-0.562256428978,-0.826962942382},
{-0.468758482208,-2.72531005927},{-0.457688147535,1.9440617124},{-0.177462424841,0.984127576979},
{-0.143413883677,-2.16302343001},{-0.0395715983674,-1.81213804254},{-0.00650726638988,-1.16335293854},
{0.,0.},{0.014251851082,2.82569238207},{0.07515443278469999,-1.27332307148},
{0.188460498661,0.982080770835},{0.236489140755,-1.90259763257},{0.435658037708,1.19883356114},
{0.463266423302,-2.51135655476},{0.536936681692,1.84051728191},{0.552970228079,-0.833201012277},
{0.57000101367,2.48930238591},{0.819094722771,0.826145410473},{0.980429430741,0.196870849384},
{1.07452338911,-0.445905696725},{1.10682582571,-2.59993791174},{1.19069959377,-1.60346152414},
{1.20720671329,-0.41188807281},{1.29691051806,1.26626746968},{1.61671016133,1.44412123974},
{1.66263959238,-0.721830854471},{1.75613753915,-2.62017797136},{1.85076611569,-0.500469429784},
{1.93463988375,0.49600695781},{1.94821037046,-0.950638865842},{2.08148213768,-2.0578913421},
{2.24267446659,1.61785681403},{2.34573423521,-1.46443857213},{2.50007782914,-0.520709489403},
{2.62520808409,-0.57642737296},{2.69215066045,1.14882961611},{2.82542242767,0.0415771398494},
{3.07468427795,-1.04545457088},{3.08967452519,0.635029909818},{3.57110311096,-1.46448166243},
{3.73750156935,0.683356735527},{3.7451089792,2.4035489347},{3.75914935688,-0.316408923663},
{3.99575860435,1.80422420916},{4.15424671232,2.90814944656},{4.1752628557,-1.22572160701},
{4.2122077198,1.5091758135},{4.3868132828,0.663116675908},{4.43454876665,-1.87546600064},
{4.47452997228,-1.01514390569},{4.61430222188,3.36680431609},{4.63277706175,0.44496485983},
{4.63476311023,-1.25746148783},{4.67650556696,0.081658387011},{4.70862655281,1.09014872195},
{4.8840249605,-2.34449319856},{4.89667279873,2.23822146072},{4.89901520775,-0.664008717866},
{4.90328284951,-0.527100535182},{5.23614167585,3.17883874818},{5.31278629755,1.32890877737},
{5.38044379352,-2.76352029012},{5.54684225191,-0.615681892157},{5.56849003944,-1.61544755135},
{5.63071601997,0.380794495437},{5.83469935037,2.92636286518},{5.88840774805,1.63003218555},
{5.89581563433,2.27961704102},{5.93875060281,1.50264435165},{5.98460353826,-2.5247602347},
{6.13896494639,2.12064886446},{6.19615396536,-0.635921951776},{6.32128422031,-0.691639835333},
{6.33026578919,2.43259078679},{6.38822679667,1.03361715374},{6.48584624952,-1.21738024067},
{6.52149856389,-0.0736353225234},{6.56022498875,-2.22363682651},{6.56763287504,-1.57405197104},
{6.77076041416,-1.16066703325},{6.78575066141,0.519817447445},{6.79032129875,2.89124565633},
{6.88464562968,0.614590062179},{7.0726918756,1.76266280095},{7.26717924718,-1.57969412481},
{7.41216075272,2.70328008841},{7.43357770557,0.568144273154},{7.4552254931,-0.431621386035},
{7.48880537442,0.853350117601},{7.79469437021,0.508995901425},{7.87133899192,-1.34093406939},
{8.01071842724,2.45080420541},{8.06442682491,1.15447352579},{8.0718347112,1.80405838125},
{8.39325204474,0.256520018424},{8.44696044241,-1.0398106612},{8.4543683287,-0.390225805733}}
PolyhedronDual[DeltoidalHexecontahedron]:=SmallRhombicosidodecahedron
PolyhedronFaces[DeltoidalHexecontahedron]:=
{{43,33,1,7},{8,2,34,44},{33,45,9,1},{2,10,46,34},{31,15,5,3},{3,6,14,31},{17,
    32,4,11},{12,4,32,16},{39,5,43,7},{40,8,44,6},{9,45,11,41},{12,46,10,
    42},{39,7,1,47},{48,2,8,40},{1,9,41,47},{48,42,10,2},{53,3,5,19},{18,6,3,
    53},{11,4,54,21},{20,54,4,12},{33,37,23,35},{24,38,34,36},{15,25,43,5},{6,
    44,26,14},{45,27,17,11},{12,16,28,46},{29,37,25,15},{14,26,38,29},{27,35,
    30,17},{16,30,36,28},{29,13,23,37},{38,24,13,29},{23,13,30,35},{36,30,13,
    24},{15,31,14,29},{25,37,33,43},{44,34,38,26},{33,35,27,45},{46,28,36,
    34},{17,30,16,32},{19,5,39,57},{58,40,6,18},{41,11,21,59},{60,20,12,
    42},{19,57,51,55},{55,52,58,18},{49,59,21,56},{56,20,60,50},{55,51,61,
    22},{22,62,52,55},{61,49,56,22},{22,56,50,62},{55,18,53,19},{57,39,47,
    51},{52,48,40,58},{47,41,59,49},{50,60,42,48},{54,20,56,21},{49,61,51,
    47},{50,48,52,62}}
PolyhedronVertices[DeltoidalHexecontahedron]:=
{{0,0,-Sqrt[5]},{0,0,Sqrt[5]},{0,-Sqrt[5],0},{0,Sqrt[5],0},
{0,(-5-3Sqrt[5])/6,(-5-Sqrt[5])/6},
{0,(-5-3Sqrt[5])/6,(5+Sqrt[5])/6},{0,(-15-Sqrt[5])/22,
(-25-9Sqrt[5])/22},{0,(-15-Sqrt[5])/22,(25+9Sqrt[5])/22},
{0,(15+Sqrt[5])/22,(-25-9Sqrt[5])/22},
{0,(15+Sqrt[5])/22,(25+9Sqrt[5])/22},
{0,(5+3Sqrt[5])/6,(-5-Sqrt[5])/6},
{0,(5+3Sqrt[5])/6,(5+Sqrt[5])/6},{-Sqrt[5],0,0},
{-Sqrt[5]/2,(-5-Sqrt[5])/4,(5-Sqrt[5])/4},
{-Sqrt[5]/2,(-5-Sqrt[5])/4,(-5+Sqrt[5])/4},
{-Sqrt[5]/2,(5+Sqrt[5])/4,(5-Sqrt[5])/4},
{-Sqrt[5]/2,(5+Sqrt[5])/4,(-5+Sqrt[5])/4},
{Sqrt[5]/2,(-5-Sqrt[5])/4,(5-Sqrt[5])/4},
{Sqrt[5]/2,(-5-Sqrt[5])/4,(-5+Sqrt[5])/4},
{Sqrt[5]/2,(5+Sqrt[5])/4,(5-Sqrt[5])/4},
{Sqrt[5]/2,(5+Sqrt[5])/4,(-5+Sqrt[5])/4},{Sqrt[5],0,0},
{(-25-9Sqrt[5])/22,0,(-15-Sqrt[5])/22},
{(-25-9Sqrt[5])/22,0,(15+Sqrt[5])/22},
{(-5-4Sqrt[5])/11,(-5-4Sqrt[5])/11,(-5-4Sqrt[5])/11},
{(-5-4Sqrt[5])/11,(-5-4Sqrt[5])/11,(5+4Sqrt[5])/11},
{(-5-4Sqrt[5])/11,(5+4Sqrt[5])/11,(-5-4Sqrt[5])/11},
{(-5-4Sqrt[5])/11,(5+4Sqrt[5])/11,(5+4Sqrt[5])/11},
{(-5-3Sqrt[5])/6,(-5-Sqrt[5])/6,0},
{(-5-3Sqrt[5])/6,(5+Sqrt[5])/6,0},
{(-15-Sqrt[5])/22,(-25-9Sqrt[5])/22,0},
{(-15-Sqrt[5])/22,(25+9Sqrt[5])/22,0},
{(-5-Sqrt[5])/6,0,(-5-3Sqrt[5])/6},
{(-5-Sqrt[5])/6,0,(5+3Sqrt[5])/6},
{(-5-Sqrt[5])/4,(5-Sqrt[5])/4,-Sqrt[5]/2},
{(-5-Sqrt[5])/4,(5-Sqrt[5])/4,Sqrt[5]/2},
{(-5-Sqrt[5])/4,(-5+Sqrt[5])/4,-Sqrt[5]/2},
{(-5-Sqrt[5])/4,(-5+Sqrt[5])/4,Sqrt[5]/2},
{(5-Sqrt[5])/4,-Sqrt[5]/2,(-5-Sqrt[5])/4},
{(5-Sqrt[5])/4,-Sqrt[5]/2,(5+Sqrt[5])/4},
{(5-Sqrt[5])/4,Sqrt[5]/2,(-5-Sqrt[5])/4},
{(5-Sqrt[5])/4,Sqrt[5]/2,(5+Sqrt[5])/4},
{(-5+Sqrt[5])/4,-Sqrt[5]/2,(-5-Sqrt[5])/4},
{(-5+Sqrt[5])/4,-Sqrt[5]/2,(5+Sqrt[5])/4},
{(-5+Sqrt[5])/4,Sqrt[5]/2,(-5-Sqrt[5])/4},
{(-5+Sqrt[5])/4,Sqrt[5]/2,(5+Sqrt[5])/4},
{(5+Sqrt[5])/6,0,(-5-3Sqrt[5])/6},
{(5+Sqrt[5])/6,0,(5+3Sqrt[5])/6},
{(5+Sqrt[5])/4,(5-Sqrt[5])/4,-Sqrt[5]/2},
{(5+Sqrt[5])/4,(5-Sqrt[5])/4,Sqrt[5]/2},
{(5+Sqrt[5])/4,(-5+Sqrt[5])/4,-Sqrt[5]/2},
{(5+Sqrt[5])/4,(-5+Sqrt[5])/4,Sqrt[5]/2},
{(15+Sqrt[5])/22,(-25-9Sqrt[5])/22,0},
{(15+Sqrt[5])/22,(25+9Sqrt[5])/22,0},
{(5+3Sqrt[5])/6,(-5-Sqrt[5])/6,0},
{(5+3Sqrt[5])/6,(5+Sqrt[5])/6,0},
{(5+4Sqrt[5])/11,(-5-4Sqrt[5])/11,(-5-4Sqrt[5])/11},
{(5+4Sqrt[5])/11,(-5-4Sqrt[5])/11,(5+4Sqrt[5])/11},
{(5+4Sqrt[5])/11,(5+4Sqrt[5])/11,(-5-4Sqrt[5])/11},
{(5+4Sqrt[5])/11,(5+4Sqrt[5])/11,(5+4Sqrt[5])/11},
{(25+9Sqrt[5])/22,0,(-15-Sqrt[5])/22},
{(25+9Sqrt[5])/22,0,(15+Sqrt[5])/22}}

(* Deltoidal Icositetrahedron *)

PolyhedronName[DeltoidalIcositetrahedron]:="deltoidal icositetrahedron"
NetFaces[DeltoidalIcositetrahedron]:={{6,9,14,11},{16,18,14,9},{20,15,14,18},{3,8,7,1},{16,9,7,8},{4,2,7,9},
{19,17,13,12},{16,8,13,17},{5,10,13,8},{27,21,22,25},{16,17,22,21},{23,24,22,17},{28,33,29,26},{35,31,29,33},
{27,25,29,31},{44,42,37,38},{35,33,37,42},{30,36,37,33},{48,43,45,50},{35,42,45,43},{47,49,45,42},{32,34,39,40},
{35,43,39,34},{46,41,39,43}}
NetVertices[DeltoidalIcositetrahedron]:={{-0.717169671553,-1.56293776056},{-0.708574552532,-1.37178918934},
{-0.49304027041,-2.53749715411},{-0.397883124879,-0.421278359951},{-0.134572525766,-2.99278531964},
{0.,0.},{0.0538725395261,-1.50184064366},{0.43818899398100003,-2.17306326828},
{0.496880137844,-0.867819179678},{0.760190736957,-3.43932613937},{0.931229264391,0.36443388583},
{0.948019728299,-3.47582165091},{1.00049784286,-2.70414490742},{1.10458418488,-0.389347926435},
{1.12157570781,0.383924494583},{1.22388543544,-1.55445102956},{1.72076557328,-2.42227020924},
{1.79664695518,-0.734728978195},{1.94489497442,-3.39682960278},{2.10733838284,0.215781851193},
{2.15511469983,-1.19001714373},{2.32846962031,-1.943798956},{2.44777087087,-3.10890205912},
{3.02053239062,-2.28918000776},{3.0841413475,-2.10872072438},{3.12273327544,-2.2961302104},
{3.15198994595,-1.1110250956},{3.5792437972,-3.18584822388},{3.85518355858,-2.04762360748},
{4.03850885676,-3.53920645952},{4.12899279768,-1.32425133158},{4.36665263887,0.0621364895147},
{4.39251496525,-2.60396334933},{4.54579913906,-0.921685918988},{5.00230966082,-1.81140393247},
{5.01551170849,-3.7524326955},{5.06798982305,-2.98075595201},{5.20655231033,-3.74170232731},
{5.27824942219,-0.67317931607},{5.36610343814,0.0952740792913},{5.55205866129,0.050192959831},
{5.69702179275,-2.53069181208},{5.81558082886,-1.22951905792},{6.15353231451,-3.42040982556},
{6.16867303719,-1.9176795344},{6.42537552443,-0.436959641065},{6.57033865589,-3.01784441297},
{6.76256083304,-0.908226556181},{6.92434476437,-2.08260130278},{6.94170733323,-1.89204896468}}
PolyhedronDual[DeltoidalIcositetrahedron]:=SmallRhombicuboctahedron
PolyhedronFaces[DeltoidalIcositetrahedron]:=
{{26,2,7,5},{6,8,3,25},{2,24,9,7},{8,10,23,3},{1,26,5,11},{11,6,25,1},{24,4,
    12,9},{10,12,4,23},{22,5,7,14},{15,8,6,21},{7,9,20,14},{15,19,10,8},{11,5,
    22,13},{13,21,6,11},{20,9,12,16},{16,12,10,19},{1,17,2,26},{25,3,17,1},{2,
    17,4,24},{23,4,17,3},{13,22,14,18},{18,15,21,13},{14,20,16,18},{18,16,19,
    15}}
PolyhedronVertices[DeltoidalIcositetrahedron]:=
{{-1,-1,0},{-1,0,-1},{-1,0,1},{-1,1,0},
{0,-1,-1},{0,-1,1},
{0,0,-Sqrt[2]},{0,0,Sqrt[2]},
{0,1,-1},{0,1,1},
{0,-Sqrt[2],0},{0,Sqrt[2],0},
{1,-1,0},{1,0,-1},{1,0,1},{1,1,0},
{-Sqrt[2],0,0},
{Sqrt[2],0,0},
{(4+Sqrt[2])/7,(4+Sqrt[2])/7,(4+Sqrt[2])/7},
{(4+Sqrt[2])/7,(4+Sqrt[2])/7,-1+(3+Sqrt[2])^(-1)},
{(4+Sqrt[2])/7,-1+(3+Sqrt[2])^(-1),(4+Sqrt[2])/7},
{(4+Sqrt[2])/7,-1+(3+Sqrt[2])^(-1),-1+(3+Sqrt[2])^(-1)},
{-1+(3+Sqrt[2])^(-1),(4+Sqrt[2])/7,(4+Sqrt[2])/7},
{-1+(3+Sqrt[2])^(-1),(4+Sqrt[2])/7,-1+(3+Sqrt[2])^(-1)},
{-1+(3+Sqrt[2])^(-1),-1+(3+Sqrt[2])^(-1),(4+Sqrt[2])/7},
{-1+(3+Sqrt[2])^(-1),-1+(3+Sqrt[2])^(-1),-1+(3+Sqrt[2])^(-1)}}

(* Disdyakis Dodecahedron *)

PolyhedronName[DisdyakisDodecahedron]:="disdyakis dodecahedron"
NetFaces[DisdyakisDodecahedron]:={{8,4,12},{12,4,9},{9,15,12},{12,15,17},{17,18,12},{12,18,11},{2,5,7},{7,5,10},
{10,15,7},{7,15,9},{9,3,7},{7,3,1},{16,21,14},{14,21,19},{19,15,14},{14,15,10},{10,6,14},{14,6,13},{25,26,22},
{22,26,20},{20,15,22},{22,15,19},{19,23,22},{22,23,24},{27,28,29},{29,28,32},{32,36,29},{29,36,31},{31,26,29},
{29,26,25},{38,45,37},{37,45,40},{40,36,37},{37,36,32},{32,30,37},{37,30,35},{50,49,44},{44,49,42},{42,36,44},
{44,36,40},{40,46,44},{44,46,48},{41,33,39},{39,33,34},{34,36,39},{39,36,42},{42,47,39},{39,47,43}}
NetVertices[DisdyakisDodecahedron]:={{-0.981950574124,-1.82661434192},{-0.918314183829,-2.20641099531},
{-0.839111324828,-0.836868440758},{-0.746765412868,-0.665087526679},{-0.460661844222,-3.09554222075},
{-0.317366293461,-3.22784050128},{-0.237738700707,-1.89714836735},{0.,0.},
{0.0836590020264,-1.2222188318},{0.286103568646,-2.43045469407},{0.351893506789,0.156416265076},
{0.469340487706,-0.581846945408},{0.605404033393,-3.61319089232},{0.85945444607,-2.91013701878},
{0.964454166201,-1.69571632785},{0.989063699762,-3.64636239367},{1.20312740914,-0.724616393187},
{1.34596665843,0.265129507972},{1.50662871985,-2.5359822104},{1.91344870367,-1.38042381629},
{1.96428105946,-3.42511343584},{2.11423303147,-2.10050160028},{2.12815757382,-3.31937348153},
{2.73162743593,-2.52198767433},{2.85950715976,-2.15874964722},{2.88866606337,-1.15917485846},
{2.96599518837,-2.52882448812},{3.52195684134,-3.36003242867},{3.60690941581,-2.1440449222},
{3.67939078623,-3.47514542921},{3.84932781454,-1.43689594127},{4.18809503093,-2.61420408892},
{4.51176912215,0.172778910569},{4.59667020813,-0.823610473959},{4.6400525374,-3.75286651202},
{4.77832925316,-1.80697205108},{4.81235818573,-3.0254483896},{5.02499346356,-3.74211609496},
{5.33753179797,-0.723858451472},{5.41269621603,-2.58000410502},{5.49781464684,0.00630301672299},
{5.68522823404,-1.38562393079},{5.83999178971,-0.170360196756},{5.96673573129,-2.07814099937},
{5.968657869,-3.41121204557},{6.11942181303,-3.28749186501},{6.54671738672,-0.877847956747},
{6.62812605774,-2.42655052472},{6.62889263948,-1.0547198814},{6.71379372547,-2.05110926593}}
PolyhedronDual[DisdyakisDodecahedron]:=GreatRhombicuboctahedron
PolyhedronFaces[DisdyakisDodecahedron]:=
{{15,3,23},{26,1,13},{2,8,23},{26,16,4},{22,4,16},{8,2,22},{25,13,1},{3,15,
    25},{23,3,11},{7,1,26},{2,23,14},{12,26,4},{12,4,22},{22,2,14},{7,25,
    1},{3,25,11},{14,23,17},{5,26,12},{23,11,17},{5,7,26},{22,14,18},{18,12,
    22},{11,25,6},{6,25,7},{8,9,23},{26,19,16},{9,15,23},{26,13,19},{10,8,
    22},{22,16,10},{15,20,25},{25,20,13},{16,19,21},{21,9,8},{13,21,19},{9,21,
    15},{21,20,15},{13,20,21},{10,21,8},{16,21,10},{24,5,12},{14,17,24},{24,7,
    5},{17,11,24},{11,6,24},{24,6,7},{18,14,24},{24,12,18}}
PolyhedronVertices[DisdyakisDodecahedron]:=
{{0,(3+6Sqrt[2])/7,(3+6Sqrt[2])/7},
{0,(-3-6Sqrt[2])/7,(-3-6Sqrt[2])/7},
{0,(3+6Sqrt[2])/7,(-3-6Sqrt[2])/7},
{0,(-3-6Sqrt[2])/7,(3+6Sqrt[2])/7},
{(3+6Sqrt[2])/7,0,(3+6Sqrt[2])/7},
{(3+6Sqrt[2])/7,(3+6Sqrt[2])/7,0},
{Sqrt[2],Sqrt[2],Sqrt[2]},
{-Sqrt[2],-Sqrt[2],-Sqrt[2]},
{(-3-6Sqrt[2])/7,0,(-3-6Sqrt[2])/7},
{(-3-6Sqrt[2])/7,(-3-6Sqrt[2])/7,0},
{Sqrt[2],Sqrt[2],-Sqrt[2]},
{Sqrt[2],-Sqrt[2],Sqrt[2]},
{-Sqrt[2],Sqrt[2],Sqrt[2]},
{Sqrt[2],-Sqrt[2],-Sqrt[2]},
{-Sqrt[2],Sqrt[2],-Sqrt[2]},
{-Sqrt[2],-Sqrt[2],Sqrt[2]},
{(3+6Sqrt[2])/7,0,(-3-6Sqrt[2])/7},
{(3+6Sqrt[2])/7,(-3-6Sqrt[2])/7,0},
{(-3-6Sqrt[2])/7,0,(3+6Sqrt[2])/7},
{(-3-6Sqrt[2])/7,(3+6Sqrt[2])/7,0},
{(-6-9Sqrt[2])/7,0,0},
{0,(-6-9Sqrt[2])/7,0},
{0,0,(-6-9Sqrt[2])/7},
{(6+9Sqrt[2])/7,0,0},
{0,(6+9Sqrt[2])/7,0},
{0,0,(6+9Sqrt[2])/7}}

(* Disdyakis Triacontahedron *)

PolyhedronName[DisdyakisTriacontahedron]:="disdyakis triacontahedron"
NetFaces[DisdyakisTriacontahedron]:={{34,35,39},{26,35,34},{22,35,26},{17,35,22},{24,35,17},{32,35,24},{38,35,32},{44,35,38},{46,35,44},{47,35,46},{34,33,25},{39,33,34},{43,33,39},{49,33,43},{42,33,49},{37,33,42},{29,33,37},{21,33,29},{20,33,21},{18,33,20},{22,10,16},{26,10,22},{19,10,26},{13,10,19},{8,10,13},{3,10,8},{2,10,3},{1,10,2},{7,10,1},{11,10,7},{24,14,31},{17,14,24},{12,14,17},{6,14,12},{5,14,6},{4,14,5},{9,14,4},{15,14,9},{23,14,15},{28,14,23},{38,40,45},{32,40,38},{30,40,32},{27,40,30},{36,40,27},{41,40,36},{50,40,41},{55,40,50},{53,40,55},{52,40,53},{46,56,48},{44,56,46},{51,56,44},{57,56,51},{59,56,57},{62,56,59},{61,56,62},{60,56,61},{58,56,60},{54,56,58},{77,68,75},{78,68,77},{73,68,78},{66,68,73},{64,68,66},{60,68,64},{61,68,60},{63,68,61},{65,68,63},{71,68,65},{83,79,89},{76,79,83},{72,79,76},{67,79,72},{69,79,67},{70,79,69},{74,79,70},{81,79,74},{87,79,81},{92,79,87},{97,106,104},{90,106,97},{93,106,90},{95,106,93},{102,106,95},{110,106,102},{115,106,110},{118,106,115},{116,106,118},{112,106,116},{100,114,99},{103,114,100},{111,114,103},{117,114,111},{120,114,117},{122,114,120},{121,114,122},{119,114,121},{113,114,119},{107,114,113},{91,96,85},{98,96,91},{105,96,98},{109,96,105},{108,96,109},{101,96,108},{94,96,101},{86,96,94},{82,96,86},{80,96,82},{77,88,78},{76,88,77},{83,88,76},{90,88,83},{97,88,90},{103,88,97},{100,88,103},{98,88,100},{91,88,98},{84,88,91}}
NetVertices[DisdyakisTriacontahedron]:={{-2.85502180042,-0.741645768258},{-2.78868588456,-0.108501343786},{-2.7001135482,0.521916975432},{-2.47288835012,-2.87240398412},{-2.40655243426,-2.23925955965},{-2.3179800979,-1.60884124043},{-2.3178539244,-1.08329323121},{-2.09632461224,0.723687232855},{-1.9357204741,-3.21405144708},{-1.79611715981,-0.230186703003},{-1.76899763044,-1.40582863168},{-1.71419116193,-1.40707098301},{-1.48581015794,0.904087945363},{-1.41398370951,-2.36094491886},{-1.38686418013,-3.53658684754},{-1.14503462396,-1.20945221595},{-1.10367670764,-1.2266702705},{-1.07371134708,0.488431309458},{-1.05214963846,0.43802907589400003},{-1.00737543122,1.12157573393},{-0.91880309486,1.75199405315},{-0.8798316600180001,-0.630712461061},{-0.853464177687,-3.18908612328},{-0.670016188151,-1.69272913997},{-0.636607542581,0.00177558494799},{-0.635156150914,-0.0429997666107},{-0.375133934991,-3.40680361185},{-0.332621439739,-2.82303263224},{-0.315014158896,1.95376431057},{-0.308798019132,-2.77365918738},{-0.25302270061,-2.17375798247},{-0.22022568277,-2.14324086816},{-0.0148067064715,0.999890374713},{0,0},{0.0499791692707,-0.998750260395},{0.16203394103,-3.74845107481},{0.295500295396,2.13416502308},{0.383563253194,-1.94147061074},{0.636275829378,0.0206248617664},{0.683770705618,-2.8953445466},{0.710890234991,-4.07098647527},{0.729160814884,1.66810615361},{0.901478793322,0.599364616655},{0.979878937404,-1.71858068109},{0.994077707486,-1.76106989823},{1.04621485326,-1.08543625662},{1.09023219809,-0.450349814213},{1.13478718962,-0.45501793740200003},{1.14615430243,1.18707731111},{1.24429023744,-3.72348575101},{1.51704681342,-2.06022814405},{1.55212515399,-2.10235654837},{1.66967259672,-2.72802012784},{1.67650392626,-0.0883620941972},{1.76513297539,-3.35743225998},{2.03878357801,-1.20712161584},{2.06590310739,-2.38276354451},{2.29468747132,-0.240419387956},{2.59930310983,-2.03526282025},{2.90713802638,-0.414133617608},{3.02468546912,-1.03979719708},{3.12014584778,-1.66920932922},{3.16417440495,-1.66093740086},{3.41477848714,-0.0299806811739},{3.78235795001,-1.81299469462},{3.93562122509,0.336072809862},{3.95019772006,-2.31102133798},{4.0040894974,-0.837886946272},{4.0677451628,-2.93668491746},{4.20723409863,-3.55782512124},{4.40550892534,-1.94320674443},{4.4578381808200005,-1.92686840155},{4.49548623226,0.0330489003744},{4.82541764369,-3.709882415},{4.93398926554,-1.55771736697},{4.97868091877,-1.56081491051},{5.00032518139,-0.924572942496},{5.04434252622,-0.28948650009},{5.04714919109,-2.73477466665},{5.258275126,0.73432280484},{5.44856861903,-3.84009446481},{5.47224903915,1.33389555311},{5.53854592594,-1.86383882},{5.55893199886,0.114359020817},{5.56809514798,0.158210763577},{5.70718671939,1.9255682687},{5.84293775924,-3.34034923249},{5.99899408436,-0.976152223616},{6.0874022199,-2.18637422047},{6.10872630837,-2.1469759731},{6.18627869304,0.00615346981785},{6.21947892564,-2.82703737214},{6.22627375111,-2.77263955257},{6.3415462513900005,1.9790500426},{6.36576268694,-3.39377975635},{6.40801024043,0.981261218168},{6.61636676913,-1.76282303666},{6.80942966838,-0.124058579993},{6.85313991118,-0.114242607341},{6.97068735392,-0.739906186813},{6.97739476966,2.01017845061},{6.98394623201,-3.54583705011},{7.11017628975,-1.3610463906},{7.13720950708,-1.39676954563},{7.20379880859,0.375686652323},{7.2056777794,-2.57072930176},{7.24926786116,0.406311114528},{7.28873274255,1.45489385456},{7.58033997498,0.888998512678},{7.60709720734,-3.67604909992},{7.72835983481,-1.51310368435},{7.77506230862,-1.54181206932},{7.88362739315,0.459792888427},{7.9500913822,-0.537995936004},{8.00146634755,-3.1763038676},{8.08640028152,-2.09709666536},{8.35151081014,-1.64331573416},{8.378007513950001,-2.66299200725},{8.51947591142,0.490921296437},{8.74587995035,-1.14357050185},{8.83081388432,-0.0643632996082},{9.12242111675,-0.630258641494}}
PolyhedronDual[DisdyakisTriacontahedron]:=GreatRhombicosidodecahedron
PolyhedronFaces[DisdyakisTriacontahedron]:=
{{15,23,5},{41,7,57},{6,23,18},{57,8,44},{5,23,1},{57,7,2},{1,23,6},{2,8,
    57},{14,25,3},{3,33,14},{25,15,5},{7,41,33},{58,53,4},{4,53,60},{6,18,
    58},{60,44,8},{34,5,1},{7,61,2},{1,6,34},{2,61,8},{25,22,3},{3,22,33},{5,
    28,25},{33,48,7},{55,58,4},{4,60,55},{58,31,6},{8,51,60},{28,5,34},{48,61,
    7},{34,6,31},{8,61,51},{13,16,24},{24,19,21},{32,26,13},{21,29,32},{16,11,
    24},{24,35,19},{32,36,26},{29,38,32},{15,11,23},{57,35,41},{36,18,23},{57,
    44,38},{17,25,14},{14,33,27},{11,15,25},{33,41,35},{18,36,58},{44,60,
    38},{42,53,58},{60,53,47},{23,11,16},{19,35,57},{26,36,23},{57,38,29},{9,
    13,24},{24,21,9},{32,13,9},{9,21,32},{11,25,17},{27,33,35},{42,58,36},{47,
    38,60},{23,16,13},{21,19,57},{13,26,23},{57,29,21},{17,14,24},{24,14,
    27},{32,53,42},{47,53,32},{24,11,17},{27,35,24},{42,36,32},{32,38,47},{25,
    37,20},{39,33,30},{40,58,45},{50,60,12},{20,37,59},{59,39,30},{62,40,
    45},{50,12,62},{59,54,10},{10,56,59},{10,54,62},{62,56,10},{59,22,20},{30,
    22,59},{45,55,62},{62,55,50},{20,22,25},{33,22,30},{37,25,28},{33,39,
    48},{40,31,58},{60,51,12},{45,58,55},{55,60,50},{43,37,34},{61,39,46},{34,
    40,49},{52,12,61},{54,43,34},{61,46,56},{34,49,54},{56,52,61},{37,28,
    34},{61,48,39},{31,40,34},{61,12,51},{59,37,43},{46,39,59},{49,40,62},{62,
    12,52},{59,43,54},{56,46,59},{54,49,62},{62,52,56}}
PolyhedronVertices[DisdyakisTriacontahedron]:=
{{0,0,(-3(5+4Sqrt[5]))/11},{0,0,(3(5+4Sqrt[5]))/11},{0,(-3(5+4Sqrt[5]))/11,0},
{0,(3(5+4Sqrt[5]))/11,0},{0,(-5+Sqrt[5])/2,(-5-Sqrt[5])/2},{0,(5-Sqrt[5])/2,(-5-Sqrt[5])/2},
{0,(-5+Sqrt[5])/2,(5+Sqrt[5])/2},{0,(5-Sqrt[5])/2,(5+Sqrt[5])/2},{(-3(5+4Sqrt[5]))/11,0,0},
{(3(5+4Sqrt[5]))/11,0,0},{-Sqrt[5],-Sqrt[5],-Sqrt[5]},{Sqrt[5],Sqrt[5],Sqrt[5]},
{(-5-Sqrt[5])/2,0,(-5+Sqrt[5])/2},{(-5+Sqrt[5])/2,(-5-Sqrt[5])/2,0},
{(-3(15+Sqrt[5]))/44,(-3(5+4Sqrt[5]))/22,(-3(25+9Sqrt[5]))/44},
{(-3(25+9Sqrt[5]))/44,(-3(15+Sqrt[5]))/44,(-3(5+4Sqrt[5]))/22},
{(-3(5+4Sqrt[5]))/22,(-3(25+9Sqrt[5]))/44,(-3(15+Sqrt[5]))/44},
{(-3(15+Sqrt[5]))/44,(3(5+4Sqrt[5]))/22,(-3(25+9Sqrt[5]))/44},
{(-3(25+9Sqrt[5]))/44,(-3(15+Sqrt[5]))/44,(3(5+4Sqrt[5]))/22},
{(3(5+4Sqrt[5]))/22,(-3(25+9Sqrt[5]))/44,(-3(15+Sqrt[5]))/44},{(-5-Sqrt[5])/2,0,(5-Sqrt[5])/2},
{(5-Sqrt[5])/2,(-5-Sqrt[5])/2,0},{(-3(5+Sqrt[5]))/10,0,(-3(5+3Sqrt[5]))/10},
{(-3(5+3Sqrt[5]))/10,(-3(5+Sqrt[5]))/10,0},{0,(-3(5+3Sqrt[5]))/10,(-3(5+Sqrt[5]))/10},
{(-3(25+9Sqrt[5]))/44,(3(15+Sqrt[5]))/44,(-3(5+4Sqrt[5]))/22},
{(-3(5+4Sqrt[5]))/22,(-3(25+9Sqrt[5]))/44,(3(15+Sqrt[5]))/44},
{(3(15+Sqrt[5]))/44,(-3(5+4Sqrt[5]))/22,(-3(25+9Sqrt[5]))/44},
{(-3(25+9Sqrt[5]))/44,(3(15+Sqrt[5]))/44,(3(5+4Sqrt[5]))/22},
{(3(5+4Sqrt[5]))/22,(-3(25+9Sqrt[5]))/44,(3(15+Sqrt[5]))/44},
{(3(15+Sqrt[5]))/44,(3(5+4Sqrt[5]))/22,(-3(25+9Sqrt[5]))/44},
{(-3(5+3Sqrt[5]))/10,(3(5+Sqrt[5]))/10,0},{0,(-3(5+3Sqrt[5]))/10,(3(5+Sqrt[5]))/10},
{(3(5+Sqrt[5]))/10,0,(-3(5+3Sqrt[5]))/10},{-Sqrt[5],-Sqrt[5],Sqrt[5]},{-Sqrt[5],Sqrt[5],-Sqrt[5]},
{Sqrt[5],-Sqrt[5],-Sqrt[5]},{-Sqrt[5],Sqrt[5],Sqrt[5]},{Sqrt[5],-Sqrt[5],Sqrt[5]},
{Sqrt[5],Sqrt[5],-Sqrt[5]},{(-3(15+Sqrt[5]))/44,(-3(5+4Sqrt[5]))/22,(3(25+9Sqrt[5]))/44},
{(-3(5+4Sqrt[5]))/22,(3(25+9Sqrt[5]))/44,(-3(15+Sqrt[5]))/44},
{(3(25+9Sqrt[5]))/44,(-3(15+Sqrt[5]))/44,(-3(5+4Sqrt[5]))/22},
{(-3(15+Sqrt[5]))/44,(3(5+4Sqrt[5]))/22,(3(25+9Sqrt[5]))/44},
{(3(5+4Sqrt[5]))/22,(3(25+9Sqrt[5]))/44,(-3(15+Sqrt[5]))/44},
{(3(25+9Sqrt[5]))/44,(-3(15+Sqrt[5]))/44,(3(5+4Sqrt[5]))/22},
{(-3(5+4Sqrt[5]))/22,(3(25+9Sqrt[5]))/44,(3(15+Sqrt[5]))/44},
{(3(15+Sqrt[5]))/44,(-3(5+4Sqrt[5]))/22,(3(25+9Sqrt[5]))/44},
{(3(25+9Sqrt[5]))/44,(3(15+Sqrt[5]))/44,(-3(5+4Sqrt[5]))/22},
{(3(5+4Sqrt[5]))/22,(3(25+9Sqrt[5]))/44,(3(15+Sqrt[5]))/44},
{(3(15+Sqrt[5]))/44,(3(5+4Sqrt[5]))/22,(3(25+9Sqrt[5]))/44},
{(3(25+9Sqrt[5]))/44,(3(15+Sqrt[5]))/44,(3(5+4Sqrt[5]))/22},{(-5+Sqrt[5])/2,(5+Sqrt[5])/2,0},
{(5+Sqrt[5])/2,0,(-5+Sqrt[5])/2},{(5-Sqrt[5])/2,(5+Sqrt[5])/2,0},{(5+Sqrt[5])/2,0,(5-Sqrt[5])/2},
{(-3(5+Sqrt[5]))/10,0,(3(5+3Sqrt[5]))/10},{0,(3(5+3Sqrt[5]))/10,(-3(5+Sqrt[5]))/10},
{(3(5+3Sqrt[5]))/10,(-3(5+Sqrt[5]))/10,0},{0,(3(5+3Sqrt[5]))/10,(3(5+Sqrt[5]))/10},
{(3(5+Sqrt[5]))/10,0,(3(5+3Sqrt[5]))/10},{(3(5+3Sqrt[5]))/10,(3(5+Sqrt[5]))/10,0}}

(* Great Rhombicosidodecahedron *)

PolyhedronName[GreatRhombicosidodecahedron]:="great rhombicosidodecahedron"
NetFaces[GreatRhombicosidodecahedron]:=
{{16,20,34,27},{10,5,9,15,20,16},{9,8,14,15},{8,4,7,13,19,14},{19,13,26,32},{7,3,1,2,6,12,23,28,24,13},{33,22,17,21,32,38,48,52,49,39},{32,26,31,37,43,38},{38,43,54,50},{31,30,36,37},{30,25,29,35,42,36},{11,18,29,25},{64,69,86,78},{57,51,56,63,69,64},{56,55,62,63},{55,50,54,61,68,62},{68,61,77,84},{54,45,40,44,53,60,72,79,73,61},{85,71,66,70,84,90,97,101,98,91},{84,77,83,89,94,90},{90,94,103,99},{83,82,88,89},{82,76,81,87,93,88},{59,67,81,76},{112,116,130,123},{106,100,105,111,116,112},{105,104,110,111},{104,99,103,109,115,110},{115,109,122,128},{103,96,92,95,102,108,119,124,120,109},{129,118,113,117,128,134,141,145,142,135},{128,122,127,133,138,134},{134,138,147,143},{127,126,132,133},{126,121,125,131,137,132},{107,114,125,121},{156,161,178,170},{150,144,149,155,161,156},{149,148,154,155},{148,143,147,153,160,154},{160,153,169,176},{147,140,136,139,146,152,166,171,167,153},{177,165,158,164,176,183,192,197,193,184},{176,169,175,182,187,183},{183,187,199,194},{175,174,181,182},{174,168,173,180,186,181},{151,159,173,168},{208,212,226,219},{202,195,201,207,212,208},{201,200,206,207},{200,194,199,205,211,206},{211,205,218,224},{199,189,185,188,198,204,215,220,216,205},{225,214,209,213,224,230,234,237,235,231},{224,218,223,229,233,230},{230,233,238,236},{223,222,228,229},{222,217,221,227,232,228},{203,210,221,217},{58,47,41,46,57,64,74,80,75,65},{173,163,157,162,172,179,190,196,191,180}}
NetVertices[GreatRhombicosidodecahedron]:={{(-1-Sqrt[5])/2,(-8-12Sqrt[3]-Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(-3-Sqrt[5])/4,(-8-12Sqrt[3]-3Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(-3-Sqrt[5])/4,(-8-12Sqrt[3]+Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{-1,-1-Sqrt[3]},{-1,0},{-1/2,(-4-6Sqrt[3]-Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/4},{-1/2,(-2-3Sqrt[3])/2},{-1/2,(-2-Sqrt[3])/2},{-1/2,-Sqrt[3]/2},{-1/2,Sqrt[3]/2},{1/2,(-6-5Sqrt[3])/2},{1/2,(-4-6Sqrt[3]-Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/4},{1/2,(-2-3Sqrt[3])/2},{1/2,(-2-Sqrt[3])/2},{1/2,-Sqrt[3]/2},{1/2,Sqrt[3]/2},{(2+Sqrt[3]-Sqrt[5])/2,(-12-8Sqrt[3]+Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/8},{1,-3(1+Sqrt[3])},{1,-1-Sqrt[3]},{1,0},{(3+2Sqrt[3]-Sqrt[5])/4,(-12-8Sqrt[3]-Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/8},{(3+2Sqrt[3]-Sqrt[5])/4,(-12-8Sqrt[3]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/8},{(3+Sqrt[5])/4,(-8-12Sqrt[3]-3Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(3+Sqrt[5])/4,(-8-12Sqrt[3]+Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(1+Sqrt[3])/2,(-5(1+Sqrt[3]))/2},{(1+Sqrt[3])/2,(-3(1+Sqrt[3]))/2},{(1+Sqrt[3])/2,(1+Sqrt[3])/2},{(1+Sqrt[5])/2,(-8-12Sqrt[3]-Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(2+Sqrt[3])/2,(-5-6Sqrt[3])/2},{(2+Sqrt[3])/2,(-5-4Sqrt[3])/2},{(2+Sqrt[3])/2,(-3-4Sqrt[3])/2},{(2+Sqrt[3])/2,(-3-2Sqrt[3])/2},{(2+Sqrt[3])/2,(-6-4Sqrt[3]+Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/4},{(2+Sqrt[3])/2,1/2},{(4+Sqrt[3])/2,(-5-6Sqrt[3])/2},{(4+Sqrt[3])/2,(-5-4Sqrt[3])/2},{(4+Sqrt[3])/2,(-3-4Sqrt[3])/2},{(4+Sqrt[3])/2,(-3-2Sqrt[3])/2},{(4+Sqrt[3])/2,(-6-4Sqrt[3]+Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/4},{(5+2Sqrt[3]-Sqrt[5])/2,(-8-12Sqrt[3]-Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(5+2Sqrt[3]-Sqrt[5])/2,(4Sqrt[3]+Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/8},{(5+Sqrt[3])/2,(-5(1+Sqrt[3]))/2},{(5+Sqrt[3])/2,(-3(1+Sqrt[3]))/2},{(9+4Sqrt[3]-Sqrt[5])/4,(-8-12Sqrt[3]-3Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(9+4Sqrt[3]-Sqrt[5])/4,(-8-12Sqrt[3]+Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(9+4Sqrt[3]-Sqrt[5])/4,(4Sqrt[3]-Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/8},{(9+4Sqrt[3]-Sqrt[5])/4,(4Sqrt[3]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/8},{(9+2Sqrt[3]+Sqrt[5])/4,(-12-8Sqrt[3]-Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/8},{(9+2Sqrt[3]+Sqrt[5])/4,(-12-8Sqrt[3]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/8},{2+Sqrt[3],-1-Sqrt[3]},{2+Sqrt[3],0},{(4+Sqrt[3]+Sqrt[5])/2,(-12-8Sqrt[3]+Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/8},{(5+2Sqrt[3])/2,(-4-6Sqrt[3]-Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/4},{(5+2Sqrt[3])/2,(-2-3Sqrt[3])/2},{(5+2Sqrt[3])/2,(-2-Sqrt[3])/2},{(5+2Sqrt[3])/2,-Sqrt[3]/2},{(5+2Sqrt[3])/2,Sqrt[3]/2},{(5+2Sqrt[3])/2,(2Sqrt[3]+Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/4},{(7+2Sqrt[3])/2,(-6-5Sqrt[3])/2},{(7+2Sqrt[3])/2,(-4-6Sqrt[3]-Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/4},{(7+2Sqrt[3])/2,(-2-3Sqrt[3])/2},{(7+2Sqrt[3])/2,(-2-Sqrt[3])/2},{(7+2Sqrt[3])/2,-Sqrt[3]/2},{(7+2Sqrt[3])/2,Sqrt[3]/2},{(7+2Sqrt[3])/2,(2Sqrt[3]+Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/4},{(8+3Sqrt[3]-Sqrt[5])/2,(-12-8Sqrt[3]+Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/8},{4+Sqrt[3],-3(1+Sqrt[3])},{4+Sqrt[3],-1-Sqrt[3]},{4+Sqrt[3],0},{(15+6Sqrt[3]-Sqrt[5])/4,(-12-8Sqrt[3]-Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/8},{(15+6Sqrt[3]-Sqrt[5])/4,(-12-8Sqrt[3]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/8},{(15+4Sqrt[3]+Sqrt[5])/4,(-8-12Sqrt[3]-3Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(15+4Sqrt[3]+Sqrt[5])/4,(-8-12Sqrt[3]+Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(15+4Sqrt[3]+Sqrt[5])/4,(4Sqrt[3]-Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/8},{(15+4Sqrt[3]+Sqrt[5])/4,(4Sqrt[3]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/8},{(7+3Sqrt[3])/2,(-5(1+Sqrt[3]))/2},{(7+3Sqrt[3])/2,(-3(1+Sqrt[3]))/2},{(7+3Sqrt[3])/2,(1+Sqrt[3])/2},{(7+2Sqrt[3]+Sqrt[5])/2,(-8-12Sqrt[3]-Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(7+2Sqrt[3]+Sqrt[5])/2,(4Sqrt[3]+Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/8},{(8+3Sqrt[3])/2,(-5-6Sqrt[3])/2},{(8+3Sqrt[3])/2,(-5-4Sqrt[3])/2},{(8+3Sqrt[3])/2,(-3-4Sqrt[3])/2},{(8+3Sqrt[3])/2,(-3-2Sqrt[3])/2},{(8+3Sqrt[3])/2,(-6-4Sqrt[3]+Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/4},{(8+3Sqrt[3])/2,1/2},{(10+3Sqrt[3])/2,(-5-6Sqrt[3])/2},{(10+3Sqrt[3])/2,(-5-4Sqrt[3])/2},{(10+3Sqrt[3])/2,(-3-4Sqrt[3])/2},{(10+3Sqrt[3])/2,(-3-2Sqrt[3])/2},{(10+3Sqrt[3])/2,(-6-4Sqrt[3]+Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/4},{(11+4Sqrt[3]-Sqrt[5])/2,(-8-12Sqrt[3]-Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(11+3Sqrt[3])/2,(-5(1+Sqrt[3]))/2},{(11+3Sqrt[3])/2,(-3(1+Sqrt[3]))/2},{(21+8Sqrt[3]-Sqrt[5])/4,(-8-12Sqrt[3]-3Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(21+8Sqrt[3]-Sqrt[5])/4,(-8-12Sqrt[3]+Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(21+6Sqrt[3]+Sqrt[5])/4,(-12-8Sqrt[3]-Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/8},{(21+6Sqrt[3]+Sqrt[5])/4,(-12-8Sqrt[3]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/8},{5+2Sqrt[3],-1-Sqrt[3]},{5+2Sqrt[3],0},{(10+3Sqrt[3]+Sqrt[5])/2,(-12-8Sqrt[3]+Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/8},{(11+4Sqrt[3])/2,(-4-6Sqrt[3]-Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/4},{(11+4Sqrt[3])/2,(-2-3Sqrt[3])/2},{(11+4Sqrt[3])/2,(-2-Sqrt[3])/2},{(11+4Sqrt[3])/2,-Sqrt[3]/2},{(11+4Sqrt[3])/2,Sqrt[3]/2},{(13+4Sqrt[3])/2,(-6-5Sqrt[3])/2},{(13+4Sqrt[3])/2,(-4-6Sqrt[3]-Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/4},{(13+4Sqrt[3])/2,(-2-3Sqrt[3])/2},{(13+4Sqrt[3])/2,(-2-Sqrt[3])/2},{(13+4Sqrt[3])/2,-Sqrt[3]/2},{(13+4Sqrt[3])/2,Sqrt[3]/2},{(14+5Sqrt[3]-Sqrt[5])/2,(-12-8Sqrt[3]+Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/8},{7+2Sqrt[3],-3(1+Sqrt[3])},{7+2Sqrt[3],-1-Sqrt[3]},{7+2Sqrt[3],0},{(27+10Sqrt[3]-Sqrt[5])/4,(-12-8Sqrt[3]-Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/8},{(27+10Sqrt[3]-Sqrt[5])/4,(-12-8Sqrt[3]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/8},{(27+8Sqrt[3]+Sqrt[5])/4,(-8-12Sqrt[3]-3Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(27+8Sqrt[3]+Sqrt[5])/4,(-8-12Sqrt[3]+Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(13+5Sqrt[3])/2,(-5(1+Sqrt[3]))/2},{(13+5Sqrt[3])/2,(-3(1+Sqrt[3]))/2},{(13+5Sqrt[3])/2,(1+Sqrt[3])/2},{(13+4Sqrt[3]+Sqrt[5])/2,(-8-12Sqrt[3]-Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(14+5Sqrt[3])/2,(-5-6Sqrt[3])/2},{(14+5Sqrt[3])/2,(-5-4Sqrt[3])/2},{(14+5Sqrt[3])/2,(-3-4Sqrt[3])/2},{(14+5Sqrt[3])/2,(-3-2Sqrt[3])/2},{(14+5Sqrt[3])/2,(-6-4Sqrt[3]+Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/4},{(14+5Sqrt[3])/2,1/2},{(16+5Sqrt[3])/2,(-5-6Sqrt[3])/2},{(16+5Sqrt[3])/2,(-5-4Sqrt[3])/2},{(16+5Sqrt[3])/2,(-3-4Sqrt[3])/2},{(16+5Sqrt[3])/2,(-3-2Sqrt[3])/2},{(16+5Sqrt[3])/2,(-6-4Sqrt[3]+Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/4},{(17+6Sqrt[3]-Sqrt[5])/2,(-8-12Sqrt[3]-Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(17+5Sqrt[3])/2,(-5(1+Sqrt[3]))/2},{(17+5Sqrt[3])/2,(-3(1+Sqrt[3]))/2},{(33+12Sqrt[3]-Sqrt[5])/4,(-8-12Sqrt[3]-3Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(33+12Sqrt[3]-Sqrt[5])/4,(-8-12Sqrt[3]+Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(33+10Sqrt[3]+Sqrt[5])/4,(-12-8Sqrt[3]-Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/8},{(33+10Sqrt[3]+Sqrt[5])/4,(-12-8Sqrt[3]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/8},{8+3Sqrt[3],-1-Sqrt[3]},{8+3Sqrt[3],0},{(16+5Sqrt[3]+Sqrt[5])/2,(-12-8Sqrt[3]+Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/8},{(17+6Sqrt[3])/2,(-4-6Sqrt[3]-Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/4},{(17+6Sqrt[3])/2,(-2-3Sqrt[3])/2},{(17+6Sqrt[3])/2,(-2-Sqrt[3])/2},{(17+6Sqrt[3])/2,-Sqrt[3]/2},{(17+6Sqrt[3])/2,Sqrt[3]/2},{(19+6Sqrt[3])/2,(-6-5Sqrt[3])/2},{(19+6Sqrt[3])/2,(-4-6Sqrt[3]-Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/4},{(19+6Sqrt[3])/2,(-2-3Sqrt[3])/2},{(19+6Sqrt[3])/2,(-2-Sqrt[3])/2},{(19+6Sqrt[3])/2,-Sqrt[3]/2},{(19+6Sqrt[3])/2,Sqrt[3]/2},{(20+7Sqrt[3]-Sqrt[5])/2,(-20-24Sqrt[3]-Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(20+7Sqrt[3]-Sqrt[5])/2,(-12-8Sqrt[3]+Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/8},{10+3Sqrt[3],-3(1+Sqrt[3])},{10+3Sqrt[3],-1-Sqrt[3]},{10+3Sqrt[3],0},{(39+14Sqrt[3]-Sqrt[5])/4,(-20-24Sqrt[3]-3Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(39+14Sqrt[3]-Sqrt[5])/4,(-20-24Sqrt[3]+Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(39+14Sqrt[3]-Sqrt[5])/4,(-12-8Sqrt[3]-Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/8},{(39+14Sqrt[3]-Sqrt[5])/4,(-12-8Sqrt[3]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/8},{(39+12Sqrt[3]+Sqrt[5])/4,(-8-12Sqrt[3]-3Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(39+12Sqrt[3]+Sqrt[5])/4,(-8-12Sqrt[3]+Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(19+7Sqrt[3])/2,(-5(1+Sqrt[3]))/2},{(19+7Sqrt[3])/2,(-3(1+Sqrt[3]))/2},{(19+7Sqrt[3])/2,(1+Sqrt[3])/2},{(19+6Sqrt[3]+Sqrt[5])/2,(-8-12Sqrt[3]-Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(20+7Sqrt[3])/2,(-10-12Sqrt[3]-Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/4},{(20+7Sqrt[3])/2,(-5-6Sqrt[3])/2},{(20+7Sqrt[3])/2,(-5-4Sqrt[3])/2},{(20+7Sqrt[3])/2,(-3-4Sqrt[3])/2},{(20+7Sqrt[3])/2,(-3-2Sqrt[3])/2},{(20+7Sqrt[3])/2,(-6-4Sqrt[3]+Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/4},{(20+7Sqrt[3])/2,1/2},{(22+7Sqrt[3])/2,(-10-12Sqrt[3]-Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/4},{(22+7Sqrt[3])/2,(-5-6Sqrt[3])/2},{(22+7Sqrt[3])/2,(-5-4Sqrt[3])/2},{(22+7Sqrt[3])/2,(-3-4Sqrt[3])/2},{(22+7Sqrt[3])/2,(-3-2Sqrt[3])/2},{(22+7Sqrt[3])/2,(-6-4Sqrt[3]+Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/4},{(23+8Sqrt[3]-Sqrt[5])/2,(-8-12Sqrt[3]-Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(23+7Sqrt[3])/2,(-5(1+Sqrt[3]))/2},{(23+7Sqrt[3])/2,(-3(1+Sqrt[3]))/2},{(45+16Sqrt[3]-Sqrt[5])/4,(-8-12Sqrt[3]-3Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(45+16Sqrt[3]-Sqrt[5])/4,(-8-12Sqrt[3]+Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(45+14Sqrt[3]+Sqrt[5])/4,(-20-24Sqrt[3]-3Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(45+14Sqrt[3]+Sqrt[5])/4,(-20-24Sqrt[3]+Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(45+14Sqrt[3]+Sqrt[5])/4,(-12-8Sqrt[3]-Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/8},{(45+14Sqrt[3]+Sqrt[5])/4,(-12-8Sqrt[3]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/8},{11+4Sqrt[3],-1-Sqrt[3]},{11+4Sqrt[3],0},{(22+7Sqrt[3]+Sqrt[5])/2,(-20-24Sqrt[3]-Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(22+7Sqrt[3]+Sqrt[5])/2,(-12-8Sqrt[3]+Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/8},{(23+8Sqrt[3])/2,(-4-6Sqrt[3]-Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/4},{(23+8Sqrt[3])/2,(-2-3Sqrt[3])/2},{(23+8Sqrt[3])/2,(-2-Sqrt[3])/2},{(23+8Sqrt[3])/2,-Sqrt[3]/2},{(23+8Sqrt[3])/2,Sqrt[3]/2},{(25+8Sqrt[3])/2,(-6-5Sqrt[3])/2},{(25+8Sqrt[3])/2,(-4-6Sqrt[3]-Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/4},{(25+8Sqrt[3])/2,(-2-3Sqrt[3])/2},{(25+8Sqrt[3])/2,(-2-Sqrt[3])/2},{(25+8Sqrt[3])/2,-Sqrt[3]/2},{(25+8Sqrt[3])/2,Sqrt[3]/2},{(26+9Sqrt[3]-Sqrt[5])/2,(-12-8Sqrt[3]+Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/8},{13+4Sqrt[3],-3(1+Sqrt[3])},{13+4Sqrt[3],-1-Sqrt[3]},{13+4Sqrt[3],0},{(51+18Sqrt[3]-Sqrt[5])/4,(-12-8Sqrt[3]-Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/8},{(51+18Sqrt[3]-Sqrt[5])/4,(-12-8Sqrt[3]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/8},{(51+16Sqrt[3]+Sqrt[5])/4,(-8-12Sqrt[3]-3Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(51+16Sqrt[3]+Sqrt[5])/4,(-8-12Sqrt[3]+Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(25+9Sqrt[3])/2,(-5(1+Sqrt[3]))/2},{(25+9Sqrt[3])/2,(-3(1+Sqrt[3]))/2},{(25+9Sqrt[3])/2,(1+Sqrt[3])/2},{(25+8Sqrt[3]+Sqrt[5])/2,(-8-12Sqrt[3]-Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(26+9Sqrt[3])/2,(-5-6Sqrt[3])/2},{(26+9Sqrt[3])/2,(-5-4Sqrt[3])/2},{(26+9Sqrt[3])/2,(-3-4Sqrt[3])/2},{(26+9Sqrt[3])/2,(-3-2Sqrt[3])/2},{(26+9Sqrt[3])/2,(-6-4Sqrt[3]+Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/4},{(26+9Sqrt[3])/2,1/2},{(28+9Sqrt[3])/2,(-5-6Sqrt[3])/2},{(28+9Sqrt[3])/2,(-5-4Sqrt[3])/2},{(28+9Sqrt[3])/2,(-3-4Sqrt[3])/2},{(28+9Sqrt[3])/2,(-3-2Sqrt[3])/2},{(28+9Sqrt[3])/2,(-6-4Sqrt[3]+Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/4},{(29+9Sqrt[3])/2,(-5(1+Sqrt[3]))/2},{(29+9Sqrt[3])/2,(-3(1+Sqrt[3]))/2},{(57+18Sqrt[3]+Sqrt[5])/4,(-12-8Sqrt[3]-Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/8},{(57+18Sqrt[3]+Sqrt[5])/4,(-12-8Sqrt[3]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/8},{14+5Sqrt[3],-1-Sqrt[3]},{(28+9Sqrt[3]+Sqrt[5])/2,(-12-8Sqrt[3]+Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/8},{(29+10Sqrt[3])/2,(-2-3Sqrt[3])/2}}
PolyhedronDual[GreatRhombicosidodecahedron]:=DisdyakisTriacontahedron
PolyhedronFaces[GreatRhombicosidodecahedron]:=
{{2,6,8,4,44,56,68,66,54,42},{109,29,17,19,31,111,103,107,105,101},{24,30,18,
    6,2,12},{7,3,15,27,31,19},{58,57,33,37,73,69,70,74,38,34},{84,116,120,88,
    87,119,115,83,91,92},{90,89,81,113,117,85,86,118,114,82},{36,40,76,72,71,
    75,39,35,59,60},{5,17,29,23,11,1},{4,8,20,32,28,16},{67,55,43,3,7,5,1,41,
    53,65},{18,30,110,102,106,108,104,112,32,20},{79,83,115,103,111,97},{38,
    74,62,48,42,54},{4,16,50,44},{23,29,109,95},{96,110,30,24},{43,49,15,
    3},{53,41,47,61,73,37},{98,112,104,116,84,80},{69,45,9,10,46,70},{26,100,
    92,91,99,25},{82,114,102,110,96,78},{55,39,75,63,49,43},{1,11,47,41},{28,
    32,112,98},{61,47,11,23,95,77,93,21,9,45},{50,16,28,98,80,100,26,14,52,
    64},{97,111,31,27},{42,48,12,2},{44,50,64,76,40,56},{77,95,109,101,113,
    81},{63,51,13,25,99,79,97,27,15,49},{46,10,22,94,78,96,24,12,48,62},{52,
    14,13,51,71,72},{22,21,93,89,90,94},{115,119,107,103},{34,38,54,66},{71,
    51,63,75},{94,90,82,78},{114,118,106,102},{35,39,55,67},{70,46,62,74},{99,
    91,83,79},{65,53,37,33},{104,108,120,116},{77,81,89,93},{76,64,52,72},{59,
    35,67,65,33,57},{106,118,86,88,120,108},{68,56,40,36},{101,105,117,
    113},{80,84,92,100},{73,61,45,69},{34,66,68,36,60,58},{105,107,119,87,85,
    117},{7,19,17,5},{6,18,20,8},{14,26,25,13},{9,21,22,10},{58,60,59,57},{85,
    87,88,86}}
PolyhedronVertices[GreatRhombicosidodecahedron]:=
{{-1,(-3-Sqrt[5])/4,(-7-3Sqrt[5])/4},
{-1,(-3-Sqrt[5])/4,(7+3Sqrt[5])/4},
{-1,(3+Sqrt[5])/4,(-7-3Sqrt[5])/4},
{-1,(3+Sqrt[5])/4,(7+3Sqrt[5])/4},
{-1/2,-1/2,-3/2-Sqrt[5]},{-1/2,-1/2,3/2+Sqrt[5]},
{-1/2,1/2,-3/2-Sqrt[5]},{-1/2,1/2,3/2+Sqrt[5]},
{-1/2,-3/2-Sqrt[5],-1/2},{-1/2,-3/2-Sqrt[5],1/2},
{-1/2,-1-Sqrt[5]/2,-2-Sqrt[5]/2},
{-1/2,-1-Sqrt[5]/2,(4+Sqrt[5])/2},
{-1/2,3/2+Sqrt[5],-1/2},{-1/2,3/2+Sqrt[5],1/2},
{-1/2,(2+Sqrt[5])/2,-2-Sqrt[5]/2},
{-1/2,(2+Sqrt[5])/2,(4+Sqrt[5])/2},
{1/2,-1/2,-3/2-Sqrt[5]},{1/2,-1/2,3/2+Sqrt[5]},
{1/2,1/2,-3/2-Sqrt[5]},{1/2,1/2,3/2+Sqrt[5]},
{1/2,-3/2-Sqrt[5],-1/2},{1/2,-3/2-Sqrt[5],1/2},
{1/2,-1-Sqrt[5]/2,-2-Sqrt[5]/2},
{1/2,-1-Sqrt[5]/2,(4+Sqrt[5])/2},
{1/2,3/2+Sqrt[5],-1/2},{1/2,3/2+Sqrt[5],1/2},
{1/2,(2+Sqrt[5])/2,-2-Sqrt[5]/2},
{1/2,(2+Sqrt[5])/2,(4+Sqrt[5])/2},
{1,(-3-Sqrt[5])/4,(-7-3Sqrt[5])/4},
{1,(-3-Sqrt[5])/4,(7+3Sqrt[5])/4},
{1,(3+Sqrt[5])/4,(-7-3Sqrt[5])/4},
{1,(3+Sqrt[5])/4,(7+3Sqrt[5])/4},
{(-7-3Sqrt[5])/4,-1,(-3-Sqrt[5])/4},
{(-7-3Sqrt[5])/4,-1,(3+Sqrt[5])/4},
{(-7-3Sqrt[5])/4,1,(-3-Sqrt[5])/4},
{(-7-3Sqrt[5])/4,1,(3+Sqrt[5])/4},
{(-5-3Sqrt[5])/4,(-5-Sqrt[5])/4,(-1-Sqrt[5])/2},
{(-5-3Sqrt[5])/4,(-5-Sqrt[5])/4,(1+Sqrt[5])/2},
{(-5-3Sqrt[5])/4,(5+Sqrt[5])/4,(-1-Sqrt[5])/2},
{(-5-3Sqrt[5])/4,(5+Sqrt[5])/4,(1+Sqrt[5])/2},
{(-5-Sqrt[5])/4,(-1-Sqrt[5])/2,(-5-3Sqrt[5])/4},
{(-5-Sqrt[5])/4,(-1-Sqrt[5])/2,(5+3Sqrt[5])/4},
{(-5-Sqrt[5])/4,(1+Sqrt[5])/2,(-5-3Sqrt[5])/4},
{(-5-Sqrt[5])/4,(1+Sqrt[5])/2,(5+3Sqrt[5])/4},
{(-3-Sqrt[5])/4,(-7-3Sqrt[5])/4,-1},
{(-3-Sqrt[5])/4,(-7-3Sqrt[5])/4,1},
{(-3-Sqrt[5])/4,(-3(1+Sqrt[5]))/4,(-3-Sqrt[5])/2},
{(-3-Sqrt[5])/4,(-3(1+Sqrt[5]))/4,(3+Sqrt[5])/2},
{(-3-Sqrt[5])/4,(3(1+Sqrt[5]))/4,(-3-Sqrt[5])/2},
{(-3-Sqrt[5])/4,(3(1+Sqrt[5]))/4,(3+Sqrt[5])/2},
{(-3-Sqrt[5])/4,(7+3Sqrt[5])/4,-1},
{(-3-Sqrt[5])/4,(7+3Sqrt[5])/4,1},
{(-3-Sqrt[5])/2,(-3-Sqrt[5])/4,(-3(1+Sqrt[5]))/4},
{(-3-Sqrt[5])/2,(-3-Sqrt[5])/4,(3(1+Sqrt[5]))/4},
{(-3-Sqrt[5])/2,(3+Sqrt[5])/4,(-3(1+Sqrt[5]))/4},
{(-3-Sqrt[5])/2,(3+Sqrt[5])/4,(3(1+Sqrt[5]))/4},
{-3/2-Sqrt[5],-1/2,-1/2},{-3/2-Sqrt[5],-1/2,1/2},
{-3/2-Sqrt[5],1/2,-1/2},{-3/2-Sqrt[5],1/2,1/2},
{(-1-Sqrt[5])/2,(-5-3Sqrt[5])/4,(-5-Sqrt[5])/4},
{(-1-Sqrt[5])/2,(-5-3Sqrt[5])/4,(5+Sqrt[5])/4},
{(-1-Sqrt[5])/2,(5+3Sqrt[5])/4,(-5-Sqrt[5])/4},
{(-1-Sqrt[5])/2,(5+3Sqrt[5])/4,(5+Sqrt[5])/4},
{-2-Sqrt[5]/2,-1/2,-1-Sqrt[5]/2},
{-2-Sqrt[5]/2,-1/2,(2+Sqrt[5])/2},
{-2-Sqrt[5]/2,1/2,-1-Sqrt[5]/2},
{-2-Sqrt[5]/2,1/2,(2+Sqrt[5])/2},
{-1-Sqrt[5]/2,-2-Sqrt[5]/2,-1/2},
{-1-Sqrt[5]/2,-2-Sqrt[5]/2,1/2},
{-1-Sqrt[5]/2,(4+Sqrt[5])/2,-1/2},
{-1-Sqrt[5]/2,(4+Sqrt[5])/2,1/2},
{(-3(1+Sqrt[5]))/4,(-3-Sqrt[5])/2,(-3-Sqrt[5])/4},
{(-3(1+Sqrt[5]))/4,(-3-Sqrt[5])/2,(3+Sqrt[5])/4},
{(-3(1+Sqrt[5]))/4,(3+Sqrt[5])/2,(-3-Sqrt[5])/4},
{(-3(1+Sqrt[5]))/4,(3+Sqrt[5])/2,(3+Sqrt[5])/4},
{(1+Sqrt[5])/2,(-5-3Sqrt[5])/4,(-5-Sqrt[5])/4},
{(1+Sqrt[5])/2,(-5-3Sqrt[5])/4,(5+Sqrt[5])/4},
{(1+Sqrt[5])/2,(5+3Sqrt[5])/4,(-5-Sqrt[5])/4},
{(1+Sqrt[5])/2,(5+3Sqrt[5])/4,(5+Sqrt[5])/4},
{(3(1+Sqrt[5]))/4,(-3-Sqrt[5])/2,(-3-Sqrt[5])/4},
{(3(1+Sqrt[5]))/4,(-3-Sqrt[5])/2,(3+Sqrt[5])/4},
{(3(1+Sqrt[5]))/4,(3+Sqrt[5])/2,(-3-Sqrt[5])/4},
{(3(1+Sqrt[5]))/4,(3+Sqrt[5])/2,(3+Sqrt[5])/4},
{3/2+Sqrt[5],-1/2,-1/2},{3/2+Sqrt[5],-1/2,1/2},
{3/2+Sqrt[5],1/2,-1/2},{3/2+Sqrt[5],1/2,1/2},
{(2+Sqrt[5])/2,-2-Sqrt[5]/2,-1/2},
{(2+Sqrt[5])/2,-2-Sqrt[5]/2,1/2},
{(2+Sqrt[5])/2,(4+Sqrt[5])/2,-1/2},
{(2+Sqrt[5])/2,(4+Sqrt[5])/2,1/2},
{(3+Sqrt[5])/4,(-7-3Sqrt[5])/4,-1},
{(3+Sqrt[5])/4,(-7-3Sqrt[5])/4,1},
{(3+Sqrt[5])/4,(-3(1+Sqrt[5]))/4,(-3-Sqrt[5])/2},
{(3+Sqrt[5])/4,(-3(1+Sqrt[5]))/4,(3+Sqrt[5])/2},
{(3+Sqrt[5])/4,(3(1+Sqrt[5]))/4,(-3-Sqrt[5])/2},
{(3+Sqrt[5])/4,(3(1+Sqrt[5]))/4,(3+Sqrt[5])/2},
{(3+Sqrt[5])/4,(7+3Sqrt[5])/4,-1},
{(3+Sqrt[5])/4,(7+3Sqrt[5])/4,1},
{(3+Sqrt[5])/2,(-3-Sqrt[5])/4,(-3(1+Sqrt[5]))/4},
{(3+Sqrt[5])/2,(-3-Sqrt[5])/4,(3(1+Sqrt[5]))/4},
{(3+Sqrt[5])/2,(3+Sqrt[5])/4,(-3(1+Sqrt[5]))/4},
{(3+Sqrt[5])/2,(3+Sqrt[5])/4,(3(1+Sqrt[5]))/4},
{(4+Sqrt[5])/2,-1/2,-1-Sqrt[5]/2},
{(4+Sqrt[5])/2,-1/2,(2+Sqrt[5])/2},
{(4+Sqrt[5])/2,1/2,-1-Sqrt[5]/2},
{(4+Sqrt[5])/2,1/2,(2+Sqrt[5])/2},
{(5+Sqrt[5])/4,(-1-Sqrt[5])/2,(-5-3Sqrt[5])/4},
{(5+Sqrt[5])/4,(-1-Sqrt[5])/2,(5+3Sqrt[5])/4},
{(5+Sqrt[5])/4,(1+Sqrt[5])/2,(-5-3Sqrt[5])/4},
{(5+Sqrt[5])/4,(1+Sqrt[5])/2,(5+3Sqrt[5])/4},
{(5+3Sqrt[5])/4,(-5-Sqrt[5])/4,(-1-Sqrt[5])/2},
{(5+3Sqrt[5])/4,(-5-Sqrt[5])/4,(1+Sqrt[5])/2},
{(5+3Sqrt[5])/4,(5+Sqrt[5])/4,(-1-Sqrt[5])/2},
{(5+3Sqrt[5])/4,(5+Sqrt[5])/4,(1+Sqrt[5])/2},
{(7+3Sqrt[5])/4,-1,(-3-Sqrt[5])/4},
{(7+3Sqrt[5])/4,-1,(3+Sqrt[5])/4},
{(7+3Sqrt[5])/4,1,(-3-Sqrt[5])/4},
{(7+3Sqrt[5])/4,1,(3+Sqrt[5])/4}}

(* Great Rhombicuboctahedron *)

PolyhedronName[GreatRhombicuboctahedron]:="great rhombicuboctahedron"
NetFaces[GreatRhombicuboctahedron]:={{6,5,9,10},{5,2,1,4,8,14,15,9},{4,3,7,8},{16,12,15,21,26,22},{15,14,20,21},{14,11,13,19,25,20},{31,30,36,37},{30,21,20,29,35,44,45,36},{29,28,34,35},{46,40,45,51,54,52},{45,44,50,51},{44,39,43,49,53,50},{58,57,61,62},{57,51,50,56,60,66,67,61},{56,55,59,60},{68,64,67,71,74,72},{67,66,70,71},{66,63,65,69,73,70},{78,77,81,82},{77,71,70,76,80,86,87,81},{76,75,79,80},{88,84,87,91,94,92},{87,86,90,91},{86,83,85,89,93,90},{32,24,23,31,37,47,48,38},{28,18,17,27,33,41,42,34}}
NetVertices[GreatRhombicuboctahedron]:={{(-1-Sqrt[2])/2,(-3-Sqrt[2])/2},{(-1-Sqrt[2])/2,(-1-Sqrt[2])/2},{-1/2,(-5-2Sqrt[2])/2},{-1/2,(-3-2Sqrt[2])/2},{-1/2,-1/2},{-1/2,1/2},{1/2,(-5-2Sqrt[2])/2},{1/2,(-3-2Sqrt[2])/2},{1/2,-1/2},{1/2,1/2},{1/Sqrt[2],(-3-Sqrt[2]-Sqrt[3])/2},{1/Sqrt[2],(-1-Sqrt[2]+Sqrt[3])/2},{(1+Sqrt[2])/2,(-3-Sqrt[2]-2Sqrt[3])/2},{(1+Sqrt[2])/2,(-3-Sqrt[2])/2},{(1+Sqrt[2])/2,(-1-Sqrt[2])/2},{(1+Sqrt[2])/2,(-1-Sqrt[2]+2Sqrt[3])/2},{(3+Sqrt[2])/2,(-7-3Sqrt[2])/2},{(3+Sqrt[2])/2,(-5-3Sqrt[2])/2},{(3+Sqrt[2])/2,(-3-Sqrt[2]-2Sqrt[3])/2},{(3+Sqrt[2])/2,(-3-Sqrt[2])/2},{(3+Sqrt[2])/2,(-1-Sqrt[2])/2},{(3+Sqrt[2])/2,(-1-Sqrt[2]+2Sqrt[3])/2},{(3+Sqrt[2])/2,(1+Sqrt[2])/2},{(3+Sqrt[2])/2,(3+Sqrt[2])/2},{(4+Sqrt[2])/2,(-3-Sqrt[2]-Sqrt[3])/2},{(4+Sqrt[2])/2,(-1-Sqrt[2]+Sqrt[3])/2},{(3+2Sqrt[2])/2,(-7-4Sqrt[2])/2},{(3+2Sqrt[2])/2,(-5-2Sqrt[2])/2},{(3+2Sqrt[2])/2,(-3-2Sqrt[2])/2},{(3+2Sqrt[2])/2,-1/2},{(3+2Sqrt[2])/2,1/2},{(3+2Sqrt[2])/2,(3+2Sqrt[2])/2},{(5+2Sqrt[2])/2,(-7-4Sqrt[2])/2},{(5+2Sqrt[2])/2,(-5-2Sqrt[2])/2},{(5+2Sqrt[2])/2,(-3-2Sqrt[2])/2},{(5+2Sqrt[2])/2,-1/2},{(5+2Sqrt[2])/2,1/2},{(5+2Sqrt[2])/2,(3+2Sqrt[2])/2},{(4+3Sqrt[2])/2,(-3-Sqrt[2]-Sqrt[3])/2},{(4+3Sqrt[2])/2,(-1-Sqrt[2]+Sqrt[3])/2},{(5+3Sqrt[2])/2,(-7-3Sqrt[2])/2},{(5+3Sqrt[2])/2,(-5-3Sqrt[2])/2},{(5+3Sqrt[2])/2,(-3-Sqrt[2]-2Sqrt[3])/2},{(5+3Sqrt[2])/2,(-3-Sqrt[2])/2},{(5+3Sqrt[2])/2,(-1-Sqrt[2])/2},{(5+3Sqrt[2])/2,(-1-Sqrt[2]+2Sqrt[3])/2},{(5+3Sqrt[2])/2,(1+Sqrt[2])/2},{(5+3Sqrt[2])/2,(3+Sqrt[2])/2},{(7+3Sqrt[2])/2,(-3-Sqrt[2]-2Sqrt[3])/2},{(7+3Sqrt[2])/2,(-3-Sqrt[2])/2},{(7+3Sqrt[2])/2,(-1-Sqrt[2])/2},{(7+3Sqrt[2])/2,(-1-Sqrt[2]+2Sqrt[3])/2},{(8+3Sqrt[2])/2,(-3-Sqrt[2]-Sqrt[3])/2},{(8+3Sqrt[2])/2,(-1-Sqrt[2]+Sqrt[3])/2},{(7+4Sqrt[2])/2,(-5-2Sqrt[2])/2},{(7+4Sqrt[2])/2,(-3-2Sqrt[2])/2},{(7+4Sqrt[2])/2,-1/2},{(7+4Sqrt[2])/2,1/2},{(9+4Sqrt[2])/2,(-5-2Sqrt[2])/2},{(9+4Sqrt[2])/2,(-3-2Sqrt[2])/2},{(9+4Sqrt[2])/2,-1/2},{(9+4Sqrt[2])/2,1/2},{(8+5Sqrt[2])/2,(-3-Sqrt[2]-Sqrt[3])/2},{(8+5Sqrt[2])/2,(-1-Sqrt[2]+Sqrt[3])/2},{(9+5Sqrt[2])/2,(-3-Sqrt[2]-2Sqrt[3])/2},{(9+5Sqrt[2])/2,(-3-Sqrt[2])/2},{(9+5Sqrt[2])/2,(-1-Sqrt[2])/2},{(9+5Sqrt[2])/2,(-1-Sqrt[2]+2Sqrt[3])/2},{(11+5Sqrt[2])/2,(-3-Sqrt[2]-2Sqrt[3])/2},{(11+5Sqrt[2])/2,(-3-Sqrt[2])/2},{(11+5Sqrt[2])/2,(-1-Sqrt[2])/2},{(11+5Sqrt[2])/2,(-1-Sqrt[2]+2Sqrt[3])/2},{(12+5Sqrt[2])/2,(-3-Sqrt[2]-Sqrt[3])/2},{(12+5Sqrt[2])/2,(-1-Sqrt[2]+Sqrt[3])/2},{(11+6Sqrt[2])/2,(-5-2Sqrt[2])/2},{(11+6Sqrt[2])/2,(-3-2Sqrt[2])/2},{(11+6Sqrt[2])/2,-1/2},{(11+6Sqrt[2])/2,1/2},{(13+6Sqrt[2])/2,(-5-2Sqrt[2])/2},{(13+6Sqrt[2])/2,(-3-2Sqrt[2])/2},{(13+6Sqrt[2])/2,-1/2},{(13+6Sqrt[2])/2,1/2},{(12+7Sqrt[2])/2,(-3-Sqrt[2]-Sqrt[3])/2},{(12+7Sqrt[2])/2,(-1-Sqrt[2]+Sqrt[3])/2},{(13+7Sqrt[2])/2,(-3-Sqrt[2]-2Sqrt[3])/2},{(13+7Sqrt[2])/2,(-3-Sqrt[2])/2},{(13+7Sqrt[2])/2,(-1-Sqrt[2])/2},{(13+7Sqrt[2])/2,(-1-Sqrt[2]+2Sqrt[3])/2},{(15+7Sqrt[2])/2,(-3-Sqrt[2]-2Sqrt[3])/2},{(15+7Sqrt[2])/2,(-3-Sqrt[2])/2},{(15+7Sqrt[2])/2,(-1-Sqrt[2])/2},{(15+7Sqrt[2])/2,(-1-Sqrt[2]+2Sqrt[3])/2},{(16+7Sqrt[2])/2,(-3-Sqrt[2]-Sqrt[3])/2},{(16+7Sqrt[2])/2,(-1-Sqrt[2]+Sqrt[3])/2}}
PolyhedronDual[GreatRhombicuboctahedron]:=DisdyakisDodecahedron
PolyhedronFaces[GreatRhombicuboctahedron]:=
	{{44,42,17,19},{14,6,3,11},{34,36,27,25},{8,16,9,1},{20,18,41,43},{12,4,5,
    13},{26,28,35,33},{2,10,15,7},{45,23,24,46},{39,29,30,40},{48,22,21,
    47},{38,32,31,37},{9,19,17,11,3,25,27,1},{2,28,26,4,12,18,20,10},{41,48,
    47,42,44,45,46,43},{35,38,37,36,34,39,40,33},{15,24,23,16,8,31,32,7},{5,
    30,29,6,14,21,22,13},{46,24,15,10,20,43},{35,28,2,7,32,38},{41,18,12,13,
    22,48},{40,30,5,4,26,33},{44,19,9,16,23,45},{37,31,8,1,27,36},{47,21,14,
    11,17,42},{34,25,3,6,29,39}}
PolyhedronVertices[GreatRhombicuboctahedron]:=
	{{-1/2,1/2+1/Sqrt[2],-1/2-Sqrt[2]},
{-1/2,1/2+1/Sqrt[2],1/2+Sqrt[2]},
{-1/2,(2-2Sqrt[2])^(-1),-1/2-Sqrt[2]},
{-1/2,(2-2Sqrt[2])^(-1),1/2+Sqrt[2]},
{-1/2,-1/2-Sqrt[2],1/2+1/Sqrt[2]},
{-1/2,-1/2-Sqrt[2],(2-2Sqrt[2])^(-1)},
{-1/2,1/2+Sqrt[2],1/2+1/Sqrt[2]},
{-1/2,1/2+Sqrt[2],(2-2Sqrt[2])^(-1)},
{1/2,1/2+1/Sqrt[2],-1/2-Sqrt[2]},
{1/2,1/2+1/Sqrt[2],1/2+Sqrt[2]},
{1/2,(2-2Sqrt[2])^(-1),-1/2-Sqrt[2]},
{1/2,(2-2Sqrt[2])^(-1),1/2+Sqrt[2]},
{1/2,-1/2-Sqrt[2],1/2+1/Sqrt[2]},
{1/2,-1/2-Sqrt[2],(2-2Sqrt[2])^(-1)},
{1/2,1/2+Sqrt[2],1/2+1/Sqrt[2]},
{1/2,1/2+Sqrt[2],(2-2Sqrt[2])^(-1)},
{1/2+1/Sqrt[2],-1/2,-1/2-Sqrt[2]},
{1/2+1/Sqrt[2],-1/2,1/2+Sqrt[2]},
{1/2+1/Sqrt[2],1/2,-1/2-Sqrt[2]},
{1/2+1/Sqrt[2],1/2,1/2+Sqrt[2]},
{1/2+1/Sqrt[2],-1/2-Sqrt[2],-1/2},
{1/2+1/Sqrt[2],-1/2-Sqrt[2],1/2},
{1/2+1/Sqrt[2],1/2+Sqrt[2],-1/2},
{1/2+1/Sqrt[2],1/2+Sqrt[2],1/2},
{(2-2Sqrt[2])^(-1),-1/2,-1/2-Sqrt[2]},
{(2-2Sqrt[2])^(-1),-1/2,1/2+Sqrt[2]},
{(2-2Sqrt[2])^(-1),1/2,-1/2-Sqrt[2]},
{(2-2Sqrt[2])^(-1),1/2,1/2+Sqrt[2]},
{(2-2Sqrt[2])^(-1),-1/2-Sqrt[2],-1/2},
{(2-2Sqrt[2])^(-1),-1/2-Sqrt[2],1/2},
{(2-2Sqrt[2])^(-1),1/2+Sqrt[2],-1/2},
{(2-2Sqrt[2])^(-1),1/2+Sqrt[2],1/2},
{-1/2-Sqrt[2],-1/2,1/2+1/Sqrt[2]},
{-1/2-Sqrt[2],-1/2,(2-2Sqrt[2])^(-1)},
{-1/2-Sqrt[2],1/2,1/2+1/Sqrt[2]},
{-1/2-Sqrt[2],1/2,(2-2Sqrt[2])^(-1)},
{-1/2-Sqrt[2],1/2+1/Sqrt[2],-1/2},
{-1/2-Sqrt[2],1/2+1/Sqrt[2],1/2},
{-1/2-Sqrt[2],(2-2Sqrt[2])^(-1),-1/2},
{-1/2-Sqrt[2],(2-2Sqrt[2])^(-1),1/2},
{1/2+Sqrt[2],-1/2,1/2+1/Sqrt[2]},
{1/2+Sqrt[2],-1/2,(2-2Sqrt[2])^(-1)},
{1/2+Sqrt[2],1/2,1/2+1/Sqrt[2]},
{1/2+Sqrt[2],1/2,(2-2Sqrt[2])^(-1)},
{1/2+Sqrt[2],1/2+1/Sqrt[2],-1/2},
{1/2+Sqrt[2],1/2+1/Sqrt[2],1/2},
{1/2+Sqrt[2],(2-2Sqrt[2])^(-1),-1/2},
{1/2+Sqrt[2],(2-2Sqrt[2])^(-1),1/2}}

(* Icosidodecahedron *)

PolyhedronName[Icosidodecahedron]:="icosidodecahedron"
NetFaces[Icosidodecahedron]:={{17,20,18,11,10},{6,10,11},{4,2,6,11,9},{8,4,9},{12,17,10},{3,7,12,10,5},{1,3,5},{23,20,17},{19,25,23,17,15},{14,19,15},{24,18,20},{30,28,24,20,26},{29,30,26},{13,11,18},{21,16,13,18,22},{27,21,22},{41,39,44,49,47},{52,47,49},{56,57,52,49,51},{54,56,51},{45,41,47},{55,50,45,47,53},{58,55,53},{35,39,41},{36,32,35,41,42},{43,36,42},{37,44,39},{30,31,37,39,33},{28,30,33},{48,49,44},{40,46,48,44,38},{34,40,38}}
NetVertices[Icosidodecahedron]:={{(-8Sqrt[3]-24Sqrt[15]-27Sqrt[2(5-Sqrt[5])]-3Sqrt[10(5-Sqrt[5])]+6Sqrt[2(5+Sqrt[5])])/96,(-16-Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])]+2Sqrt[6(5+Sqrt[5])])/32},{(-5Sqrt[3]-3Sqrt[15]-3Sqrt[2(5+Sqrt[5])])/24,(1-Sqrt[5]+Sqrt[6(5+Sqrt[5])])/8},{(-8Sqrt[3]-24Sqrt[15]-3Sqrt[2(5-Sqrt[5])]-3Sqrt[10(5-Sqrt[5])]+6Sqrt[2(5+Sqrt[5])])/96,(-24-8Sqrt[5]-Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])]+2Sqrt[6(5+Sqrt[5])])/32},{(-4Sqrt[3]-3Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/48,(4+Sqrt[6(5+Sqrt[5])]+Sqrt[30(5+Sqrt[5])])/16},{(4Sqrt[3]-12Sqrt[15]-15Sqrt[2(5-Sqrt[5])]-3Sqrt[10(5-Sqrt[5])]+6Sqrt[2(5+Sqrt[5])])/96,(-20-4Sqrt[5]+2Sqrt[6(5-Sqrt[5])]-Sqrt[6(5+Sqrt[5])]+Sqrt[30(5+Sqrt[5])])/32},{-(1/Sqrt[3]),0},{(-20Sqrt[3]-12Sqrt[15]+6Sqrt[2(5-Sqrt[5])]+15Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/96,(-28-4Sqrt[5]-4Sqrt[6(5-Sqrt[5])]-2Sqrt[30(5-Sqrt[5])]-Sqrt[6(5+Sqrt[5])]+Sqrt[30(5+Sqrt[5])])/32},{(-4Sqrt[3]+12Sqrt[2(5-Sqrt[5])]-3Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/48,(8+4Sqrt[5]+Sqrt[6(5+Sqrt[5])]+Sqrt[30(5+Sqrt[5])])/16},{(Sqrt[3]+3Sqrt[15]-3Sqrt[2(5+Sqrt[5])])/24,(3+Sqrt[5]+Sqrt[6(5+Sqrt[5])])/8},{1/(2Sqrt[3]),-1/2},{1/(2Sqrt[3]),1/2},{(7Sqrt[3]-3Sqrt[15]+3Sqrt[2(5+Sqrt[5])])/24,(-6-2Sqrt[5]-Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])])/16},{(14Sqrt[3]-6Sqrt[15]+3Sqrt[2(5-Sqrt[5])]+3Sqrt[10(5-Sqrt[5])])/48,(6+2Sqrt[5]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])])/16},{(-8Sqrt[3]-6Sqrt[2(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+9Sqrt[10(5+Sqrt[5])])/96,(-8-16Sqrt[5]-2Sqrt[6(5-Sqrt[5])]-3Sqrt[6(5+Sqrt[5])]-Sqrt[30(5+Sqrt[5])])/32},{(4Sqrt[3]-12Sqrt[15]+6Sqrt[10(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+9Sqrt[10(5+Sqrt[5])])/96,(-12-12Sqrt[5]+2Sqrt[30(5-Sqrt[5])]-3Sqrt[6(5+Sqrt[5])]-Sqrt[30(5+Sqrt[5])])/32},{(40Sqrt[3]+6Sqrt[10(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/96,(16+8Sqrt[5]+4Sqrt[6(5-Sqrt[5])]+2Sqrt[30(5-Sqrt[5])]-Sqrt[6(5+Sqrt[5])]+Sqrt[30(5+Sqrt[5])])/32},{(8Sqrt[3]+3Sqrt[2(5-Sqrt[5])]+3Sqrt[10(5-Sqrt[5])]+6Sqrt[2(5+Sqrt[5])])/48,(-1-Sqrt[5])/4},{(4Sqrt[3]+3Sqrt[2(5-Sqrt[5])]+3Sqrt[10(5-Sqrt[5])])/24,(1+Sqrt[5])/4},{(-8Sqrt[3]+6Sqrt[2(5-Sqrt[5])]+12Sqrt[10(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+9Sqrt[10(5+Sqrt[5])])/96,(-16-8Sqrt[5]-2Sqrt[6(5-Sqrt[5])]-3Sqrt[6(5+Sqrt[5])]-Sqrt[30(5+Sqrt[5])])/32},{(4Sqrt[3]+9Sqrt[2(5-Sqrt[5])]+3Sqrt[10(5-Sqrt[5])])/24,0},{(52Sqrt[3]+12Sqrt[15]+9Sqrt[2(5-Sqrt[5])]+9Sqrt[10(5-Sqrt[5])]-6Sqrt[2(5+Sqrt[5])])/96,(20+12Sqrt[5]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]+2Sqrt[6(5+Sqrt[5])])/32},{(64Sqrt[3]+15Sqrt[2(5-Sqrt[5])]+15Sqrt[10(5-Sqrt[5])]-6Sqrt[2(5+Sqrt[5])])/96,(24+8Sqrt[5]-2Sqrt[6(5-Sqrt[5])]-Sqrt[6(5+Sqrt[5])]+Sqrt[30(5+Sqrt[5])])/32},{(14Sqrt[3]+6Sqrt[15]+6Sqrt[2(5-Sqrt[5])]+6Sqrt[10(5-Sqrt[5])]-3Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/48,(-1-Sqrt[5]-Sqrt[6(5-Sqrt[5])])/8},{(7Sqrt[3]+3Sqrt[15]+6Sqrt[2(5-Sqrt[5])]+3Sqrt[10(5-Sqrt[5])])/24,(1+Sqrt[5]+Sqrt[6(5-Sqrt[5])])/8},{(40Sqrt[3]+9Sqrt[2(5-Sqrt[5])]+15Sqrt[10(5-Sqrt[5])]-3Sqrt[2(5+Sqrt[5])]+9Sqrt[10(5+Sqrt[5])])/96,(-8Sqrt[5]-3Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])]-Sqrt[6(5+Sqrt[5])]-Sqrt[30(5+Sqrt[5])])/32},{(4Sqrt[3]+12Sqrt[15]+30Sqrt[2(5-Sqrt[5])]+12Sqrt[10(5-Sqrt[5])]+9Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/96,(-4+4Sqrt[5]+2Sqrt[6(5-Sqrt[5])]-3Sqrt[6(5+Sqrt[5])]-Sqrt[30(5+Sqrt[5])])/32},{(52Sqrt[3]+12Sqrt[15]+21Sqrt[2(5-Sqrt[5])]+21Sqrt[10(5-Sqrt[5])]-6Sqrt[2(5+Sqrt[5])])/96,(28+4Sqrt[5]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]+2Sqrt[6(5+Sqrt[5])])/32},{(40Sqrt[3]+24Sqrt[15]+27Sqrt[2(5-Sqrt[5])]+9Sqrt[10(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/96,(8+8Sqrt[5]+3Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-Sqrt[6(5+Sqrt[5])]-Sqrt[30(5+Sqrt[5])])/32},{(52Sqrt[3]+12Sqrt[15]+30Sqrt[2(5-Sqrt[5])]+12Sqrt[10(5-Sqrt[5])]+9Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/96,(-20+4Sqrt[5]+2Sqrt[6(5-Sqrt[5])]-3Sqrt[6(5+Sqrt[5])]-Sqrt[30(5+Sqrt[5])])/32},{(52Sqrt[3]+12Sqrt[15]+30Sqrt[2(5-Sqrt[5])]+12Sqrt[10(5-Sqrt[5])]+9Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/96,(3+Sqrt[5]-Sqrt[6(5+Sqrt[5])])/8},{(64Sqrt[3]+24Sqrt[15]+21Sqrt[2(5-Sqrt[5])]+15Sqrt[10(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/96,(8-3Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])]-Sqrt[6(5+Sqrt[5])]-Sqrt[30(5+Sqrt[5])])/32},{(76Sqrt[3]+12Sqrt[15]+12Sqrt[2(5-Sqrt[5])]+18Sqrt[10(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+9Sqrt[10(5+Sqrt[5])])/96,(12+12Sqrt[5]+2Sqrt[6(5-Sqrt[5])]-Sqrt[6(5+Sqrt[5])]+Sqrt[30(5+Sqrt[5])])/32},{(40Sqrt[3]+24Sqrt[15]+39Sqrt[2(5-Sqrt[5])]+21Sqrt[10(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/96,(16+3Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-Sqrt[6(5+Sqrt[5])]-Sqrt[30(5+Sqrt[5])])/32},{(88Sqrt[3]+24Sqrt[15]+21Sqrt[2(5-Sqrt[5])]+21Sqrt[10(5-Sqrt[5])]+6Sqrt[2(5+Sqrt[5])])/96,(-16+5Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-8Sqrt[6(5+Sqrt[5])]-2Sqrt[30(5+Sqrt[5])])/32},{(88Sqrt[3]+24Sqrt[15]+9Sqrt[2(5-Sqrt[5])]+21Sqrt[10(5-Sqrt[5])]+6Sqrt[10(5+Sqrt[5])])/96,(8+8Sqrt[5]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32},{(64Sqrt[3]+24Sqrt[15]+15Sqrt[2(5-Sqrt[5])]+21Sqrt[10(5-Sqrt[5])]+9Sqrt[2(5+Sqrt[5])]+9Sqrt[10(5+Sqrt[5])])/96,(16+8Sqrt[5]+3Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]+Sqrt[6(5+Sqrt[5])]+Sqrt[30(5+Sqrt[5])])/32},{(76Sqrt[3]+36Sqrt[15]+27Sqrt[2(5-Sqrt[5])]+15Sqrt[10(5-Sqrt[5])]+6Sqrt[10(5+Sqrt[5])])/96,(4-4Sqrt[5]-Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32},{(88Sqrt[3]+24Sqrt[15]+45Sqrt[2(5-Sqrt[5])]+21Sqrt[10(5-Sqrt[5])]+6Sqrt[2(5+Sqrt[5])])/96,(-8+8Sqrt[5]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-10Sqrt[6(5+Sqrt[5])])/32},{(88Sqrt[3]+24Sqrt[15]+33Sqrt[2(5-Sqrt[5])]+21Sqrt[10(5-Sqrt[5])]+6Sqrt[10(5+Sqrt[5])])/96,(Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32},{(100Sqrt[3]+36Sqrt[15]+33Sqrt[2(5-Sqrt[5])]+21Sqrt[10(5-Sqrt[5])]+6Sqrt[2(5+Sqrt[5])])/96,(-12+4Sqrt[5]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-8Sqrt[6(5+Sqrt[5])]-2Sqrt[30(5+Sqrt[5])])/32},{(100Sqrt[3]+36Sqrt[15]+33Sqrt[2(5-Sqrt[5])]+21Sqrt[10(5-Sqrt[5])]+6Sqrt[2(5+Sqrt[5])])/96,(4+4Sqrt[5]+5Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32},{(112Sqrt[3]+24Sqrt[15]+18Sqrt[2(5-Sqrt[5])]+24Sqrt[10(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+9Sqrt[10(5+Sqrt[5])])/96,(8Sqrt[5]+4Sqrt[6(5-Sqrt[5])]+2Sqrt[30(5-Sqrt[5])]-Sqrt[6(5+Sqrt[5])]+Sqrt[30(5+Sqrt[5])])/32},{(112Sqrt[3]+24Sqrt[15]+15Sqrt[2(5-Sqrt[5])]+21Sqrt[10(5-Sqrt[5])]+9Sqrt[2(5+Sqrt[5])]+9Sqrt[10(5+Sqrt[5])])/96,(32+8Sqrt[5]+3Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]+Sqrt[6(5+Sqrt[5])]+Sqrt[30(5+Sqrt[5])])/32},{(76Sqrt[3]+36Sqrt[15]+39Sqrt[2(5-Sqrt[5])]+27Sqrt[10(5-Sqrt[5])]+6Sqrt[10(5+Sqrt[5])])/96,(-4+4Sqrt[5]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-6Sqrt[6(5+Sqrt[5])])/32},{(100Sqrt[3]+36Sqrt[15]+57Sqrt[2(5-Sqrt[5])]+21Sqrt[10(5-Sqrt[5])]+6Sqrt[2(5+Sqrt[5])])/96,(12+12Sqrt[5]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-4Sqrt[6(5+Sqrt[5])]+2Sqrt[30(5+Sqrt[5])])/32},{(112Sqrt[3]+48Sqrt[15]+33Sqrt[2(5-Sqrt[5])]+21Sqrt[10(5-Sqrt[5])]+6Sqrt[10(5+Sqrt[5])])/96,(-16+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-10Sqrt[6(5+Sqrt[5])])/32},{(112Sqrt[3]+48Sqrt[15]+45Sqrt[2(5-Sqrt[5])]+21Sqrt[10(5-Sqrt[5])]+6Sqrt[2(5+Sqrt[5])])/96,(8+8Sqrt[5]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32},{(124Sqrt[3]+36Sqrt[15]+33Sqrt[2(5-Sqrt[5])]+21Sqrt[10(5-Sqrt[5])]+12Sqrt[2(5+Sqrt[5])]+6Sqrt[10(5+Sqrt[5])])/96,(-20+4Sqrt[5]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-6Sqrt[6(5+Sqrt[5])])/32},{(124Sqrt[3]+36Sqrt[15]+39Sqrt[2(5-Sqrt[5])]+27Sqrt[10(5-Sqrt[5])]+6Sqrt[10(5+Sqrt[5])])/96,(12+4Sqrt[5]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-6Sqrt[6(5+Sqrt[5])])/32},{(88Sqrt[3]+48Sqrt[15]+54Sqrt[2(5-Sqrt[5])]+24Sqrt[10(5-Sqrt[5])]+9Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/96,(16+8Sqrt[5]+2Sqrt[30(5-Sqrt[5])]-3Sqrt[6(5+Sqrt[5])]+3Sqrt[30(5+Sqrt[5])])/32},{(68Sqrt[3]+24Sqrt[15]+15Sqrt[2(5-Sqrt[5])]+15Sqrt[10(5-Sqrt[5])]-3Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/48,(2-Sqrt[6(5-Sqrt[5])]-Sqrt[6(5+Sqrt[5])])/8},{(112Sqrt[3]+48Sqrt[15]+45Sqrt[2(5-Sqrt[5])]+33Sqrt[10(5-Sqrt[5])]+6Sqrt[10(5+Sqrt[5])])/96,(16+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32},{(124Sqrt[3]+60Sqrt[15]+48Sqrt[2(5-Sqrt[5])]+18Sqrt[10(5-Sqrt[5])]+9Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/96,(4+4Sqrt[5]-2Sqrt[6(5-Sqrt[5])]-3Sqrt[6(5+Sqrt[5])]+3Sqrt[30(5+Sqrt[5])])/32},{(68Sqrt[3]+24Sqrt[15]+27Sqrt[2(5-Sqrt[5])]+15Sqrt[10(5-Sqrt[5])]-3Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/48,(-2Sqrt[5]-Sqrt[6(5-Sqrt[5])]-Sqrt[6(5+Sqrt[5])])/8},{(136Sqrt[3]+48Sqrt[15]+51Sqrt[2(5-Sqrt[5])]+21Sqrt[10(5-Sqrt[5])]+15Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/96,(8Sqrt[5]-Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-Sqrt[6(5+Sqrt[5])]+3Sqrt[30(5+Sqrt[5])])/32},{(74Sqrt[3]+30Sqrt[15]+21Sqrt[2(5-Sqrt[5])]+15Sqrt[10(5-Sqrt[5])]-3Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/48,(1-Sqrt[5]-Sqrt[6(5+Sqrt[5])])/8},{(160Sqrt[3]+48Sqrt[15]+51Sqrt[2(5-Sqrt[5])]+33Sqrt[10(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/96,(3Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-Sqrt[6(5+Sqrt[5])]-Sqrt[30(5+Sqrt[5])])/32},{(124Sqrt[3]+60Sqrt[15]+57Sqrt[2(5-Sqrt[5])]+27Sqrt[10(5-Sqrt[5])]+15Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/96,(-4+12Sqrt[5]-3Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])]-Sqrt[6(5+Sqrt[5])]+3Sqrt[30(5+Sqrt[5])])/32}}
PolyhedronDual[Icosidodecahedron]:=RhombicTriacontahedron
PolyhedronFaces[Icosidodecahedron]:=
	{{30,24,29,7,8},{26,24,30},{25,29,24},{5,7,29},{14,8,7},{6,30,8},{16,2,6},{19,
    21,26},{20,18,25},{1,15,5},{22,23,14},{2,19,26,30,6},{21,20,25,24,26},{18,
    1,5,29,25},{15,22,14,7,5},{23,16,6,8,14},{12,13,4,17,3},{3,17,9},{17,4,
    10},{4,13,28},{13,12,11},{12,3,27},{27,1,18},{9,22,15},{10,16,23},{28,19,
    2},{11,20,21},{27,3,9,15,1},{9,17,10,23,22},{10,4,28,2,16},{28,13,11,21,
    19},{11,12,27,18,20}}
PolyhedronVertices[Icosidodecahedron]:=
{{0,(-1-Sqrt[5])/2,0},
{0,(1+Sqrt[5])/2,0},
{Sqrt[1/8-1/(8Sqrt[5])],(-1-Sqrt[5])/4,-Sqrt[1+2/Sqrt[5]]},
{Sqrt[1/8-1/(8Sqrt[5])],(1+Sqrt[5])/4,-Sqrt[1+2/Sqrt[5]]},
{Sqrt[1/8+1/(8Sqrt[5])],(-3-Sqrt[5])/4,Sqrt[(5+Sqrt[5])/10]},
{Sqrt[1/8+1/(8Sqrt[5])],(3+Sqrt[5])/4,Sqrt[(5+Sqrt[5])/10]},
{Sqrt[1/4+1/(2Sqrt[5])],-1/2,Sqrt[1+2/Sqrt[5]]},
{Sqrt[1/4+1/(2Sqrt[5])],1/2,Sqrt[1+2/Sqrt[5]]},
{Sqrt[5/8+11/(8Sqrt[5])],(-1-Sqrt[5])/4,Root[1-5#1^2+5#1^4&,1,0]},
{Sqrt[5/8+11/(8Sqrt[5])],(1+Sqrt[5])/4,Root[1-5#1^2+5#1^4&,1,0]},
{-Sqrt[1+2/Sqrt[5]],0,Root[1-5#1^2+5#1^4&,1,0]},
{-Sqrt[1+2/Sqrt[5]]/2,-1/2,-Sqrt[1+2/Sqrt[5]]},
{-Sqrt[1+2/Sqrt[5]]/2,1/2,-Sqrt[1+2/Sqrt[5]]},
{Sqrt[1+2/Sqrt[5]],0,Sqrt[(5+Sqrt[5])/10]},
{Sqrt[5/8+Sqrt[5]/8],-(1+Sqrt[5])^2/8,0},
{Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,0},
{Sqrt[(5+Sqrt[5])/10],0,-Sqrt[1+2/Sqrt[5]]},
{-Sqrt[(5+Sqrt[5])/2]/2,-(1+Sqrt[5])^2/8,0},
{-Sqrt[(5+Sqrt[5])/2]/2,(3+Sqrt[5])/4,0},
{-Sqrt[5+2Sqrt[5]]/2,-1/2,0},
{-Sqrt[5+2Sqrt[5]]/2,1/2,0},
{Sqrt[5+2Sqrt[5]]/2,-1/2,0},
{Sqrt[5+2Sqrt[5]]/2,1/2,0},
{Root[1-5#1^2+5#1^4&,1,0],0,Sqrt[1+2/Sqrt[5]]},
{Root[1-100#1^2+80#1^4&,1,0],(-1-Sqrt[5])/4,Sqrt[(5+Sqrt[5])/10]},
{Root[1-100#1^2+80#1^4&,1,0],(1+Sqrt[5])/4,Sqrt[(5+Sqrt[5])/10]},
{Root[1-20#1^2+80#1^4&,1,0],(-3-Sqrt[5])/4,Root[1-5#1^2+5#1^4&,1,0]},
{Root[1-20#1^2+80#1^4&,1,0],(3+Sqrt[5])/4,Root[1-5#1^2+5#1^4&,1,0]},
{Root[1-20#1^2+80#1^4&,2,0],(-1-Sqrt[5])/4,Sqrt[1+2/Sqrt[5]]},
{Root[1-20#1^2+80#1^4&,2,0],(1+Sqrt[5])/4,Sqrt[1+2/Sqrt[5]]}}

(* Pentagonal Hexecontahedron *)

PolyhedronFaces[PentagonalHexecontahedron]:=PolyhedronFaces[PentagonalHexecontahedron,"dextro"]
PolyhedronName[PentagonalHexecontahedron]:=PolyhedronName[PentagonalHexecontahedron,"dextro"]
PolyhedronDual[PentagonalHexecontahedron]:=PolyhedronDual[PentagonalHexecontahedron,"dextro"]
PolyhedronVertices[PentagonalHexecontahedron]:=PolyhedronVertices[PentagonalHexecontahedron,"dextro"]

NetFaces[PentagonalHexecontahedron]:={{49,65,69,62,55},{49,55,50,41,36},{49,36,26,23,33},{49,33,30,43,48},
{49,48,57,63,61},{58,74,81,80,70},{58,70,68,59,53},{58,53,46,38,45},{58,45,35,44,51},{58,51,55,62,67},
{18,28,27,20,14},{18,14,10,4,8},{18,8,2,7,12},{18,12,16,22,25},{18,25,36,41,32},{17,13,9,3,6},{17,6,1,5,11},
{17,11,15,21,24},{17,24,34,39,29},{17,29,33,23,19},{54,40,31,37,47},{54,47,52,60,66},{54,66,77,78,71},
{54,71,75,64,56},{54,56,48,43,42},{84,79,83,86,88},{84,88,93,94,91},{84,91,92,87,85},{84,85,82,73,72},
{84,72,61,63,76},{134,118,114,121,128},{134,128,133,142,147},{134,147,157,160,150},{134,150,153,140,135},
{134,135,126,120,122},{125,109,102,103,113},{125,113,115,124,130},{125,130,137,145,138},{125,138,148,139,132},
{125,132,128,121,116},{165,155,156,163,169},{165,169,173,179,175},{165,175,181,176,171},{165,171,167,161,158},
{165,158,147,142,151},{166,170,174,180,177},{166,177,182,178,172},{166,172,168,162,159},{166,159,149,144,154},
{166,154,150,160,164},{129,143,152,146,136},{129,136,131,123,117},{129,117,106,105,112},{129,112,108,119,127},
{129,127,135,140,141},{99,104,100,97,95},{99,95,90,89,92},{99,92,91,96,98},{99,98,101,110,111},
{99,111,122,120,107}}
NetVertices[PentagonalHexecontahedron]:={{-5.78978791158,-2.80761669086},
{-5.62044484574,1.74245918164},{-5.61659031376,-1.05249040074},{-5.44724724792,3.49758547176},
{-5.31891465278,-3.68981756036},{-5.23389293563,-1.9763641566},{-5.1495715869400005,0.860258312137},
{-5.0645498698,2.5737117159},{-4.98236552247,-0.279341699569},{-4.81302245663,4.27073417293},
{-4.31891496984,-3.69061388707},{-4.14957190401,0.859461985426},{-4.0014982338,-0.474019284345},
{-3.83215516796,4.07605658815},{-3.7643445957,-4.52275071483},{-3.59500152986,0.0273251576639},
{-3.49249862184,-2.14820666942},{-3.32315555601,2.40186920308},{-3.31500500407,-0.407379273907},
{-3.12563306362,4.78374756314},{-2.76902314758,-4.42613179884},{-2.59968008175,0.123944073655},
{-2.38989400094,-0.0276825108488},{-2.38485484459,-3.5028687188},{-2.21551177875,1.0472071537},
{-2.19203385945,0.952547750064},{-2.16839442405,4.49444808696},{-1.97209570009,3.51390394671},
{-1.81666800484,-1.64464361633},{-1.72024133332,-1.6592556802},{-1.65241191737,-4.70866571759},
{-1.647324939,2.90543225617},{-1.61880786335,-0.6644133554180001},{-1.4036787461,-3.30975356207},
{-1.35320542483,3.33070829436},{-1.23433568027,1.24032231042},{-1.18153865857,-5.59086658708},
{-1.18000782701,5.08583458449},{-1.11127390266,-2.35345893826},{-1.09651694143,-3.87741318332},
{-0.941930836821,2.19661693424},{-0.894839437395,-3.21783753958},{-0.890797435108,-2.2178457085},
{-0.882332166028,2.44850742486},{-0.797310448886,4.16196082862},{-0.545783035725,5.85898328566},
{-0.181538975637,-5.59166291379},{-0.0070729080754,-1.74983827232},{0.,0.},
{0.0392452616672,2.38973209096},{0.117667516904,2.44771109815},{0.37303139851,-6.42379974156},
{0.435084252947,5.66430570088},{0.644877372364,-4.04925569614},{0.672237891051,1.61557427039},
{0.8223709901340001,-2.30842830063},{0.872839399411,-2.22497439085},{0.944083864904,3.99011831581},
{1.14160635729,6.37199667587},{1.36835284662,-6.32718082557},{1.61338389116,-0.677477988604},
{1.66755933916,1.71219318638},{1.70677180253,-1.67310778828},{1.74748199327,-1.92873153757},
{1.74985201191,-0.00139345433874},{1.75252114962,-5.40391774552},{2.05172764216,2.63545626643},
{2.09884499686,6.08269719969},{2.22212971331,0.880056358619},{2.29514372082,5.10215305944},
{2.32070798937,-3.54569264305},{2.44731629428,-0.125611386034},{2.45135829657,0.874380445041},
{2.48916524201,3.16871248223},{2.51856813085,-2.56546238214},{2.62878313184,-2.06027068502},
{2.7336972481,-5.2108025888},{3.02610209155,-4.25450796499},{3.16061675604,-2.49943676025},
{3.21964882551,4.7209833972},{3.31949772829,3.72598078584},{3.3350828236,1.34238788122},
{3.71518713019,-3.33157358801},{3.98703310404,-0.957029542596},{4.16452672181,0.783797852916},
{4.7105085783,-3.23495467202},{5.08963772495,1.16349461597},{5.09467688129,-2.31169159198},
{5.15532976035,1.23557909334},{5.44773460379,2.19187371715},{5.66286372104,-0.453466489509},
{5.86072386253,0.526763771404},{6.0758529797800005,-2.11857643525},{6.36825782322,-1.16228181144},
{6.4289107022800005,2.38498887387},{6.43394985862,-1.09019733408},{6.81307900527,3.30825195392},
{7.35906086176,-0.71050057102},{7.53655447953,1.03032682449},{7.80840045338,3.40487086991},
{8.18850475997,-1.26909059933},{8.20408985528,-3.65268350395},{8.30393875807,-4.64768611531},
{8.36297082753,2.57273404214},{8.49748549202,4.32780524688},{8.78989033547,5.28409987069},
{8.89480445173,2.13356796692},{9.00501945272,2.63875966404},{9.03442234156,-3.09541520033},
{9.072229287,-0.801083163146},{9.07627128929,0.19890866793},{9.20287959421,3.61898992495},
{9.22844386275,-5.02885577754},{9.30145787026,-0.806759076724},{9.42474258672,-6.00939991779},
{9.47185994142,-2.56215898453},{9.77106643396,5.47721502742},{9.77373557166,0.0746907362343},
{9.7761055903,2.00202881947},{9.81681578104,1.74640507018},{9.85602824441,-1.63889590449},
{9.91020369241,0.7507752705},{10.1552347369,6.40047810746},{10.3819812263,-6.29869939397},
{10.5795037187,-3.91682103391},{10.6507481842,2.29827167275},{10.7012165934,2.38172558252},
{10.8513496925,-1.5422769885},{10.8787102112,4.12255297804},{11.0885033306,-5.59100841899},
{11.1505561851,6.49709702345},{11.4059200667,-2.37441381626},{11.4843423219,-2.31643480906},
{11.5235875836,0.0732972818956},{11.5306604916,1.82313555422},{11.7051265592,5.66496019569},
{12.0693706193,-5.78568600376},{12.3208980325,-4.08866354673},{12.4059197496,-2.37521014297},
{12.4143850187,2.2911429904},{12.418427021,3.29113482147},{12.4655184204,-2.12331965234},
{12.620104525,3.95071046522},{12.6348614862,2.42675622016},{12.7035954106,-5.01253730259},
{12.7051262421,5.66416386898},{12.7579232638,-1.16702502853},{12.8767930084,-3.25741101247},
{12.9272663297,3.38305084397},{13.1423954469,0.737710637314},{13.1709125226,-2.83213497427},
{13.1759995009,4.78196299948},{13.2438289169,1.73255296209},{13.3402555884,1.71794089823},
{13.4956832837,-3.44060666482},{13.6919820076,-4.42115080506},{13.715621443,-0.879250468169},
{13.7390993623,-0.973909871805},{13.9084424282,3.57616600069},{13.9134815845,0.100979792744},
{14.1232676653,-0.050646791759},{14.2926107312,4.49942908074},{14.6492206472,-4.71045028124},
{14.8385925876,0.480676555802},{14.8467431396,-2.32857192118},{15.0160862054,2.22150395131},
{15.1185891134,0.0459721242317},{15.2879321793,4.59604799673},{15.3557427515,-4.00275930626},
{15.5250858174,0.54731656624},{15.6731594876,-0.786164703531},{15.8425025534,3.76391116897},
{16.3366100402,-4.19743689103},{16.505953106,0.352638981464},{16.5881374534,-2.500414434},
{16.6731591705,-0.786961030241},{16.7574805192,2.0496614385},{16.8425022363,3.76311484226},
{16.9708348315,-3.42428818987},{17.1401778973,1.12578768263},{17.1440324293,-1.66916189974},
{17.3133754952,2.88091397276}}

PolyhedronName[PentagonalHexecontahedron,"dextro"]:="pentagonal hexecontahedron (dextro)"
PolyhedronDual[PentagonalHexecontahedron,"dextro"]:=Sequence[SnubDodecahedron,"laevo"]
PolyhedronFaces[PentagonalHexecontahedron,"dextro"]:=
{{29,33,17,5,11},{12,30,35,18,6},{61,67,34,27,7},{8,62,68,36,28},{34,15,5,17,27},{18,28,36,16,
6},{75,22,31,86,77},{32,88,78,76,24},{7,27,17,33,55},{35,56,8,28,18},{90,75,77,48,57},{76,78,
50,58,92},{65,86,67,61,70},{62,72,66,88,68},{9,5,15,31,22},{6,16,32,24,10},{45,7,55,81,69},
{82,71,46,8,56},{29,11,21,59,85},{12,23,60,87,30},{79,3,25,90,83},{26,92,84,80,4},{89,79,83,
1,64},{80,84,2,63,91},{85,81,55,33,29},{30,87,82,56,35},{21,11,5,13,37},{6,14,38,23,12},
{86,31,15,34,67},{36,68,88,32,16},{53,69,81,85,43},{87,44,54,71,82},{47,73,89,54,44},{74,91,
53,43,49},{85,59,73,47,41},{60,74,49,42,87},{73,59,21,37,89},{23,38,91,74,60},{13,5,9,25,3},{6,
10,26,4,14},{37,13,3,79,89},{14,4,80,91,38},{51,70,61,7,39},{40,52,72,62,8},{25,9,22,75,90},
{10,24,76,92,26},{1,83,90,52,40},{92,51,39,2,84},{48,77,86,19,20},{88,20,19,50,78},{50,19,86,
65,58},{66,57,48,20,88},{92,58,65,70,51},{52,90,57,66,72},{2,39,7,45,63},{46,64,1,40,8},
{49,43,85,41,42},{42,41,47,44,87},{91,63,45,69,53},{54,89,64,46,71}}
PolyhedronVertices[PentagonalHexecontahedron,"dextro"]:=
{{0,0,Root[729-7776#1^2+34992#1^4-79488#1^6+85248#1^8-33792#1^10+4096#1^12&,1,0]},
{0,0,Root[729-7776#1^2+34992#1^4-79488#1^6+85248#1^8-33792#1^10+4096#1^12&,8,0]},
{Root[1-24#1^2+243#1^4-1242#1^6+2997#1^8-2673#1^10+729#1^12&,1,0],0,Root[15625-300000#1^2+2430000#1^4-9936000#1^6+19180800#1^8-13685760#1^10+2985984#1^12&,1,0]},
{Root[1-24#1^2+243#1^4-1242#1^6+2997#1^8-2673#1^10+729#1^12&,8,0],0,Root[15625-300000#1^2+2430000#1^4-9936000#1^6+19180800#1^8-13685760#1^10+2985984#1^12&,8,0]},
{Root[1+24#1^2-414#1^4-43470#1^6+828873#1^8-3503574#1^10+700569#1^12&,1,0],0,Root[1+1776#1^2+1313136#1^4-40556160#1^6+429919488#1^8-1881916416#1^10+2869530624#1^12&,2,0]},
{Root[1+24#1^2-414#1^4-43470#1^6+828873#1^8-3503574#1^10+700569#1^12&,8,0],0,Root[1+1776#1^2+1313136#1^4-40556160#1^6+429919488#1^8-1881916416#1^10+2869530624#1^12&,7,0]},
{Root[1+156#1^2+9711#1^4-190890#1^6+1024893#1^8-1780461#1^10+700569#1^12&,1,0],0,Root[1-336#1^2+17136#1^4+1745280#1^6+28470528#1^8-8939787264#1^10+2869530624#1^12&,8,0]},
{Root[1+156#1^2+9711#1^4-190890#1^6+1024893#1^8-1780461#1^10+700569#1^12&,8,0],0,Root[1-336#1^2+17136#1^4+1745280#1^6+28470528#1^8-8939787264#1^10+2869530624#1^12&,1,0]},
{Root[116281-4894656#1^2+36660528#1^4-70670016#1^6+59222016#1^8-22146048#1^10+2985984#1^12&,1,0],Root[1-128#1^2+1392#1^4-4672#1^6+4096#1^8-1024#1^10+4096#1^12&,8,0],Root[14641-263280#1^2+1504368#1^4-2699136#1^6+435456#1^8-1741824#1^10+2985984#1^12&,1,0]},
{Root[116281-4894656#1^2+36660528#1^4-70670016#1^6+59222016#1^8-22146048#1^10+2985984#1^12&,8,0],Root[1-128#1^2+1392#1^4-4672#1^6+4096#1^8-1024#1^10+4096#1^12&,8,0],Root[14641-263280#1^2+1504368#1^4-2699136#1^6+435456#1^8-1741824#1^10+2985984#1^12&,8,0]},
{Root[434281-5981244#1^2+24945984#1^4-47186496#1^6+44727552#1^8-19906560#1^10+2985984#1^12&,1,0],Root[1-212#1^2+4992#1^4-17344#1^6+24832#1^8-16384#1^10+4096#1^12&,2,0],Root[1-2064#1^2+130608#1^4-1828224#1^6+5412096#1^8+7216128#1^10+2985984#1^12&,2,0]},
{Root[434281-5981244#1^2+24945984#1^4-47186496#1^6+44727552#1^8-19906560#1^10+2985984#1^12&,8,0],Root[1-212#1^2+4992#1^4-17344#1^6+24832#1^8-16384#1^10+4096#1^12&,2,0],Root[1-2064#1^2+130608#1^4-1828224#1^6+5412096#1^8+7216128#1^10+2985984#1^12&,7,0]},
{Root[3721-998784#1^2+9821664#1^4-28826496#1^6+34359552#1^8-17418240#1^10+2985984#1^12&,1,0],Root[1-48#1^2+672#1^4-3456#1^6+4352#1^8+6144#1^10+4096#1^12&,2,0],Root[212521-2455056#1^2+7899696#1^4-12666240#1^6+16443648#1^8-11695104#1^10+2985984#1^12&,2,0]},
{Root[3721-998784#1^2+9821664#1^4-28826496#1^6+34359552#1^8-17418240#1^10+2985984#1^12&,8,0],Root[1-48#1^2+672#1^4-3456#1^6+4352#1^8+6144#1^10+4096#1^12&,2,0],Root[212521-2455056#1^2+7899696#1^4-12666240#1^6+16443648#1^8-11695104#1^10+2985984#1^12&,7,0]},
{Root[44521-890592#1^2+6120288#1^4-18508608#1^6+25754112#1^8-16920576#1^10+2985984#1^12&,1,0],Root[1-720#1^2+6112#1^4-14528#1^6+16896#1^8-12288#1^10+4096#1^12&,7,0],Root[961-74064#1^2+1300464#1^4-6742656#1^6+1181952#1^8+1244160#1^10+2985984#1^12&,5,0]},
{Root[44521-890592#1^2+6120288#1^4-18508608#1^6+25754112#1^8-16920576#1^10+2985984#1^12&,8,0],Root[1-720#1^2+6112#1^4-14528#1^6+16896#1^8-12288#1^10+4096#1^12&,7,0],Root[961-74064#1^2+1300464#1^4-6742656#1^6+1181952#1^8+1244160#1^10+2985984#1^12&,4,0]},
{Root[1-24828#1^2+830016#1^4-6725376#1^6+17604864#1^8-16174080#1^10+2985984#1^12&,1,0],Root[1-84#1^2+2176#1^4-17152#1^6-16128#1^8+3072#1^10+4096#1^12&,2,0],Root[39601-600384#1^2+3539952#1^4-10247040#1^6+15116544#1^8-10699776#1^10+2985984#1^12&,7,0]},
{Root[1-24828#1^2+830016#1^4-6725376#1^6+17604864#1^8-16174080#1^10+2985984#1^12&,8,0],Root[1-84#1^2+2176#1^4-17152#1^6-16128#1^8+3072#1^10+4096#1^12&,2,0],Root[39601-600384#1^2+3539952#1^4-10247040#1^6+15116544#1^8-10699776#1^10+2985984#1^12&,2,0]},
{Root[1-15864#1^2+1723824#1^4-20903616#1^6+30108672#1^8-16174080#1^10+2985984#1^12&,3,0],Root[1-40#1^2+624#1^4-4672#1^6+16384#1^8-21504#1^10+4096#1^12&,8,0],Root[1-2064#1^2+130608#1^4-1828224#1^6+5412096#1^8+7216128#1^10+2985984#1^12&,7,0]},
{Root[1-15864#1^2+1723824#1^4-20903616#1^6+30108672#1^8-16174080#1^10+2985984#1^12&,6,0],Root[1-40#1^2+624#1^4-4672#1^6+16384#1^8-21504#1^10+4096#1^12&,8,0],Root[1-2064#1^2+130608#1^4-1828224#1^6+5412096#1^8+7216128#1^10+2985984#1^12&,2,0]},
{Root[15625-300000#1^2+2430000#1^4-9936000#1^6+19180800#1^8-13685760#1^10+2985984#1^12&,1,0],Root[1-32#1^2+432#1^4-2944#1^6+9472#1^8-11264#1^10+4096#1^12&,1,0],Root[1-96#1^2+3888#1^4-79488#1^6+767232#1^8-2737152#1^10+2985984#1^12&,1,0]},
{Root[15625-300000#1^2+2430000#1^4-9936000#1^6+19180800#1^8-13685760#1^10+2985984#1^12&,1,0],Root[1-32#1^2+432#1^4-2944#1^6+9472#1^8-11264#1^10+4096#1^12&,8,0],Root[1-96#1^2+3888#1^4-79488#1^6+767232#1^8-2737152#1^10+2985984#1^12&,1,0]},
{Root[15625-300000#1^2+2430000#1^4-9936000#1^6+19180800#1^8-13685760#1^10+2985984#1^12&,8,0],Root[1-32#1^2+432#1^4-2944#1^6+9472#1^8-11264#1^10+4096#1^12&,1,0],Root[1-96#1^2+3888#1^4-79488#1^6+767232#1^8-2737152#1^10+2985984#1^12&,8,0]},
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(*
PolyhedronFaces[PentagonalHexecontahedron,"dextro"]:=
{{29,33,17,11,6},{5,30,35,18,12},{61,67,34,27,13},{14,62,68,36,28},{34,15,11,17,27},{18,28,36,16,12},
{75,22,31,8,77},{32,10,78,76,24},{13,27,17,33,55},{35,56,14,28,18},{92,75,77,48,57},{76,78,50,58,90},
{65,8,67,61,70},{62,72,66,10,68},{86,11,15,31,22},{12,16,32,24,85},{45,13,55,81,69},{82,71,46,14,56},
{29,6,21,59,7},{5,23,60,9,30},{79,3,25,92,83},{26,90,84,80,4},{91,79,83,1,64},{80,84,2,63,89},
{7,81,55,33,29},{30,9,82,56,35},{21,6,11,88,37},{12,87,38,23,5},{8,31,15,34,67},{36,68,10,32,16},
{53,69,81,7,43},{9,44,54,71,82},{47,73,91,54,44},{74,89,53,43,49},{7,59,73,47,41},{60,74,49,42,9},
{73,59,21,37,91},{23,38,89,74,60},{88,11,86,25,3},{12,85,26,4,87},{37,88,3,79,91},{87,4,80,89,38},
{51,70,61,13,39},{40,52,72,62,14},{25,86,22,75,92},{85,24,76,90,26},{1,83,92,52,40},{90,51,39,2,84},
{48,77,8,19,20},{10,20,19,50,78},{50,19,8,65,58},{66,57,48,20,10},{90,58,65,70,51},{52,92,57,66,72},
{2,39,13,45,63},{46,64,1,40,14},{49,43,7,41,42},{42,41,47,44,9},{89,63,45,69,53},{54,91,64,46,71}}
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Root[15625-300000#1^2+2430000#1^4-9936000#1^6+19180800#1^8-13685760#1^10+2985984#1^12&,8,0]},
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Root[1-32#1^2+432#1^4-2944#1^6+9472#1^8-11264#1^10+4096#1^12&,8,0],
Root[15625-300000#1^2+2430000#1^4-9936000#1^6+19180800#1^8-13685760#1^10+2985984#1^12&,8,0]},
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Root[1-32#1^2+432#1^4-2944#1^6+9472#1^8-11264#1^10+4096#1^12&,1,0],
Root[15625-300000#1^2+2430000#1^4-9936000#1^6+19180800#1^8-13685760#1^10+2985984#1^12&,1,0]},
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Root[15625-300000#1^2+2430000#1^4-9936000#1^6+19180800#1^8-13685760#1^10+2985984#1^12&,1,0]},
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Root[50712847-14719519#1-16919466#1^2-6949449#1^3-7715885#1^4+1032652#1^5-1650384#1^6-281720#1^7-36438168#1^8-28772184#1^9+49944993#1^10+30795213#1^11+1016#1^12&,2,0]}}
*)

PolyhedronName[PentagonalHexecontahedron,"laevo"]:="pentagonal hexecontahedron (laevo)"
PolyhedronDual[PentagonalHexecontahedron,"laevo"]:=Sequence[SnubDodecahedron,"dextro"]
PolyhedronFaces[PentagonalHexecontahedron,"laevo"]:=
{{29,11,5,17,34},{18,36,30,12,6},{61,7,27,33,67},{35,68,62,8,28},{33,27,17,5,15},{16,35,
28,18,6},{85,31,21,75,77},{32,23,76,78,87},{7,55,34,17,27},{28,8,56,36,18},{47,77,75,89,
57},{76,91,58,49,78},{65,69,61,67,85},{87,66,71,62,68},{31,15,5,9,21},{6,10,23,32,16},
{45,70,81,55,7},{8,46,72,82,56},{59,22,11,29,86},{12,30,88,60,24},{89,25,3,79,83},{26,4,
80,84,91},{1,83,79,90,64},{80,92,63,2,84},{86,29,34,55,81},{56,82,88,30,36},{13,5,11,
22,37},{6,12,24,38,14},{85,67,33,15,31},{32,87,68,35,16},{53,43,86,81,70},{72,54,44,88,
82},{54,90,73,48,44},{74,50,43,53,92},{48,73,59,86,41},{60,88,42,50,74},{37,22,59,73,90},
{24,60,74,92,38},{25,9,5,13,3},{6,14,4,26,10},{79,3,13,37,90},{14,38,92,80,4},{51,39,7,
61,69},{62,71,52,40,8},{75,21,9,25,89},{10,26,91,76,23},{52,89,83,1,40},{91,84,2,39,51},
{19,85,77,47,20},{87,78,49,19,20},{49,58,65,85,19},{20,47,57,66,87},{91,51,69,65,58},
{66,57,89,52,71},{2,63,45,7,39},{40,1,64,46,8},{50,42,41,86,43},{44,48,41,42,88},{92,
53,70,45,63},{46,64,90,54,72}}
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(*
PolyhedronFaces[PentagonalHexecontahedron,"laevo"]:=
{{29,6,11,17,34},{18,36,30,5,12},{61,13,27,33,67},{35,68,62,14,28},{33,27,17,11,15},
{16,35,28,18,12},{7,31,21,75,77},{32,23,76,78,9},{13,55,34,17,27},{28,14,56,36,18},
{47,77,75,91,57},{76,89,58,49,78},{65,69,61,67,7},{9,66,71,62,68},{31,15,11,86,21},
{12,85,23,32,16},{45,70,81,55,13},{14,46,72,82,56},{59,22,6,29,8},{5,30,10,60,24},
{91,25,3,79,83},{26,4,80,84,89},{1,83,79,92,64},{80,90,63,2,84},{8,29,34,55,81},
{56,82,10,30,36},{88,11,6,22,37},{12,5,24,38,87},{7,67,33,15,31},{32,9,68,35,16},
{53,43,8,81,70},{72,54,44,10,82},{54,92,73,48,44},{74,50,43,53,90},{48,73,59,8,41},
{60,10,42,50,74},{37,22,59,73,92},{24,60,74,90,38},{25,86,11,88,3},{12,87,4,26,85},
{79,3,88,37,92},{87,38,90,80,4},{51,39,13,61,69},{62,71,52,40,14},{75,21,86,25,91},
{85,26,89,76,23},{52,91,83,1,40},{89,84,2,39,51},{19,7,77,47,20},{9,78,49,19,20},
{49,58,65,7,19},{20,47,57,66,9},{89,51,69,65,58},{66,57,91,52,71},{2,63,45,13,39},
{40,1,64,46,14},{50,42,41,8,43},{44,48,41,42,10},{90,53,70,45,63},{46,64,92,54,72}}
PolyhedronVertices[PentagonalHexecontahedron,"laevo"]:=
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{Root[1-24#1^2+243#1^4-1242#1^6+2997#1^8-2673#1^10+729#1^12&,1,0],0,
Root[15625-300000#1^2+2430000#1^4-9936000#1^6+19180800#1^8-13685760#1^10+2985984#1^12&,1,0]},
{Root[1-24#1^2+243#1^4-1242#1^6+2997#1^8-2673#1^10+729#1^12&,8,0],0,
Root[15625-300000#1^2+2430000#1^4-9936000#1^6+19180800#1^8-13685760#1^10+2985984#1^12&,8,0]},
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Root[-2187711+8841998#1-12614098#1^2+3217970#1^3+9686313#1^4-3042903#1^5+30885087#1^6+21760483#1^7-15557618#1^8+38208092#1^9-15899559#1^10+17321368#1^11+4079327#1^12&,2,0]},
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Root[-2187711-8841998#1-12614098#1^2-3217970#1^3+9686313#1^4+3042903#1^5+30885087#1^6-21760483#1^7-15557618#1^8-38208092#1^9-15899559#1^10-17321368#1^11+4079327#1^12&,1,0]},
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Root[-2187711-8841998#1-12614098#1^2-3217970#1^3+9686313#1^4+3042903#1^5+30885087#1^6-21760483#1^7-15557618#1^8-38208092#1^9-15899559#1^10-17321368#1^11+4079327#1^12&,1,0]},
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Root[-2187711-8841998#1-12614098#1^2-3217970#1^3+9686313#1^4+3042903#1^5+30885087#1^6-21760483#1^7-15557618#1^8-38208092#1^9-15899559#1^10-17321368#1^11+4079327#1^12&,1,0]},
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{Root[292681-2155404#1^2+4638528#1^4-1905984#1^6-808704#1^8-3981312#1^10+2985984#1^12&,1,0],Root[1-212#1^2+4992#1^4-17344#1^6+24832#1^8-16384#1^10+4096#1^12&,2,0],Root[3721-752784#1^2+7512624#1^4-16004736#1^6+11135232#1^8-8709120#1^10+2985984#1^12&,8,0]},
{Root[292681-2155404#1^2+4638528#1^4-1905984#1^6-808704#1^8-3981312#1^10+2985984#1^12&,8,0],Root[1-212#1^2+4992#1^4-17344#1^6+24832#1^8-16384#1^10+4096#1^12&,2,0],Root[3721-752784#1^2+7512624#1^4-16004736#1^6+11135232#1^8-8709120#1^10+2985984#1^12&,1,0]},
{Root[1-504#1^2+65664#1^4-2647296#1^6+8335872#1^8-3732480#1^10+2985984#1^12&,3,0],Root[1-328#1^2+2944#1^4-8960#1^6+9728#1^8-3072#1^10+4096#1^12&,7,0],
Root[1-218304#1^2+422064#1^4-245376#1^6+26065152#1^8-18662400#1^10+2985984#1^12&,8,0]},{Root[1-504#1^2+65664#1^4-2647296#1^6+8335872#1^8-3732480#1^10+2985984#1^12&,6,0],Root[1-328#1^2+2944#1^4-8960#1^6+9728#1^8-3072#1^10+4096#1^12&,7,0],Root[1-218304#1^2+422064#1^4-245376#1^6+26065152#1^8-18662400#1^10+2985984#1^12&,1,0]},
{Root[19321-437004#1^2+3336912#1^4-9884160#1^6+10036224#1^8-3234816#1^10+2985984#1^12&,2,0],Root[1-100#1^2+1872#1^4-9472#1^6+9216#1^8-13312#1^10+4096#1^12&,1,0],Root[212521-2455056#1^2+7899696#1^4-12666240#1^6+16443648#1^8-11695104#1^10+2985984#1^12&,7,0]},
{Root[19321-437004#1^2+3336912#1^4-9884160#1^6+10036224#1^8-3234816#1^10+2985984#1^12&,7,0],Root[1-100#1^2+1872#1^4-9472#1^6+9216#1^8-13312#1^10+4096#1^12&,1,0],Root[212521-2455056#1^2+7899696#1^4-12666240#1^6+16443648#1^8-11695104#1^10+2985984#1^12&,2,0]},
{Root[1-22884#1^2+631728#1^4-2329344#1^6-7962624#1^8-2737152#1^10+2985984#1^12&,1,0],Root[361-11292#1^2+54448#1^4-92160#1^6+72704#1^8-27648#1^10+4096#1^12&,2,0],Root[14641-263280#1^2+1504368#1^4-2699136#1^6+435456#1^8-1741824#1^10+2985984#1^12&,8,0]},
{Root[1-22884#1^2+631728#1^4-2329344#1^6-7962624#1^8-2737152#1^10+2985984#1^12&,8,0],Root[361-11292#1^2+54448#1^4-92160#1^6+72704#1^8-27648#1^10+4096#1^12&,2,0],Root[14641-263280#1^2+1504368#1^4-2699136#1^6+435456#1^8-1741824#1^10+2985984#1^12&,1,0]},
{Root[1-96#1^2+3888#1^4-79488#1^6+767232#1^8-2737152#1^10+2985984#1^12&,1,0],
Root[1-32#1^2+432#1^4-2944#1^6+9472#1^8-11264#1^10+4096#1^12&,1,0],
Root[15625-300000#1^2+2430000#1^4-9936000#1^6+19180800#1^8-13685760#1^10+2985984#1^12&,8,0]},
{Root[1-96#1^2+3888#1^4-79488#1^6+767232#1^8-2737152#1^10+2985984#1^12&,1,0],
Root[1-32#1^2+432#1^4-2944#1^6+9472#1^8-11264#1^10+4096#1^12&,8,0],
Root[15625-300000#1^2+2430000#1^4-9936000#1^6+19180800#1^8-13685760#1^10+2985984#1^12&,8,0]},
{Root[1-96#1^2+3888#1^4-79488#1^6+767232#1^8-2737152#1^10+2985984#1^12&,8,0],
Root[1-32#1^2+432#1^4-2944#1^6+9472#1^8-11264#1^10+4096#1^12&,1,0],
Root[15625-300000#1^2+2430000#1^4-9936000#1^6+19180800#1^8-13685760#1^10+2985984#1^12&,1,0]},
{Root[1-96#1^2+3888#1^4-79488#1^6+767232#1^8-2737152#1^10+2985984#1^12&,8,0],
Root[1-32#1^2+432#1^4-2944#1^6+9472#1^8-11264#1^10+4096#1^12&,8,0],
Root[15625-300000#1^2+2430000#1^4-9936000#1^6+19180800#1^8-13685760#1^10+2985984#1^12&,1,0]},
{Root[14641-1163052#1^2+6839568#1^4-1012608#1^6-12918528#1^8-1990656#1^10+2985984#1^12&,2,0],Root[1-36#1^2+272#1^4+384#1^6-768#1^8-12288#1^10+4096#1^12&,8,0],Root[961-148608#1^2+2974896#1^4-14103936#1^6+22166784#1^8-13685760#1^10+2985984#1^12&,1,0]},
{Root[14641-1163052#1^2+6839568#1^4-1012608#1^6-12918528#1^8-1990656#1^10+2985984#1^12&,7,0],Root[1-36#1^2+272#1^4+384#1^6-768#1^8-12288#1^10+4096#1^12&,8,0],Root[961-148608#1^2+2974896#1^4-14103936#1^6+22166784#1^8-13685760#1^10+2985984#1^12&,8,0]},
{Root[841-542844#1^2+3713184#1^4+558144#1^6-5764608#1^8-1244160#1^10+2985984#1^12&,1,0],Root[121-1108#1^2-96#1^4+9920#1^6+4608#1^8-13312#1^10+4096#1^12&,1,0],Root[961-148608#1^2+2974896#1^4-14103936#1^6+22166784#1^8-13685760#1^10+2985984#1^12&,1,0]},
{Root[841-542844#1^2+3713184#1^4+558144#1^6-5764608#1^8-1244160#1^10+2985984#1^12&,8,0],Root[121-1108#1^2-96#1^4+9920#1^6+4608#1^8-13312#1^10+4096#1^12&,1,0],Root[961-148608#1^2+2974896#1^4-14103936#1^6+22166784#1^8-13685760#1^10+2985984#1^12&,8,0]},
{Root[7921-369780#1^2+4366368#1^4-5170176#1^6-1741824#1^8-497664#1^10+2985984#1^12&,2,0],Root[1-188#1^2+8928#1^4-8960#1^6+16384#1^8-18432#1^10+4096#1^12&,1,0],Root[39601-600384#1^2+3539952#1^4-10247040#1^6+15116544#1^8-10699776#1^10+2985984#1^12&,2,0]},
{Root[7921-369780#1^2+4366368#1^4-5170176#1^6-1741824#1^8-497664#1^10+2985984#1^12&,7,0],Root[1-188#1^2+8928#1^4-8960#1^6+16384#1^8-18432#1^10+4096#1^12&,1,0],Root[39601-600384#1^2+3539952#1^4-10247040#1^6+15116544#1^8-10699776#1^10+2985984#1^12&,7,0]},
{Root[130321-1654824#1^2+4097664#1^4+48384#1^6-6594048#1^8+1244160#1^10+2985984#1^12&,1,0],Root[1-56#1^2+1024#1^4-7168#1^6+18944#1^8-15360#1^10+4096#1^12&,5,0],Root[121-155616#1^2+1668528#1^4+1102464#1^6+14121216#1^8-14681088#1^10+2985984#1^12&,1,0]},
{Root[130321-1654824#1^2+4097664#1^4+48384#1^6-6594048#1^8+1244160#1^10+2985984#1^12&,8,0],Root[1-56#1^2+1024#1^4-7168#1^6+18944#1^8-15360#1^10+4096#1^12&,5,0],Root[121-155616#1^2+1668528#1^4+1102464#1^6+14121216#1^8-14681088#1^10+2985984#1^12&,8,0]},
{Root[1-1044#1^2+270144#1^4-3476736#1^6-4831488#1^8+1244160#1^10+2985984#1^12&,1,0],Root[121-3420#1^2+21504#1^4-44288#1^6+40704#1^8-19456#1^10+4096#1^12&,8,0],Root[212521-2455056#1^2+7899696#1^4-12666240#1^6+16443648#1^8-11695104#1^10+2985984#1^12&,7,0]},
{Root[1-1044#1^2+270144#1^4-3476736#1^6-4831488#1^8+1244160#1^10+2985984#1^12&,8,0],Root[121-3420#1^2+21504#1^4-44288#1^6+40704#1^8-19456#1^10+4096#1^12&,8,0],Root[212521-2455056#1^2+7899696#1^4-12666240#1^6+16443648#1^8-11695104#1^10+2985984#1^12&,2,0]},
{Root[5041-143496#1^2+1133568#1^4-2958336#1^6+124416#1^8+5225472#1^10+2985984#1^12&,1,0],Root[1-56#1^2+1024#1^4-7168#1^6+18944#1^8-15360#1^10+4096#1^12&,4,0],Root[1-218304#1^2+422064#1^4-245376#1^6+26065152#1^8-18662400#1^10+2985984#1^12&,1,0]},
{Root[5041-143496#1^2+1133568#1^4-2958336#1^6+124416#1^8+5225472#1^10+2985984#1^12&,8,0],Root[1-56#1^2+1024#1^4-7168#1^6+18944#1^8-15360#1^10+4096#1^12&,4,0],Root[1-218304#1^2+422064#1^4-245376#1^6+26065152#1^8-18662400#1^10+2985984#1^12&,8,0]},
{Root[4451309+4239891#1+25585637#1^2+26195962#1^3+12670977#1^4+31234656#1^5-19026992#1^6-8300265#1^7+20344044#1^8+4631411#1^9+1859159#1^10-14607929#1^11+4772350#1^12&,3,0],Root[1-128#1^2+1392#1^4-4672#1^6+4096#1^8-1024#1^10+4096#1^12&,1,0],Root[14641-263280#1^2+1504368#1^4-2699136#1^6+435456#1^8-1741824#1^10+2985984#1^12&,8,0]},
{Root[4451309-4239891#1+25585637#1^2-26195962#1^3+12670977#1^4-31234656#1^5-19026992#1^6+8300265#1^7+20344044#1^8-4631411#1^9+1859159#1^10+14607929#1^11+4772350#1^12&,2,0],Root[1-128#1^2+1392#1^4-4672#1^6+4096#1^8-1024#1^10+4096#1^12&,1,0],Root[14641-263280#1^2+1504368#1^4-2699136#1^6+435456#1^8-1741824#1^10+2985984#1^12&,1,0]},
{Root[-733597-6256840#1-12734255#1^2+6126819#1^3+22979407#1^4-18322356#1^5+25812303#1^6+13537551#1^7-15428379#1^8+19460333#1^9-22685448#1^10-1745412#1^11+4903405#1^12&,6,0],Root[1-48#1^2+672#1^4-3456#1^6+4352#1^8+6144#1^10+4096#1^12&,7,0],Root[212521-2455056#1^2+7899696#1^4-12666240#1^6+16443648#1^8-11695104#1^10+2985984#1^12&,7,0]},
{Root[-733597+6256840#1-12734255#1^2-6126819#1^3+22979407#1^4+18322356#1^5+25812303#1^6-13537551#1^7-15428379#1^8-19460333#1^9-22685448#1^10+1745412#1^11+4903405#1^12&,1,0],Root[1-48#1^2+672#1^4-3456#1^6+4352#1^8+6144#1^10+4096#1^12&,7,0],Root[212521-2455056#1^2+7899696#1^4-12666240#1^6+16443648#1^8-11695104#1^10+2985984#1^12&,2,0]},
{Root[302813-10499516#1-9739290#1^2+41433808#1^3-11574965#1^4-15958684#1^5+29215167#1^6-732721#1^7-5842077#1^8+16196953#1^9+7887217#1^10-4707171#1^11+14303515#1^12&,4,0],Root[1+208#1^2+17264#1^4-452480#1^6+3239168#1^8-7502848#1^10+3936256#1^12&,1,0],
Root[50712847+14719519#1-16919466#1^2+6949449#1^3-7715885#1^4-1032652#1^5-1650384#1^6+281720#1^7-36438168#1^8+28772184#1^9+49944993#1^10-30795213#1^11+1016#1^12&,1,0]},
{Root[302813-10499516#1-9739290#1^2+41433808#1^3-11574965#1^4-15958684#1^5+29215167#1^6-732721#1^7-5842077#1^8+16196953#1^9+7887217#1^10-4707171#1^11+14303515#1^12&,4,0],Root[1+208#1^2+17264#1^4-452480#1^6+3239168#1^8-7502848#1^10+3936256#1^12&,8,0],
Root[50712847+14719519#1-16919466#1^2+6949449#1^3-7715885#1^4-1032652#1^5-1650384#1^6+281720#1^7-36438168#1^8+28772184#1^9+49944993#1^10-30795213#1^11+1016#1^12&,1,0]},
{Root[302813+10499516#1-9739290#1^2-41433808#1^3-11574965#1^4+15958684#1^5+29215167#1^6+732721#1^7-5842077#1^8-16196953#1^9+7887217#1^10+4707171#1^11+14303515#1^12&,1,0],Root[1+208#1^2+17264#1^4-452480#1^6+3239168#1^8-7502848#1^10+3936256#1^12&,1,0],
Root[50712847-14719519#1-16919466#1^2-6949449#1^3-7715885#1^4+1032652#1^5-1650384#1^6-281720#1^7-36438168#1^8-28772184#1^9+49944993#1^10+30795213#1^11+1016#1^12&,2,0]},
{Root[302813+10499516#1-9739290#1^2-41433808#1^3-11574965#1^4+15958684#1^5+29215167#1^6+732721#1^7-5842077#1^8-16196953#1^9+7887217#1^10+4707171#1^11+14303515#1^12&,1,0],Root[1+208#1^2+17264#1^4-452480#1^6+3239168#1^8-7502848#1^10+3936256#1^12&,8,0],
Root[50712847-14719519#1-16919466#1^2-6949449#1^3-7715885#1^4+1032652#1^5-1650384#1^6-281720#1^7-36438168#1^8-28772184#1^9+49944993#1^10+30795213#1^11+1016#1^12&,2,0]}}
*)

(* Pentagonal Icositetrahedron *)

PolyhedronName[PentagonalIcositetrahedron]:=PolyhedronName[PentagonalIcositetrahedron,"dextro"]
PolyhedronFaces[PentagonalIcositetrahedron]:=
	PolyhedronFaces[PentagonalIcositetrahedron,"dextro"]
PolyhedronVertices[PentagonalIcositetrahedron]:=
	PolyhedronVertices[PentagonalIcositetrahedron,"dextro"]

NetFaces[PentagonalIcositetrahedron]:={{60,57,64,69,67},{73,74,72,67,69},{62,67,72,71,66},{58,56,50,47,52},
{41,44,48,52,47},{60,52,48,54,57},{53,55,61,65,59},{70,68,63,59,65},{51,59,63,57,54},{49,54,48,45,43},
{39,35,40,43,45},{46,43,40,38,42},{25,18,13,14,20},{5,16,21,20,14},{28,20,21,26,33},{29,36,40,35,32},
{37,30,24,32,35},{25,32,24,19,18},{8,3,1,4,6},{2,7,11,6,4},{10,6,11,18,19},{15,19,24,27,22},{34,31,23,22,27},
{12,22,23,17,9}}
NetVertices[PentagonalIcositetrahedron]:={{-10.548175435,-1.32023134861},
{-10.154023521,-3.48802246942},{-10.1178760126,-0.417545131235},{-9.90938933094,-2.08961572149},
{-9.08737168902,-4.86858137149},{-8.942965191919999,-1.83266361794},{-8.73447851028,-3.50473420819},
{-8.69833100189,-0.43425687001200003},{-8.36000784507,1.49970577605},{-8.332093466449999,-0.551171107385},
{-8.30417908784,-2.60204799082},{-8.22046419767,0.0869372564203},{-8.19979959047,-2.851928242},
{-8.10150473022,-3.84708557652},{-8.07228253547,-0.267803587364},{-7.92071229567,-5.67746528126},
{-7.49796320342,2.00653813078},{-7.33775494882,-2.34509588727},{-7.1654340731,-1.36005501716},
{-7.15696251426,-4.17547559201},{-7.05866765402,-5.17063292653},{-6.84848656875,0.45171760858800003},
{-6.67616569303,1.4367584787},{-6.19900993408,-1.10310291361},{-6.17109555546,-3.15397979704},
{-6.11412543806,-5.49902294201},{-6.02668905836,-0.118062043493},{-5.93316654754,-3.45595439605},
{-5.80485802003,-3.27089403441},{-5.76871051163,-0.200416696229},{-5.70974155401,1.69371058225},
{-5.56022383,-1.87248728648},{-5.41967852671,-4.77947899803},{-4.80289309164,0.601459152459},
{-4.59379969098,-1.61553518293},{-4.38531300934,-3.28760577319},{-4.34916550094,-0.217128435006},
{-4.12733446261,-3.36996042593},{-3.98292796551,-0.334042672379},{-3.95501358689,-2.38491955581},
{-3.43030874369,0.421315759488},{-3.30553695222,-3.93974007801},{-2.98858944787,-2.12796745226},
{-2.84977514326,-0.874202692079},{-2.81626857216,-1.14292658215},{-2.0817409855,-3.22021888205},
{-2.05833111477,0.786096111655},{-1.84984443314,-0.885974478599},{-1.82193005452,-2.93685136203},
{-1.62803169232,1.68878232903},{-1.45569251909,-3.05376559941},{-1.41954501069,0.0167117387766},
{-1.40812513732,-3.43525769035},{-1.21105832905,-1.65535885148},{-0.827591536883,-4.73077614192},
{-0.628100982203,1.67701054251},{-0.244634190033,-1.39840674793},{-0.0475673817687,0.381492090944},
{-0.036147508398,-3.07047733818},{0.,0.},{0.172339173237,-4.74254792844},
{0.174253708405,0.342687199014},{0.39415191405,-2.16779112081},{0.577163320352,-1.96818640001},
{0.602638595685,-3.83986171106},{1.1601206672,1.36418299398},{1.34091310176,-0.466196710756},
{1.39408262417,-2.17956290733},{1.439207962,-1.46135404528},{1.97461622461,-3.47508135889},
{2.10466288316,1.0357929785},{2.20295774341,0.040635643975},{2.78012106376,-1.92755075603},
{3.02475525379,-0.529144008105}}

PolyhedronDual[PentagonalIcositetrahedron,"dextro"]:=Sequence[SnubCube,"dextro"]
PolyhedronName[PentagonalIcositetrahedron,"dextro"]="pentagonal icositetrahedron (dextro)"
PolyhedronFaces[PentagonalIcositetrahedron,"dextro"]:=
{{5,13,8,21,17},{22,20,6,13,7},{15,18,7,13,5},{8,13,6,16,19},{14,12,29,26,9},
{11,30,27,10,14},{9,23,25,11,14},{14,10,24,28,12},{35,33,18,15,3},{34,19,16,4,36},
{31,17,21,1,37},{38,32,20,22,2},{3,31,37,25,23},{28,24,4,32,38},{26,29,2,33,35},
{1,34,36,27,30},{15,5,17,31,3},{4,16,6,20,32},{2,22,7,18,33},{21,8,19,34,1},
{9,26,35,3,23},{27,36,4,24,10},{25,37,1,30,11},{12,28,38,2,29}}
PolyhedronVertices[PentagonalIcositetrahedron,"dextro"]:=
{{0,0,Root[-1-24#1^2-224#1^4+128#1^6&,1,0]},
{0,0,Root[-1-24#1^2-224#1^4+128#1^6&,2,0]},
{0,Root[-1-24#1^2-224#1^4+128#1^6&,1,0],0},
{0,Root[-1-24#1^2-224#1^4+128#1^6&,2,0],0},
{Root[-49+208#1^2-288#1^4+128#1^6&,1,0],Root[-1+16#1^2-96#1^4+128#1^6&,1,0],Root[-1+16#1^2+96#1^4+128#1^6&,1,0]},
{Root[-49+208#1^2-288#1^4+128#1^6&,1,0],Root[-1+16#1^2-96#1^4+128#1^6&,2,0],Root[-1+16#1^2+96#1^4+128#1^6&,2,0]},
{Root[-49+208#1^2-288#1^4+128#1^6&,1,0],Root[-1+16#1^2+96#1^4+128#1^6&,1,0],Root[-1+16#1^2-96#1^4+128#1^6&,2,0]},
{Root[-49+208#1^2-288#1^4+128#1^6&,1,0],Root[-1+16#1^2+96#1^4+128#1^6&,2,0],Root[-1+16#1^2-96#1^4+128#1^6&,1,0]},
{Root[-49+208#1^2-288#1^4+128#1^6&,2,0],Root[-1+16#1^2-96#1^4+128#1^6&,1,0],Root[-1+16#1^2+96#1^4+128#1^6&,2,0]},
{Root[-49+208#1^2-288#1^4+128#1^6&,2,0],Root[-1+16#1^2-96#1^4+128#1^6&,2,0],Root[-1+16#1^2+96#1^4+128#1^6&,1,0]},
{Root[-49+208#1^2-288#1^4+128#1^6&,2,0],Root[-1+16#1^2+96#1^4+128#1^6&,1,0],Root[-1+16#1^2-96#1^4+128#1^6&,1,0]},
{Root[-49+208#1^2-288#1^4+128#1^6&,2,0],Root[-1+16#1^2+96#1^4+128#1^6&,2,0],Root[-1+16#1^2-96#1^4+128#1^6&,2,0]},
{Root[-1-24#1^2-224#1^4+128#1^6&,1,0],0,0},
{Root[-1-24#1^2-224#1^4+128#1^6&,2,0],0,0},
{Root[-1+16#1^2-96#1^4+128#1^6&,1,0],Root[-49+208#1^2-288#1^4+128#1^6&,1,0],Root[-1+16#1^2+96#1^4+128#1^6&,2,0]},
{Root[-1+16#1^2-96#1^4+128#1^6&,1,0],Root[-49+208#1^2-288#1^4+128#1^6&,2,0],Root[-1+16#1^2+96#1^4+128#1^6&,1,0]},
{Root[-1+16#1^2-96#1^4+128#1^6&,1,0],Root[-1+16#1^2-96#1^4+128#1^6&,1,0],Root[-1+16#1^2-96#1^4+128#1^6&,1,0]},
{Root[-1+16#1^2-96#1^4+128#1^6&,1,0],Root[-1+16#1^2-96#1^4+128#1^6&,1,0],Root[-1+16#1^2-96#1^4+128#1^6&,2,0]},
{Root[-1+16#1^2-96#1^4+128#1^6&,1,0],Root[-1+16#1^2-96#1^4+128#1^6&,2,0],Root[-1+16#1^2-96#1^4+128#1^6&,1,0]},
{Root[-1+16#1^2-96#1^4+128#1^6&,1,0],Root[-1+16#1^2-96#1^4+128#1^6&,2,0],Root[-1+16#1^2-96#1^4+128#1^6&,2,0]},
{Root[-1+16#1^2-96#1^4+128#1^6&,1,0],Root[-1+16#1^2+96#1^4+128#1^6&,1,0],Root[-49+208#1^2-288#1^4+128#1^6&,1,0]},
{Root[-1+16#1^2-96#1^4+128#1^6&,1,0],Root[-1+16#1^2+96#1^4+128#1^6&,2,0],Root[-49+208#1^2-288#1^4+128#1^6&,2,0]},
{Root[-1+16#1^2-96#1^4+128#1^6&,2,0],Root[-49+208#1^2-288#1^4+128#1^6&,1,0],Root[-1+16#1^2+96#1^4+128#1^6&,1,0]},
{Root[-1+16#1^2-96#1^4+128#1^6&,2,0],Root[-49+208#1^2-288#1^4+128#1^6&,2,0],Root[-1+16#1^2+96#1^4+128#1^6&,2,0]},
{Root[-1+16#1^2-96#1^4+128#1^6&,2,0],Root[-1+16#1^2-96#1^4+128#1^6&,1,0],Root[-1+16#1^2-96#1^4+128#1^6&,1,0]},
{Root[-1+16#1^2-96#1^4+128#1^6&,2,0],Root[-1+16#1^2-96#1^4+128#1^6&,1,0],Root[-1+16#1^2-96#1^4+128#1^6&,2,0]},
{Root[-1+16#1^2-96#1^4+128#1^6&,2,0],Root[-1+16#1^2-96#1^4+128#1^6&,2,0],Root[-1+16#1^2-96#1^4+128#1^6&,1,0]},
{Root[-1+16#1^2-96#1^4+128#1^6&,2,0],Root[-1+16#1^2-96#1^4+128#1^6&,2,0],Root[-1+16#1^2-96#1^4+128#1^6&,2,0]},
{Root[-1+16#1^2-96#1^4+128#1^6&,2,0],Root[-1+16#1^2+96#1^4+128#1^6&,1,0],Root[-49+208#1^2-288#1^4+128#1^6&,2,0]},
{Root[-1+16#1^2-96#1^4+128#1^6&,2,0],Root[-1+16#1^2+96#1^4+128#1^6&,2,0],Root[-49+208#1^2-288#1^4+128#1^6&,1,0]},
{Root[-1+16#1^2+96#1^4+128#1^6&,1,0],Root[-49+208#1^2-288#1^4+128#1^6&,1,0],Root[-1+16#1^2-96#1^4+128#1^6&,1,0]},
{Root[-1+16#1^2+96#1^4+128#1^6&,1,0],Root[-49+208#1^2-288#1^4+128#1^6&,2,0],Root[-1+16#1^2-96#1^4+128#1^6&,2,0]},
{Root[-1+16#1^2+96#1^4+128#1^6&,1,0],Root[-1+16#1^2-96#1^4+128#1^6&,1,0],Root[-49+208#1^2-288#1^4+128#1^6&,2,0]},
{Root[-1+16#1^2+96#1^4+128#1^6&,1,0],Root[-1+16#1^2-96#1^4+128#1^6&,2,0],Root[-49+208#1^2-288#1^4+128#1^6&,1,0]},
{Root[-1+16#1^2+96#1^4+128#1^6&,2,0],Root[-49+208#1^2-288#1^4+128#1^6&,1,0],Root[-1+16#1^2-96#1^4+128#1^6&,2,0]},
{Root[-1+16#1^2+96#1^4+128#1^6&,2,0],Root[-49+208#1^2-288#1^4+128#1^6&,2,0],Root[-1+16#1^2-96#1^4+128#1^6&,1,0]},
{Root[-1+16#1^2+96#1^4+128#1^6&,2,0],Root[-1+16#1^2-96#1^4+128#1^6&,1,0],Root[-49+208#1^2-288#1^4+128#1^6&,1,0]},
{Root[-1+16#1^2+96#1^4+128#1^6&,2,0],Root[-1+16#1^2-96#1^4+128#1^6&,2,0],Root[-49+208#1^2-288#1^4+128#1^6&,2,0]}}

PolyhedronDual[PentagonalIcositetrahedron,"laevo"]:=Sequence[SnubCube,"laevo"]
PolyhedronName[PentagonalIcositetrahedron,"laevo"]:="pentagonal icositetrahedron (laevo)"
PolyhedronFaces[PentagonalIcositetrahedron,"laevo"]:=
{{18,21,8,13,5},{7,13,6,19,22},{5,13,7,17,15},{20,16,6,13,8},{9,25,29,12,14},
{14,10,28,30,11},{14,11,26,23,9},{12,27,24,10,14},{3,15,17,33,35},{36,4,16,20,34},
{37,2,21,18,31},{1,22,19,32,38},{23,26,37,31,3},{38,32,4,24,27},{35,33,1,29,25},
{30,28,36,34,2},{3,31,18,5,15},{32,19,6,16,4},{33,17,7,22,1},{2,34,20,8,21},
{23,3,35,25,9},{10,24,4,36,28},{11,30,2,37,26},{29,1,38,27,12}}
PolyhedronVertices[PentagonalIcositetrahedron,"laevo"]:=
{{0,0,Root[-1-24#1^2-224#1^4+128#1^6&,1,0]},
{0,0,Root[-1-24#1^2-224#1^4+128#1^6&,2,0]},
{0,Root[-1-24#1^2-224#1^4+128#1^6&,1,0],0},
{0,Root[-1-24#1^2-224#1^4+128#1^6&,2,0],0},
{Root[-49+208#1^2-288#1^4+128#1^6&,1,0],Root[-1+16#1^2-96#1^4+128#1^6&,1,0],Root[-1+16#1^2+96#1^4+128#1^6&,2,0]},
{Root[-49+208#1^2-288#1^4+128#1^6&,1,0],Root[-1+16#1^2-96#1^4+128#1^6&,2,0],Root[-1+16#1^2+96#1^4+128#1^6&,1,0]},
{Root[-49+208#1^2-288#1^4+128#1^6&,1,0],Root[-1+16#1^2+96#1^4+128#1^6&,1,0],Root[-1+16#1^2-96#1^4+128#1^6&,1,0]},
{Root[-49+208#1^2-288#1^4+128#1^6&,1,0],Root[-1+16#1^2+96#1^4+128#1^6&,2,0],Root[-1+16#1^2-96#1^4+128#1^6&,2,0]},
{Root[-49+208#1^2-288#1^4+128#1^6&,2,0],Root[-1+16#1^2-96#1^4+128#1^6&,1,0],Root[-1+16#1^2+96#1^4+128#1^6&,1,0]},
{Root[-49+208#1^2-288#1^4+128#1^6&,2,0],Root[-1+16#1^2-96#1^4+128#1^6&,2,0],Root[-1+16#1^2+96#1^4+128#1^6&,2,0]},
{Root[-49+208#1^2-288#1^4+128#1^6&,2,0],Root[-1+16#1^2+96#1^4+128#1^6&,1,0],Root[-1+16#1^2-96#1^4+128#1^6&,2,0]},
{Root[-49+208#1^2-288#1^4+128#1^6&,2,0],Root[-1+16#1^2+96#1^4+128#1^6&,2,0],Root[-1+16#1^2-96#1^4+128#1^6&,1,0]},
{Root[-1-24#1^2-224#1^4+128#1^6&,1,0],0,0},
{Root[-1-24#1^2-224#1^4+128#1^6&,2,0],0,0},
{Root[-1+16#1^2-96#1^4+128#1^6&,1,0],Root[-49+208#1^2-288#1^4+128#1^6&,1,0],Root[-1+16#1^2+96#1^4+128#1^6&,1,0]},
{Root[-1+16#1^2-96#1^4+128#1^6&,1,0],Root[-49+208#1^2-288#1^4+128#1^6&,2,0],Root[-1+16#1^2+96#1^4+128#1^6&,2,0]},
{Root[-1+16#1^2-96#1^4+128#1^6&,1,0],Root[-1+16#1^2-96#1^4+128#1^6&,1,0],Root[-1+16#1^2-96#1^4+128#1^6&,1,0]},
{Root[-1+16#1^2-96#1^4+128#1^6&,1,0],Root[-1+16#1^2-96#1^4+128#1^6&,1,0],Root[-1+16#1^2-96#1^4+128#1^6&,2,0]},
{Root[-1+16#1^2-96#1^4+128#1^6&,1,0],Root[-1+16#1^2-96#1^4+128#1^6&,2,0],Root[-1+16#1^2-96#1^4+128#1^6&,1,0]},
{Root[-1+16#1^2-96#1^4+128#1^6&,1,0],Root[-1+16#1^2-96#1^4+128#1^6&,2,0],Root[-1+16#1^2-96#1^4+128#1^6&,2,0]},
{Root[-1+16#1^2-96#1^4+128#1^6&,1,0],Root[-1+16#1^2+96#1^4+128#1^6&,1,0],Root[-49+208#1^2-288#1^4+128#1^6&,2,0]},
{Root[-1+16#1^2-96#1^4+128#1^6&,1,0],Root[-1+16#1^2+96#1^4+128#1^6&,2,0],Root[-49+208#1^2-288#1^4+128#1^6&,1,0]},
{Root[-1+16#1^2-96#1^4+128#1^6&,2,0],Root[-49+208#1^2-288#1^4+128#1^6&,1,0],Root[-1+16#1^2+96#1^4+128#1^6&,2,0]},
{Root[-1+16#1^2-96#1^4+128#1^6&,2,0],Root[-49+208#1^2-288#1^4+128#1^6&,2,0],Root[-1+16#1^2+96#1^4+128#1^6&,1,0]},
{Root[-1+16#1^2-96#1^4+128#1^6&,2,0],Root[-1+16#1^2-96#1^4+128#1^6&,1,0],Root[-1+16#1^2-96#1^4+128#1^6&,1,0]},
{Root[-1+16#1^2-96#1^4+128#1^6&,2,0],Root[-1+16#1^2-96#1^4+128#1^6&,1,0],Root[-1+16#1^2-96#1^4+128#1^6&,2,0]},
{Root[-1+16#1^2-96#1^4+128#1^6&,2,0],Root[-1+16#1^2-96#1^4+128#1^6&,2,0],Root[-1+16#1^2-96#1^4+128#1^6&,1,0]},
{Root[-1+16#1^2-96#1^4+128#1^6&,2,0],Root[-1+16#1^2-96#1^4+128#1^6&,2,0],Root[-1+16#1^2-96#1^4+128#1^6&,2,0]},
{Root[-1+16#1^2-96#1^4+128#1^6&,2,0],Root[-1+16#1^2+96#1^4+128#1^6&,1,0],Root[-49+208#1^2-288#1^4+128#1^6&,1,0]},
{Root[-1+16#1^2-96#1^4+128#1^6&,2,0],Root[-1+16#1^2+96#1^4+128#1^6&,2,0],Root[-49+208#1^2-288#1^4+128#1^6&,2,0]},
{Root[-1+16#1^2+96#1^4+128#1^6&,1,0],Root[-49+208#1^2-288#1^4+128#1^6&,1,0],Root[-1+16#1^2-96#1^4+128#1^6&,2,0]},
{Root[-1+16#1^2+96#1^4+128#1^6&,1,0],Root[-49+208#1^2-288#1^4+128#1^6&,2,0],Root[-1+16#1^2-96#1^4+128#1^6&,1,0]},
{Root[-1+16#1^2+96#1^4+128#1^6&,1,0],Root[-1+16#1^2-96#1^4+128#1^6&,1,0],Root[-49+208#1^2-288#1^4+128#1^6&,1,0]},
{Root[-1+16#1^2+96#1^4+128#1^6&,1,0],Root[-1+16#1^2-96#1^4+128#1^6&,2,0],Root[-49+208#1^2-288#1^4+128#1^6&,2,0]},
{Root[-1+16#1^2+96#1^4+128#1^6&,2,0],Root[-49+208#1^2-288#1^4+128#1^6&,1,0],Root[-1+16#1^2-96#1^4+128#1^6&,1,0]},
{Root[-1+16#1^2+96#1^4+128#1^6&,2,0],Root[-49+208#1^2-288#1^4+128#1^6&,2,0],Root[-1+16#1^2-96#1^4+128#1^6&,2,0]},
{Root[-1+16#1^2+96#1^4+128#1^6&,2,0],Root[-1+16#1^2-96#1^4+128#1^6&,1,0],Root[-49+208#1^2-288#1^4+128#1^6&,2,0]},
{Root[-1+16#1^2+96#1^4+128#1^6&,2,0],Root[-1+16#1^2-96#1^4+128#1^6&,2,0],Root[-49+208#1^2-288#1^4+128#1^6&,1,0]}}

(* Pentakis Dodecahedron *)

PolyhedronName[PentakisDodecahedron]:="pentakis dodecahedron"
NetFaces[PentakisDodecahedron]:={{16,20,12},{16,12,10},{16,10,17},{16,17,24},{16,24,22},{15,12,20},{15,20,21},
  {15,21,14},{15,14,7},{15,7,9},{5,10,12},{5,12,4},{5,4,1},{5,1,2},{5,2,8},{11,17,10},{11,10,3},{11,3,6},
  {11,6,13},{11,13,18},{25,24,17},{25,17,19},{25,19,27},{25,27,30},{25,30,26},{28,22,24},{28,24,31},{28,31,32},
  {28,32,29},{28,29,23},{34,42,38},{34,38,32},{34,32,31},{34,31,33},{34,33,39},{45,50,42},{45,42,36},{45,36,44},
  {45,44,51},{45,51,52},{58,53,50},{58,50,55},{58,55,61},{58,61,62},{58,62,56},{57,47,53},{57,53,59},{57,59,60},
  {57,60,54},{57,54,48},{43,40,47},{43,47,49},{43,49,41},{43,41,35},{43,35,37},{46,38,42},{46,42,50},{46,50,53},
  {46,53,47},{46,47,40}}
NetVertices[PentakisDodecahedron]:={{-2.26049084372,0.629397262407},{-2.02733656302,-0.343042494797},
  {-1.64334617913,-0.90301180923},{-1.43998091638,1.20102953857},{-1.43139209459,0.314013121388},
  {-1.41019189843,-1.87545156643},{-1.05854385087,1.76274122002},{-1.0368233344,-0.480459902509},
  {-0.825389570175,0.790301462816},{-0.8228362518010001,-0.331379533066},{-0.814247430003,-1.21839595025},
  {-0.608555842786,0.645392654454},{-0.41967866982,-2.01286897415},{-0.238033923543,2.33437349618},
  {-0.229445101745,1.447357079},{0.,0.},{0.00858882179785,-0.8870164171830001},
  {0.0693949164729,-1.14062644641},{0.241743102496,-1.85945617439},{0.379110741041,0.801964424547},
  {0.593391150056,1.77873661207},{0.59594446843,0.657055616185},{0.723493723379,0.884534440774},
  {0.829098749128,-0.315384141019},{0.837687570926,-1.2024005582},{1.08742229771,-0.351222281663},
  {1.23225633111,-1.9968735821},{1.42504321756,0.341671475166},{1.67477794434,1.19284975171},
  {1.7213299174,-1.12463105436},{1.81961197774,-0.452801548731},{2.30868556403,0.419440979005},
  {2.45351959744,-1.22621032143},{2.70325432422,-0.375032044892},{2.80841438067,1.54324341797},
  {3.04327948705,-1.56265871365},{3.19001100441,0.618914463131},{3.29919879265,0.282023571293},
  {3.4048038184,-0.917895010501},{3.4267480476,0.509502395881},{3.53000672872,2.23556161304},
  {3.53235307335,-0.690416185912},{3.67648277437,1.36068067242},{3.67718710675,-2.33606748635},
  {3.92692183353,-1.48488920981},{4.12829754178,-0.0333605697268},{4.37803226855,0.817817706813},
  {4.39360622132,1.0781499562},{4.43774887715,1.81603307769},{4.52286630196,-0.827833593624},
  {4.62847132771,-2.02775217542},{4.6881879363,-1.02953680454},{5.01193988825,0.0444089341121},
  {5.1151985693799995,1.77046815127},{5.15677392166,-1.60124236632},{5.26003260278,0.124816850836},
  {5.26167461503,0.895587210652},{5.40650864843,-0.750064089785},{5.96322410921,0.352724245044},
  {6.0229407178,1.35093961592},{6.10805814261,-1.29292705539},{6.1677747512,-0.294711684514}}
PolyhedronDual[PentakisDodecahedron]:=TruncatedIcosahedron
PolyhedronFaces[PentakisDodecahedron]:=
{{7,9,19},{21,10,8},{31,14,24},{25,15,32},{6,14,17},{24,19,31},{6,16,15},{21,
    25,32},{9,12,3},{7,19,24},{3,11,10},{8,25,21},{13,14,6},{6,15,13},{7,24,
    18},{8,20,25},{4,1,22},{22,1,5},{9,1,4},{1,10,5},{26,17,31},{32,16,27},{3,
    1,9},{3,10,1},{28,14,13},{13,15,30},{18,12,7},{8,11,20},{16,17,29},{11,12,
    2},{19,26,31},{27,21,32},{29,17,26},{27,16,29},{28,13,23},{23,13,30},{23,
    18,28},{30,20,23},{2,18,23},{23,20,2},{19,9,4},{5,10,21},{24,14,28},{30,
    15,25},{26,22,29},{22,27,29},{4,26,19},{5,21,27},{18,24,28},{25,20,30},{2,
    12,18},{20,11,2},{3,12,11},{4,22,26},{5,27,22},{6,17,16},{17,14,31},{15,
    16,32},{7,12,9},{10,11,8}}
PolyhedronVertices[PentakisDodecahedron]:=
{{Root[81-45#1^2+5#1^4&,2,0],0,Sqrt[9/4+9/(2Sqrt[5])]},
{(9Sqrt[13+22/Sqrt[5]])/19,0,(9Sqrt[13+22/Sqrt[5]])/38},{0,0,(9Sqrt[65+22Sqrt[5]])/38},
{(-9Sqrt[61/2+131/(2Sqrt[5])])/38,(-9(9+Sqrt[5]))/76,(9Sqrt[13+22/Sqrt[5]])/38},
{(-9Sqrt[61/2+131/(2Sqrt[5])])/38,(9(9+Sqrt[5]))/76,(9Sqrt[13+22/Sqrt[5]])/38},
{0,0,(-9Sqrt[65+22Sqrt[5]])/38},{(9Sqrt[(85+Sqrt[5])/10])/38,(-9(7+5Sqrt[5]))/76,
(9Sqrt[13+22/Sqrt[5]])/38},{(9Sqrt[(85+Sqrt[5])/10])/38,(9(7+5Sqrt[5]))/76,(9Sqrt[13+22/Sqrt[5]])/38},
{-Sqrt[9/4-9/(2Sqrt[5])],-3/2,Sqrt[9/4+9/(2Sqrt[5])]},{-Sqrt[9/4-9/(2Sqrt[5])],3/2,
Sqrt[9/4+9/(2Sqrt[5])]},{Sqrt[9/8+9/(8Sqrt[5])],(3(-1+Sqrt[5]))/4,Sqrt[9/4+9/(2Sqrt[5])]},
{Sqrt[9/8+9/(8Sqrt[5])],(-3(-1+Sqrt[5]))/4,Sqrt[9/4+9/(2Sqrt[5])]},
{Sqrt[9/2-9/(2Sqrt[5])],0,(-3Sqrt[1+2/Sqrt[5]])/2},{Sqrt[9/4-9/(2Sqrt[5])],-3/2,
(-3Sqrt[1+2/Sqrt[5]])/2},{Sqrt[9/4-9/(2Sqrt[5])],3/2,(-3Sqrt[1+2/Sqrt[5]])/2},
{(-3Sqrt[(5+Sqrt[5])/10])/2,(3(-1+Sqrt[5]))/4,(-3Sqrt[1+2/Sqrt[5]])/2},
{(-3Sqrt[(5+Sqrt[5])/10])/2,(-3(-1+Sqrt[5]))/4,(-3Sqrt[1+2/Sqrt[5]])/2},
{Sqrt[9/4+9/(2Sqrt[5])],-3/2,Sqrt[9/4-9/(2Sqrt[5])]},{-Sqrt[9/8-9/(8Sqrt[5])],(-3(1+Sqrt[5]))/4,
Sqrt[9/4-9/(2Sqrt[5])]},{Sqrt[9/4+9/(2Sqrt[5])],3/2,Sqrt[9/4-9/(2Sqrt[5])]},
{-Sqrt[9/8-9/(8Sqrt[5])],(3(1+Sqrt[5]))/4,Sqrt[9/4-9/(2Sqrt[5])]},
{-3Sqrt[(5+Sqrt[5])/10],0,Sqrt[9/4-9/(2Sqrt[5])]},{3Sqrt[(5+Sqrt[5])/10],0,-Sqrt[9/4-9/(2Sqrt[5])]},
{Sqrt[9/8-9/(8Sqrt[5])],(-3(1+Sqrt[5]))/4,-Sqrt[9/4-9/(2Sqrt[5])]},
{Sqrt[9/8-9/(8Sqrt[5])],(3(1+Sqrt[5]))/4,-Sqrt[9/4-9/(2Sqrt[5])]},
{(-3Sqrt[1+2/Sqrt[5]])/2,-3/2,-Sqrt[9/4-9/(2Sqrt[5])]},{(-3Sqrt[1+2/Sqrt[5]])/2,3/2,
-Sqrt[9/4-9/(2Sqrt[5])]},{(9Sqrt[61/2+131/(2Sqrt[5])])/38,(-9(9+Sqrt[5]))/76,
(-9Sqrt[13+22/Sqrt[5]])/38},{(-9Sqrt[13+22/Sqrt[5]])/19,0,(-9Sqrt[13+22/Sqrt[5]])/38},
{(9Sqrt[61/2+131/(2Sqrt[5])])/38,(9(9+Sqrt[5]))/76,(-9Sqrt[13+22/Sqrt[5]])/38},
{(-9Sqrt[(85+Sqrt[5])/10])/38,(-9(7+5Sqrt[5]))/76,(-9Sqrt[13+22/Sqrt[5]])/38},
{(-9Sqrt[(85+Sqrt[5])/10])/38,(9(7+5Sqrt[5]))/76,(-9Sqrt[13+22/Sqrt[5]])/38}}

(* Rhombic Dodecahedron *)

PolyhedronName[RhombicDodecahedron]:="rhombic dodecahedron"
NetFaces[RhombicDodecahedron]:={{3,5,2,1},{7,4,2,5},{7,5,6,8},{11,8,6,9},{10,8,11,13},{15,12,10,13},{15,13,14,16},
{19,16,14,17},{18,16,19,21},{23,20,18,21},{23,21,22,24},{26,24,22,25}}
NetVertices[RhombicDodecahedron]:={{-0.824527579823,0.565821765321},{-0.565908230162,1.53180107772},
{0.,0.},{0.242396172715,2.1205658799},{0.258619349661,0.9659793123980001},
{0.544277327566,0.0076476789542},{1.06692375254,1.55474411458},{1.35258173044,0.5964124811379999},
{1.36880490739,-0.558174086367},{1.6112010801,1.56239179354},{2.17710931027,0.0305907158168},
{2.41950548298,2.15115659572},{2.43572865993,0.996570028215},{2.72138663783,0.038238394771},
{3.2440330628,1.5853348304},{3.52969104071,0.627003196955},{3.54591421766,-0.52758337055},
{3.78831039037,1.59298250935},{4.35421862053,0.0611814316336},{4.59661479325,2.18174731154},
{4.61283797019,1.02716074403},{4.8984959481,0.0688291105878},{5.42114237307,1.61592554622},
{5.70680035097,0.657593912772},{5.72302352792,-0.496992654733},{6.5313279308,0.09177214745040001}}
PolyhedronDual[RhombicDodecahedron]:=Cuboctahedron
PolyhedronFaces[RhombicDodecahedron]:=
{Reverse@{4,3,1,2},Reverse@{5,7,2,1},{6,8,3,1},Reverse@{7,9,4,2},{8,10,4,3},{11,6,1,5},
Reverse@{14,10,4,9},Reverse@{11,12,7,5},{11,13,8,6},Reverse@{12,14,9,7},{13,14,10,8},{14,13,11,12}}
PolyhedronVertices[RhombicDodecahedron]:=
{{-3/4, -3/4, 0}, {-3/4, 0, -3/(4*Sqrt[2])}, {-3/4, 0, 3/(4*Sqrt[2])}, 
 {-3/4, 3/4, 0}, {0, -3/4, -3/(4*Sqrt[2])}, {0, -3/4, 3/(4*Sqrt[2])}, 
 {0, 0, -3/(2*Sqrt[2])}, {0, 0, 3/(2*Sqrt[2])}, {0, 3/4, -3/(4*Sqrt[2])}, 
 {0, 3/4, 3/(4*Sqrt[2])}, {3/4, -3/4, 0}, {3/4, 0, -3/(4*Sqrt[2])}, 
 {3/4, 0, 3/(4*Sqrt[2])}, {3/4, 3/4, 0}}

(* Rhombic Hexecontahedron *)

PolyhedronName[RhombicHexecontahedron]:="rhombic hexecontahedron"
PolyhedronFaces[RhombicHexecontahedron]:=
{{9,10,44,43},{6,40,39,5},{7,41,42,8},{45,46,12,11},{61,13,21,35},{36,24,15,
    63},{28,38,30,19},{17,25,37,26},{61,51,59,47},{43,57,37,9},{63,47,59,
    52},{38,58,45,11},{26,53,60,49},{39,55,35,5},{28,49,60,54},{36,56,41,
    7},{5,3,25,17},{41,56,52,31},{7,19,30,4},{51,55,39,31},{9,3,21,13},{45,58,
    54,33},{11,15,24,4},{53,57,43,33},{5,17,18,6},{47,48,62,61},{63,64,48,
    47},{8,20,19,7},{26,49,50,27},{13,14,10,9},{11,12,16,15},{29,50,49,28},{2,
    27,50,29},{52,59,51,31},{1,22,48,23},{54,60,53,33},{2,29,20,8},{35,21,3,
    5},{7,4,24,36},{6,18,27,2},{9,37,25,3},{23,16,12,1},{1,10,14,22},{4,30,38,
    11},{61,35,55,51},{12,46,34,1},{63,52,56,36},{34,44,10,1},{26,37,57,
    53},{8,42,32,2},{28,54,58,38},{32,40,6,2},{61,62,14,13},{32,42,41,31},{31,
    39,40,32},{15,16,64,63},{33,34,46,45},{27,18,17,26},{28,19,20,29},{43,44,
    34,33}}
PolyhedronVertices[RhombicHexecontahedron]:=
{{0,0,-Sqrt[25/16+(5Sqrt[5])/8]},{0,0,Sqrt[25/16+(5Sqrt[5])/8]},{0,(-5-3Sqrt[5])/4,0},
{0,(5+3Sqrt[5])/4,0},{-Sqrt[5/32+Sqrt[5]/32],(-5-3Sqrt[5])/8,Sqrt[5/16+Sqrt[5]/8]},
{-Sqrt[5/32+Sqrt[5]/32],(-5-3Sqrt[5])/8,Sqrt[25/8+(11Sqrt[5])/8]},
{-Sqrt[5/32+Sqrt[5]/32],(5+3Sqrt[5])/8,Sqrt[5/16+Sqrt[5]/8]},
{-Sqrt[5/32+Sqrt[5]/32],(5+3Sqrt[5])/8,Sqrt[25/8+(11Sqrt[5])/8]},
{Sqrt[5/32+Sqrt[5]/32],(-5-3Sqrt[5])/8,-Sqrt[5/16+Sqrt[5]/8]},
{Sqrt[5/32+Sqrt[5]/32],(-5-3Sqrt[5])/8,-Sqrt[25/8+(11Sqrt[5])/8]},
{Sqrt[5/32+Sqrt[5]/32],(5+3Sqrt[5])/8,-Sqrt[5/16+Sqrt[5]/8]},
{Sqrt[5/32+Sqrt[5]/32],(5+3Sqrt[5])/8,-Sqrt[25/8+(11Sqrt[5])/8]},
{-Sqrt[5/16+Sqrt[5]/8],(-5-2Sqrt[5])/4,-Sqrt[5/4+Sqrt[5]/2]},
{-Sqrt[5/16+Sqrt[5]/8],(-5-2Sqrt[5])/4,-Sqrt[85/16+(19Sqrt[5])/8]},
{-Sqrt[5/16+Sqrt[5]/8],(5+2Sqrt[5])/4,-Sqrt[5/4+Sqrt[5]/2]},
{-Sqrt[5/16+Sqrt[5]/8],(5+2Sqrt[5])/4,-Sqrt[85/16+(19Sqrt[5])/8]},
{Sqrt[5/16+Sqrt[5]/8],(-5-2Sqrt[5])/4,Sqrt[5/4+Sqrt[5]/2]},
{Sqrt[5/16+Sqrt[5]/8],(-5-2Sqrt[5])/4,Sqrt[85/16+(19Sqrt[5])/8]},
{Sqrt[5/16+Sqrt[5]/8],(5+2Sqrt[5])/4,Sqrt[5/4+Sqrt[5]/2]},
{Sqrt[5/16+Sqrt[5]/8],(5+2Sqrt[5])/4,Sqrt[85/16+(19Sqrt[5])/8]},
{-Sqrt[25/32+(11Sqrt[5])/32],(-15-7Sqrt[5])/8,-Sqrt[5/16+Sqrt[5]/8]},
{-Sqrt[25/32+(11Sqrt[5])/32],(-5-Sqrt[5])/8,-Sqrt[25/8+(11Sqrt[5])/8]},
{-Sqrt[25/32+(11Sqrt[5])/32],(5+Sqrt[5])/8,-Sqrt[25/8+(11Sqrt[5])/8]},
{-Sqrt[25/32+(11Sqrt[5])/32],(15+7Sqrt[5])/8,-Sqrt[5/16+Sqrt[5]/8]},
{Sqrt[25/32+(11Sqrt[5])/32],(-15-7Sqrt[5])/8,Sqrt[5/16+Sqrt[5]/8]},
{Sqrt[25/32+(11Sqrt[5])/32],(-5-Sqrt[5])/8,Sqrt[5/16+Sqrt[5]/8]},
{Sqrt[25/32+(11Sqrt[5])/32],(-5-Sqrt[5])/8,Sqrt[25/8+(11Sqrt[5])/8]},
{Sqrt[25/32+(11Sqrt[5])/32],(5+Sqrt[5])/8,Sqrt[5/16+Sqrt[5]/8]},
{Sqrt[25/32+(11Sqrt[5])/32],(5+Sqrt[5])/8,Sqrt[25/8+(11Sqrt[5])/8]},
{Sqrt[25/32+(11Sqrt[5])/32],(15+7Sqrt[5])/8,Sqrt[5/16+Sqrt[5]/8]},
{-Sqrt[5/4+Sqrt[5]/2],0,Sqrt[5/16+Sqrt[5]/8]},{-Sqrt[5/4+Sqrt[5]/2],0,
Sqrt[25/8+(11Sqrt[5])/8]},{Sqrt[5/4+Sqrt[5]/2],0,-Sqrt[5/16+Sqrt[5]/8]},
{Sqrt[5/4+Sqrt[5]/2],0,-Sqrt[25/8+(11Sqrt[5])/8]},{-Sqrt[25/16+(5Sqrt[5])/8],(-5-2Sqrt[5])/4,
0},{-Sqrt[25/16+(5Sqrt[5])/8],(5+2Sqrt[5])/4,0},{Sqrt[25/16+(5Sqrt[5])/8],(-5-2Sqrt[5])/4,
0},{Sqrt[25/16+(5Sqrt[5])/8],(5+2Sqrt[5])/4,0},{-Sqrt[65/32+(29Sqrt[5])/32],
(-5-3Sqrt[5])/8,Sqrt[5/4+Sqrt[5]/2]},{-Sqrt[65/32+(29Sqrt[5])/32],(-5-3Sqrt[5])/8,
Sqrt[85/16+(19Sqrt[5])/8]},{-Sqrt[65/32+(29Sqrt[5])/32],(5+3Sqrt[5])/8,Sqrt[5/4+Sqrt[5]/2]},
{-Sqrt[65/32+(29Sqrt[5])/32],(5+3Sqrt[5])/8,Sqrt[85/16+(19Sqrt[5])/8]},
{Sqrt[65/32+(29Sqrt[5])/32],(-5-3Sqrt[5])/8,-Sqrt[5/4+Sqrt[5]/2]},
{Sqrt[65/32+(29Sqrt[5])/32],(-5-3Sqrt[5])/8,-Sqrt[85/16+(19Sqrt[5])/8]},
{Sqrt[65/32+(29Sqrt[5])/32],(5+3Sqrt[5])/8,-Sqrt[5/4+Sqrt[5]/2]},
{Sqrt[65/32+(29Sqrt[5])/32],(5+3Sqrt[5])/8,-Sqrt[85/16+(19Sqrt[5])/8]},
{-Sqrt[25/8+(11Sqrt[5])/8],0,-Sqrt[5/4+Sqrt[5]/2]},{-Sqrt[25/8+(11Sqrt[5])/8],0,
-Sqrt[85/16+(19Sqrt[5])/8]},{Sqrt[25/8+(11Sqrt[5])/8],0,Sqrt[5/4+Sqrt[5]/2]},
{Sqrt[25/8+(11Sqrt[5])/8],0,Sqrt[85/16+(19Sqrt[5])/8]},
{-Sqrt[125/32+(55Sqrt[5])/32],(-5-Sqrt[5])/8,0},{-Sqrt[125/32+(55Sqrt[5])/32],(5+Sqrt[5])/8,
0},{Sqrt[125/32+(55Sqrt[5])/32],(-5-Sqrt[5])/8,0},{Sqrt[125/32+(55Sqrt[5])/32],
(5+Sqrt[5])/8,0},{-Sqrt[85/16+(19Sqrt[5])/8],(-5-2Sqrt[5])/4,Sqrt[5/16+Sqrt[5]/8]},
{-Sqrt[85/16+(19Sqrt[5])/8],(5+2Sqrt[5])/4,Sqrt[5/16+Sqrt[5]/8]},
{Sqrt[85/16+(19Sqrt[5])/8],(-5-2Sqrt[5])/4,-Sqrt[5/16+Sqrt[5]/8]},
{Sqrt[85/16+(19Sqrt[5])/8],(5+2Sqrt[5])/4,-Sqrt[5/16+Sqrt[5]/8]},
{-Sqrt[65/8+(29Sqrt[5])/8],0,-Sqrt[5/16+Sqrt[5]/8]},{Sqrt[65/8+(29Sqrt[5])/8],0,
Sqrt[5/16+Sqrt[5]/8]},{Root[5-400#1^2+256#1^4&,1,0],(-5-Sqrt[5])/8,
-Sqrt[5/16+Sqrt[5]/8]},{Root[5-400#1^2+256#1^4&,1,0],(-5-Sqrt[5])/8,
-Sqrt[25/8+(11Sqrt[5])/8]},{Root[5-400#1^2+256#1^4&,1,0],(5+Sqrt[5])/8,
-Sqrt[5/16+Sqrt[5]/8]},{Root[5-400#1^2+256#1^4&,1,0],(5+Sqrt[5])/8,
-Sqrt[25/8+(11Sqrt[5])/8]}}

(* Rhombic Triacontahedron *)

PolyhedronName[RhombicTriacontahedron]:="rhombic triacontahedron"
NetFaces[RhombicTriacontahedron]:={{4,1,3,6},{3,7,10,6},{3,2,5,7},{10,7,9,13},{9,14,17,13},{9,8,11,14},
  {16,12,15,18},{15,19,22,18},{15,13,17,19},{22,19,21,25},{21,26,29,25},{21,20,23,26},{28,24,27,30},{27,31,34,30},
  {27,25,29,31},{34,31,33,37},{33,38,41,37},{33,32,35,38},{40,36,39,42},{39,43,46,42},{39,37,41,43},{46,43,45,49},
  {45,50,53,49},{45,44,47,50},{52,48,51,54},{51,55,58,54},{51,49,53,55},{58,55,57,60},{57,61,62,60},{57,56,59,61}}
NetVertices[RhombicTriacontahedron]:={{-0.848521685476,-0.529160608206},{-0.840481007634,-2.52106300735},
  {-0.754696482912,-1.52474929404},{0.,0.},{0.0122850625731,-3.0433560568},
  {0.0938252025637,-0.995588685836},{0.0980695872942,-2.04704234349},{0.106110265137,-4.03894474263},
  {0.191894789858,-3.04263102933},{0.94659127277,-1.51788173529},{0.958876335344,-4.56123779208},
  {1.03237579749,-0.521568021982},{1.04041647533,-2.51347042112},{1.04466086006,-3.56492407878},
  {1.12620100006,-1.51715670782},{1.88089748297,0.00759258622409},{1.89318254554,-3.03576347057},
  {1.97472268553,-0.987996099612},{1.97896707026,-2.03944975727},{1.9870077481,-4.03135215641},
  {2.07279227283,-3.0350384431},{2.82748875574,-1.51028914906},{2.83977381831,-4.55364520586},
  {2.91327328046,-0.513975435758},{2.9213139583,-2.5058778349},{2.92555834303,-3.55733149255},
  {3.00709848302,-1.50956412159},{3.76179496594,0.0151851724482},{3.77408002851,-3.02817088435},
  {3.8556201685,-0.980403513387},{3.85986455323,-2.03185717104},{3.86790523107,-4.02375957018},
  {3.95368975579,-3.02744585688},{4.70838623871,-1.50269656284},{4.72067130128,-4.54605261963},
  {4.79417076343,-0.506382849533},{4.80221144127,-2.49828524867},{4.806455826,-3.54973890633},
  {4.88799596599,-1.50197153537},{5.6426924489,0.0227777586723},{5.65497751148,-3.02057829812},
  {5.73651765147,-0.972810927163},{5.7407620362,-2.02426458482},{5.74880271404,-4.01616698396},
  {5.83458723876,-3.01985327066},{6.58928372167,-1.49510397661},{6.60156878425,-4.53846003341},
  {6.6750682464,-0.498790263309},{6.68310892424,-2.49069266245},{6.68735330897,-3.54214632011},
  {6.76889344896,-1.49437894914},{7.52358993187,0.0303703448963},{7.53587499444,-3.0129857119},
  {7.61741513444,-0.965218340939},{7.62165951917,-2.0166719986},{7.62970019701,-4.00857439774},
  {7.71548472173,-3.01226068443},{8.47018120464,-1.48751139039},{8.48246626721,-4.53086744719},
  {8.56400640721,-2.48310007623},{8.56825079194,-3.53455373388},{9.41677247741,-3.00539312568}}
PolyhedronDual[RhombicTriacontahedron]:=Icosidodecahedron
PolyhedronFaces[RhombicTriacontahedron]:={{16,15,11,12},{14,13,17,18},{10,28,20,30},{8,
      5,6,25},{12,28,31,16},{32,30,14,18},{6,3,11,15},{8,17,13,4},{11,21,19,
      27},{13,29,19,22},{7,16,23,26},{24,18,9,26},{12,11,27,28},{30,29,13,
      14},{7,6,15,16},{18,17,8,9},{2,22,19,21},{23,1,24,26},{3,2,21,11},{4,13,
      22,2},{16,31,1,23},{1,32,18,24},{31,28,10,1},{10,30,32,1},{6,5,2,3},{8,
      4,2,5},{28,27,19,20},{20,19,29,30},{26,25,6,7},{9,8,25,26}}
PolyhedronVertices[RhombicTriacontahedron]:=
	{{0,0,-Sqrt[5(5+2Sqrt[5])]/4},{0,0,
      Sqrt[5(5+2Sqrt[5])]/4},{Sqrt[5/32-Sqrt[5]/32],(-5-Sqrt[5])/8,
      Sqrt[25/32+(11Sqrt[5])/32]},{Sqrt[5/32-Sqrt[5]/32],(5+Sqrt[5])/8,
      Sqrt[25/32+(11Sqrt[5])/32]},{Sqrt[5/8+Sqrt[5]/8],0,
      Sqrt[25/32+(11Sqrt[5])/32]},{Sqrt[25/32+(11Sqrt[5])/32],(-5-Sqrt[5])/
        8,Sqrt[5+2Sqrt[5]]/4},{Sqrt[25/32+(11Sqrt[5])/32],(-5-Sqrt[5])/8,
      Root[5-80#1^2+256#1^4&,2,0]},{Sqrt[25/32+(11Sqrt[5])/32],(5+Sqrt[5])/
        8,Sqrt[5+2Sqrt[5]]/4},{Sqrt[25/32+(11Sqrt[5])/32],(5+Sqrt[5])/8,
      Root[5-80#1^2+256#1^4&,2,0]},{-Sqrt[(5+Sqrt[5])/2]/2,0,
      Root[5-400#1^2+256#1^4&,1,0]},{-Sqrt[(5+Sqrt[5])/2]/4,(-5-3Sqrt[5])/
        8,Sqrt[5+2Sqrt[5]]/4},{-Sqrt[(5+Sqrt[5])/2]/4,(-5-3Sqrt[5])/8,
      Root[5-80#1^2+256#1^4&,2,0]},{-Sqrt[(5+Sqrt[5])/2]/4,(5+3Sqrt[5])/8,
      Sqrt[5+2Sqrt[5]]/4},{-Sqrt[(5+Sqrt[5])/2]/4,(5+3Sqrt[5])/8,
      Root[5-80#1^2+256#1^4&,2,0]},{Sqrt[(5+Sqrt[5])/2]/4,(-5-3Sqrt[5])/8,
      Sqrt[5/32-Sqrt[5]/32]},{Sqrt[(5+Sqrt[5])/2]/4,(-5-3Sqrt[5])/
        8,-Sqrt[5+2Sqrt[5]]/4},{Sqrt[(5+Sqrt[5])/2]/4,(5+3Sqrt[5])/8,
      Sqrt[5/32-Sqrt[5]/32]},{Sqrt[(5+Sqrt[5])/2]/4,(5+3Sqrt[5])/
        8,-Sqrt[5+2Sqrt[5]]/4},{-Sqrt[5+2Sqrt[5]]/2,0,
      Sqrt[5+2Sqrt[5]]/4},{-Sqrt[5+2Sqrt[5]]/2,0,
      Root[5-80#1^2+256#1^4&,2,0]},{-Sqrt[5+2Sqrt[5]]/4,-Sqrt[5]/4,
      Sqrt[25/32+(11Sqrt[5])/32]},{-Sqrt[5+2Sqrt[5]]/4,Sqrt[5]/4,
      Sqrt[25/32+(11Sqrt[5])/32]},{Sqrt[5+2Sqrt[5]]/4,-Sqrt[5]/4,
      Root[5-400#1^2+256#1^4&,1,0]},{Sqrt[5+2Sqrt[5]]/4,Sqrt[5]/4,
      Root[5-400#1^2+256#1^4&,1,0]},{Sqrt[5+2Sqrt[5]]/2,0,
      Sqrt[5/32-Sqrt[5]/32]},{Sqrt[5+2Sqrt[5]]/2,
      0,-Sqrt[5+2Sqrt[5]]/4},{Root[5-400#1^2+256#1^4&,1,0],(-5-Sqrt[5])/8,
      Sqrt[5/32-Sqrt[5]/32]},{Root[5-400#1^2+256#1^4&,1,0],(-5-Sqrt[5])/
        8,-Sqrt[5+2Sqrt[5]]/4},{Root[5-400#1^2+256#1^4&,1,0],(5+Sqrt[5])/8,
      Sqrt[5/32-Sqrt[5]/32]},{Root[5-400#1^2+256#1^4&,1,0],(5+Sqrt[5])/
        8,-Sqrt[5+2Sqrt[5]]/4},{Root[5-80#1^2+256#1^4&,2,0],(-5-Sqrt[5])/8,
      Root[5-400#1^2+256#1^4&,1,0]},{Root[5-80#1^2+256#1^4&,2,
        0],(5+Sqrt[5])/8,Root[5-400#1^2+256#1^4&,1,0]}}

(* Small Rhombicosidodecahedron *)

PolyhedronName[SmallRhombicosidodecahedron]:="small rhombicosidodecahedron"
NetFaces[SmallRhombicosidodecahedron]:={{11,8,14,17},{17,14,21},{14,13,20,21},{5,1,4,13,14},{13,16,20},{20,16,25,29},{7,10,16,13},{7,3,10},{3,2,9,10},{10,9,18,22,19},{2,6,9},{9,6,12,15},{33,30,37,40},{40,37,45},{37,36,44,45},{27,23,26,36,37},{36,39,44},{44,39,50,54},{29,32,39,36},{29,25,32},{25,24,31,32},{32,31,41,46,42},{24,28,31},{31,28,35,38},{58,55,61,64},{64,61,68},{61,60,67,68},{52,47,51,60,61},{60,63,67},{67,63,72,76},{54,57,63,60},{54,50,57},{50,49,56,57},{57,56,65,69,66},{49,53,56},{56,53,59,62},{80,77,83,86},{86,83,90},{83,82,89,90},{74,70,73,82,83},{82,85,89},{89,85,95,100},{76,79,85,82},{76,72,79},{72,71,78,79},{79,78,87,92,88},{71,75,78},{78,75,81,84},{104,101,108,111},{111,108,115},{108,107,114,115},{98,93,97,107,108},{107,110,114},{114,110,117,118},{100,103,110,107},{100,95,103},{95,94,102,103},{103,102,112,116,113},{94,99,102},{102,99,106,109},{34,33,40,48,43},{99,91,96,105,106}}
NetVertices[SmallRhombicosidodecahedron]:={{(-8-4Sqrt[2(5-Sqrt[5])]-2Sqrt[10(5-Sqrt[5])]+Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/16,0},{(-2-Sqrt[3])/2,(-4-Sqrt[3])/2},{(-2-Sqrt[3])/2,(-2-Sqrt[3])/2},{(-4-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])])/8,(-1-Sqrt[5])/4},{(-8-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])]-2Sqrt[2(5+Sqrt[5])])/16,(1+Sqrt[5])/4},{(-1-Sqrt[3])/2,-2-Sqrt[3]},{(-1-Sqrt[3])/2,-1},{(-1-Sqrt[3])/2,1},{-Sqrt[3]/2,(-4-Sqrt[3])/2},{-Sqrt[3]/2,(-2-Sqrt[3])/2},{-Sqrt[3]/2,(2+Sqrt[3])/2},{-1/2,(-5-2Sqrt[3])/2},{-1/2,-1/2},{-1/2,1/2},{0,(-5-Sqrt[3])/2},{0,(-1-Sqrt[3])/2},{0,(1+Sqrt[3])/2},{(-8Sqrt[3]-2Sqrt[2(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/16,(-7-2Sqrt[3]-Sqrt[5])/4},{(-8Sqrt[3]+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])]+2Sqrt[2(5+Sqrt[5])])/16,(-5-2Sqrt[3]+Sqrt[5])/4},{1/2,-1/2},{1/2,1/2},{(-8Sqrt[3]+2Sqrt[2(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/16,(-3-Sqrt[3])/2},{(8+16Sqrt[3]-4Sqrt[2(5-Sqrt[5])]-2Sqrt[10(5-Sqrt[5])]+Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/16,0},{Sqrt[3]/2,(-4-Sqrt[3])/2},{Sqrt[3]/2,(-2-Sqrt[3])/2},{(4+8Sqrt[3]-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])])/8,(-1-Sqrt[5])/4},{(8+16Sqrt[3]-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])]-2Sqrt[2(5+Sqrt[5])])/16,(1+Sqrt[5])/4},{(1+Sqrt[3])/2,-2-Sqrt[3]},{(1+Sqrt[3])/2,-1},{(1+Sqrt[3])/2,1},{(2+Sqrt[3])/2,(-4-Sqrt[3])/2},{(2+Sqrt[3])/2,(-2-Sqrt[3])/2},{(2+Sqrt[3])/2,(2+Sqrt[3])/2},{(32+20Sqrt[3]-4Sqrt[15]+3Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/32,(28+16Sqrt[3]+4Sqrt[5]+3Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]+3Sqrt[6(5+Sqrt[5])]-Sqrt[30(5+Sqrt[5])])/32},{(1+2Sqrt[3])/2,(-5-2Sqrt[3])/2},{(1+2Sqrt[3])/2,-1/2},{(1+2Sqrt[3])/2,1/2},{1+Sqrt[3],(-5-Sqrt[3])/2},{1+Sqrt[3],(-1-Sqrt[3])/2},{1+Sqrt[3],(1+Sqrt[3])/2},{(16+8Sqrt[3]-2Sqrt[2(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/16,(-7-2Sqrt[3]-Sqrt[5])/4},{(16+8Sqrt[3]+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])]+2Sqrt[2(5+Sqrt[5])])/16,(-5-2Sqrt[3]+Sqrt[5])/4},{(16+12Sqrt[3]+3Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])])/16,(12+8Sqrt[3]+3Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])])/16},{(3+2Sqrt[3])/2,-1/2},{(3+2Sqrt[3])/2,1/2},{(16+8Sqrt[3]+2Sqrt[2(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/16,(-3-Sqrt[3])/2},{(24+32Sqrt[3]-4Sqrt[2(5-Sqrt[5])]-2Sqrt[10(5-Sqrt[5])]+Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/16,0},{(32+28Sqrt[3]+4Sqrt[15]+4Sqrt[2(5-Sqrt[5])]+2Sqrt[10(5-Sqrt[5])]+Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/32,(20+16Sqrt[3]-4Sqrt[5]+4Sqrt[6(5-Sqrt[5])]+2Sqrt[30(5-Sqrt[5])]+Sqrt[6(5+Sqrt[5])]-Sqrt[30(5+Sqrt[5])])/32},{(2+3Sqrt[3])/2,(-4-Sqrt[3])/2},{(2+3Sqrt[3])/2,(-2-Sqrt[3])/2},{(12+16Sqrt[3]-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])])/8,(-1-Sqrt[5])/4},{(24+32Sqrt[3]-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])]-2Sqrt[2(5+Sqrt[5])])/16,(1+Sqrt[5])/4},{(3(1+Sqrt[3]))/2,-2-Sqrt[3]},{(3(1+Sqrt[3]))/2,-1},{(3(1+Sqrt[3]))/2,1},{(4+3Sqrt[3])/2,(-4-Sqrt[3])/2},{(4+3Sqrt[3])/2,(-2-Sqrt[3])/2},{(4+3Sqrt[3])/2,(2+Sqrt[3])/2},{(3+4Sqrt[3])/2,(-5-2Sqrt[3])/2},{(3+4Sqrt[3])/2,-1/2},{(3+4Sqrt[3])/2,1/2},{2(1+Sqrt[3]),(-5-Sqrt[3])/2},{2(1+Sqrt[3]),(-1-Sqrt[3])/2},{2(1+Sqrt[3]),(1+Sqrt[3])/2},{(32+24Sqrt[3]-2Sqrt[2(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/16,(-7-2Sqrt[3]-Sqrt[5])/4},{(32+24Sqrt[3]+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])]+2Sqrt[2(5+Sqrt[5])])/16,(-5-2Sqrt[3]+Sqrt[5])/4},{(5+4Sqrt[3])/2,-1/2},{(5+4Sqrt[3])/2,1/2},{(32+24Sqrt[3]+2Sqrt[2(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/16,(-3-Sqrt[3])/2},{(40+48Sqrt[3]-4Sqrt[2(5-Sqrt[5])]-2Sqrt[10(5-Sqrt[5])]+Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/16,0},{(4+5Sqrt[3])/2,(-4-Sqrt[3])/2},{(4+5Sqrt[3])/2,(-2-Sqrt[3])/2},{(20+24Sqrt[3]-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])])/8,(-1-Sqrt[5])/4},{(40+48Sqrt[3]-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])]-2Sqrt[2(5+Sqrt[5])])/16,(1+Sqrt[5])/4},{(5(1+Sqrt[3]))/2,-2-Sqrt[3]},{(5(1+Sqrt[3]))/2,-1},{(5(1+Sqrt[3]))/2,1},{(6+5Sqrt[3])/2,(-4-Sqrt[3])/2},{(6+5Sqrt[3])/2,(-2-Sqrt[3])/2},{(6+5Sqrt[3])/2,(2+Sqrt[3])/2},{(5+6Sqrt[3])/2,(-5-2Sqrt[3])/2},{(5+6Sqrt[3])/2,-1/2},{(5+6Sqrt[3])/2,1/2},{3(1+Sqrt[3]),(-5-Sqrt[3])/2},{3(1+Sqrt[3]),(-1-Sqrt[3])/2},{3(1+Sqrt[3]),(1+Sqrt[3])/2},{(48+40Sqrt[3]-2Sqrt[2(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/16,(-7-2Sqrt[3]-Sqrt[5])/4},{(48+40Sqrt[3]+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])]+2Sqrt[2(5+Sqrt[5])])/16,(-5-2Sqrt[3]+Sqrt[5])/4},{(7+6Sqrt[3])/2,-1/2},{(7+6Sqrt[3])/2,1/2},{(56+58Sqrt[3]-2Sqrt[15]-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])])/16,(-34-16Sqrt[3]+2Sqrt[5]-Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])])/16},{(48+40Sqrt[3]+2Sqrt[2(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/16,(-3-Sqrt[3])/2},{(56+64Sqrt[3]-4Sqrt[2(5-Sqrt[5])]-2Sqrt[10(5-Sqrt[5])]+Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/16,0},{(6+7Sqrt[3])/2,(-4-Sqrt[3])/2},{(6+7Sqrt[3])/2,(-2-Sqrt[3])/2},{(112+120Sqrt[3]-3Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])]-Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/32,(-72-32Sqrt[3]-3Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])]-Sqrt[6(5+Sqrt[5])]-Sqrt[30(5+Sqrt[5])])/32},{(28+32Sqrt[3]-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])])/8,(-1-Sqrt[5])/4},{(56+64Sqrt[3]-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])]-2Sqrt[2(5+Sqrt[5])])/16,(1+Sqrt[5])/4},{(7(1+Sqrt[3]))/2,-2-Sqrt[3]},{(7(1+Sqrt[3]))/2,-1},{(7(1+Sqrt[3]))/2,1},{(8+7Sqrt[3])/2,(-4-Sqrt[3])/2},{(8+7Sqrt[3])/2,(-2-Sqrt[3])/2},{(8+7Sqrt[3])/2,(2+Sqrt[3])/2},{(112+124Sqrt[3]+4Sqrt[15]+Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])]-Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/32,(-76-32Sqrt[3]-4Sqrt[5]+Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])]-Sqrt[6(5+Sqrt[5])]-Sqrt[30(5+Sqrt[5])])/32},{(7+8Sqrt[3])/2,(-5-2Sqrt[3])/2},{(7+8Sqrt[3])/2,-1/2},{(7+8Sqrt[3])/2,1/2},{4(1+Sqrt[3]),(-5-Sqrt[3])/2},{4(1+Sqrt[3]),(-1-Sqrt[3])/2},{4(1+Sqrt[3]),(1+Sqrt[3])/2},{(64+56Sqrt[3]-2Sqrt[2(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/16,(-7-2Sqrt[3]-Sqrt[5])/4},{(64+56Sqrt[3]+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])]+2Sqrt[2(5+Sqrt[5])])/16,(-5-2Sqrt[3]+Sqrt[5])/4},{(9+8Sqrt[3])/2,-1/2},{(9+8Sqrt[3])/2,1/2},{(64+56Sqrt[3]+2Sqrt[2(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/16,(-3-Sqrt[3])/2},{(8+9Sqrt[3])/2,(-2-Sqrt[3])/2},{(9(1+Sqrt[3]))/2,-1}}
PolyhedronDual[SmallRhombicosidodecahedron]:=DeltoidalHexecontahedron
PolyhedronFaces[SmallRhombicosidodecahedron]:=
{{36,23,27},{37,28,24},{40,8,7},{35,5,6},{38,29,25},{39,26,30},{10,14,2},{9,1,
    13},{12,4,16},{11,15,3},{54,45,41},{55,42,46},{58,19,20},{53,18,17},{56,
    43,47},{57,48,44},{34,32,22},{33,21,31},{59,51,49},{60,50,52},{27,31,21,
    36},{23,36,1,9},{10,2,37,24},{37,22,32,28},{8,40,30,26},{25,29,40,7},{35,
    27,23,5},{6,24,28,35},{3,38,25,11},{21,33,29,38},{39,30,34,22},{12,26,39,
    4},{55,14,10,42},{41,9,13,54},{57,44,12,16},{15,11,43,56},{45,54,59,
    49},{50,60,55,46},{48,58,20,44},{43,19,58,47},{53,17,41,45},{46,42,18,
    53},{59,56,47,51},{52,48,57,60},{31,32,34,33},{17,18,6,5},{1,3,15,13},{14,
    16,4,2},{7,8,20,19},{51,52,50,49},{3,1,36,21,38},{22,37,2,4,39},{29,33,34,
    30,40},{27,35,28,32,31},{42,10,24,6,18},{41,17,5,23,9},{20,8,26,12,
    44},{11,25,7,19,43},{56,59,54,13,15},{57,16,14,55,60},{58,48,52,51,
    47},{49,50,46,53,45}}
PolyhedronVertices[SmallRhombicosidodecahedron]:=
{{-1/2,-1/2,-1-Sqrt[5]/2},{-1/2,-1/2,(2+Sqrt[5])/2},{-1/2,1/2,-1-Sqrt[5]/2},
{-1/2,1/2,(2+Sqrt[5])/2},{-1/2,-1-Sqrt[5]/2,-1/2},{-1/2,-1-Sqrt[5]/2,1/2},
{-1/2,(2+Sqrt[5])/2,-1/2},{-1/2,(2+Sqrt[5])/2,1/2},
{0,(-3+Sqrt[5])^(-1),(-5-Sqrt[5])/4},{0,(-3+Sqrt[5])^(-1),(5+Sqrt[5])/4},
{0,(3+Sqrt[5])/4,(-5-Sqrt[5])/4},{0,(3+Sqrt[5])/4,(5+Sqrt[5])/4},
{1/2,-1/2,-1-Sqrt[5]/2},{1/2,-1/2,(2+Sqrt[5])/2},{1/2,1/2,-1-Sqrt[5]/2},
{1/2,1/2,(2+Sqrt[5])/2},{1/2,-1-Sqrt[5]/2,-1/2},{1/2,-1-Sqrt[5]/2,1/2},
{1/2,(2+Sqrt[5])/2,-1/2},{1/2,(2+Sqrt[5])/2,1/2},
{(-5-Sqrt[5])/4,0,(-3+Sqrt[5])^(-1)},{(-5-Sqrt[5])/4,0,(3+Sqrt[5])/4},
{(-1-Sqrt[5])/4,(-1-Sqrt[5])/2,(-3+Sqrt[5])^(-1)},
{(-1-Sqrt[5])/4,(-1-Sqrt[5])/2,(3+Sqrt[5])/4},{(-1-Sqrt[5])/4,(1+Sqrt[5])/2,
(-3+Sqrt[5])^(-1)},{(-1-Sqrt[5])/4,(1+Sqrt[5])/2,(3+Sqrt[5])/4},
{(-1-Sqrt[5])/2,(-3+Sqrt[5])^(-1),(-1-Sqrt[5])/4},
{(-1-Sqrt[5])/2,(-3+Sqrt[5])^(-1),(1+Sqrt[5])/4},
{(-1-Sqrt[5])/2,(3+Sqrt[5])/4,(-1-Sqrt[5])/4},{(-1-Sqrt[5])/2,(3+Sqrt[5])/4,
(1+Sqrt[5])/4},{-1-Sqrt[5]/2,-1/2,-1/2},{-1-Sqrt[5]/2,-1/2,1/2},
{-1-Sqrt[5]/2,1/2,-1/2},{-1-Sqrt[5]/2,1/2,1/2},{(-3+Sqrt[5])^(-1),(-5-Sqrt[5])/4,
0},{(-3+Sqrt[5])^(-1),(-1-Sqrt[5])/4,(-1-Sqrt[5])/2},
{(-3+Sqrt[5])^(-1),(-1-Sqrt[5])/4,(1+Sqrt[5])/2},
{(-3+Sqrt[5])^(-1),(1+Sqrt[5])/4,(-1-Sqrt[5])/2},
{(-3+Sqrt[5])^(-1),(1+Sqrt[5])/4,(1+Sqrt[5])/2},{(-3+Sqrt[5])^(-1),(5+Sqrt[5])/4,
0},{(1+Sqrt[5])/4,(-1-Sqrt[5])/2,(-3+Sqrt[5])^(-1)},
{(1+Sqrt[5])/4,(-1-Sqrt[5])/2,(3+Sqrt[5])/4},{(1+Sqrt[5])/4,(1+Sqrt[5])/2,
(-3+Sqrt[5])^(-1)},{(1+Sqrt[5])/4,(1+Sqrt[5])/2,(3+Sqrt[5])/4},
{(1+Sqrt[5])/2,(-3+Sqrt[5])^(-1),(-1-Sqrt[5])/4},
{(1+Sqrt[5])/2,(-3+Sqrt[5])^(-1),(1+Sqrt[5])/4},
{(1+Sqrt[5])/2,(3+Sqrt[5])/4,(-1-Sqrt[5])/4},{(1+Sqrt[5])/2,(3+Sqrt[5])/4,
(1+Sqrt[5])/4},{(2+Sqrt[5])/2,-1/2,-1/2},{(2+Sqrt[5])/2,-1/2,1/2},
{(2+Sqrt[5])/2,1/2,-1/2},{(2+Sqrt[5])/2,1/2,1/2},
{(3+Sqrt[5])/4,(-5-Sqrt[5])/4,0},{(3+Sqrt[5])/4,(-1-Sqrt[5])/4,(-1-Sqrt[5])/2},
{(3+Sqrt[5])/4,(-1-Sqrt[5])/4,(1+Sqrt[5])/2},{(3+Sqrt[5])/4,(1+Sqrt[5])/4,
(-1-Sqrt[5])/2},{(3+Sqrt[5])/4,(1+Sqrt[5])/4,(1+Sqrt[5])/2},
{(3+Sqrt[5])/4,(5+Sqrt[5])/4,0},{(5+Sqrt[5])/4,0,(-3+Sqrt[5])^(-1)},
{(5+Sqrt[5])/4,0,(3+Sqrt[5])/4}}

(* Small Rhombicuboctahedron *)

PolyhedronName[SmallRhombicuboctahedron]:="small rhombicuboctahedron"
NetFaces[SmallRhombicuboctahedron]:={{4,3,7,8},{3,2,6,7},{2,1,5,6},{10,7,14},{7,6,13,14},{6,9,13},{15,14,20,21},{14,13,19,20},{13,12,18,19},{24,20,27},{20,19,26,27},{19,23,26},{28,27,31,32},{27,26,30,31},{26,25,29,30},{34,31,37},{31,30,36,37},{30,33,36},{38,37,41,42},{37,36,40,41},{36,35,39,40},{44,41,46},{41,40,45,46},{40,43,45},{16,15,21,22},{12,11,17,18}}
NetVertices[SmallRhombicuboctahedron]:={{-1/2,-5/2},{-1/2,-3/2},{-1/2,-1/2},{-1/2,1/2},{1/2,-5/2},{1/2,-3/2},{1/2,-1/2},{1/2,1/2},{1,(-3-Sqrt[3])/2},{1,(-1+Sqrt[3])/2},{3/2,-7/2},{3/2,-5/2},{3/2,-3/2},{3/2,-1/2},{3/2,1/2},{3/2,3/2},{5/2,-7/2},{5/2,-5/2},{5/2,-3/2},{5/2,-1/2},{5/2,1/2},{5/2,3/2},{3,(-3-Sqrt[3])/2},{3,(-1+Sqrt[3])/2},{7/2,-5/2},{7/2,-3/2},{7/2,-1/2},{7/2,1/2},{9/2,-5/2},{9/2,-3/2},{9/2,-1/2},{9/2,1/2},{5,(-3-Sqrt[3])/2},{5,(-1+Sqrt[3])/2},{11/2,-5/2},{11/2,-3/2},{11/2,-1/2},{11/2,1/2},{13/2,-5/2},{13/2,-3/2},{13/2,-1/2},{13/2,1/2},{7,(-3-Sqrt[3])/2},{7,(-1+Sqrt[3])/2},{15/2,-3/2},{15/2,-1/2}}
PolyhedronDual[SmallRhombicuboctahedron]:=DeltoidalIcositetrahedron
PolyhedronFaces[SmallRhombicuboctahedron]:=
	{{3,11,9,1},{2,10,12,4},{24,22,21,23},{19,17,18,20},{5,13,14,6},{8,16,15,
    7},{13,21,22,14},{16,24,23,15},{6,18,17,5},{7,19,20,8},{6,14,10,2},{4,12,
    16,8},{22,24,12,10},{2,4,20,18},{1,9,13,5},{7,15,11,3},{9,11,23,21},{17,
    19,3,1},{22,10,14},{16,12,24},{6,2,18},{20,4,8},{13,9,21},{23,11,15},{17,
    1,5},{7,3,19}}
PolyhedronVertices[SmallRhombicuboctahedron]:={{-1/2, -1/2, -1/2 - 1/Sqrt[2]}, 
{-1/2,-1/2,1/2+1/Sqrt[2]},
{-1/2,1/2,-1/2-1/Sqrt[2]},
{-1/2,1/2,1/2+1/Sqrt[2]},{-1/2,-1/2-1/Sqrt[2],
-1/2},{-1/2,-1/2-1/Sqrt[2],1/2},
{-1/2,1/2+1/Sqrt[2],-1/2},{-1/2,1/2+1/Sqrt[2],
1/2},{1/2,-1/2,-1/2-1/Sqrt[2]},
{1/2,-1/2,1/2+1/Sqrt[2]},{1/2,1/2,-1/2-1/Sqrt[2]},
{1/2,1/2,1/2+1/Sqrt[2]},{1/2,-1/2-1/Sqrt[2],-1/2},
{1/2,-1/2-1/Sqrt[2],1/2},{1/2,1/2+1/Sqrt[2],-1/2},
{1/2,1/2+1/Sqrt[2],1/2},{-1/2-1/Sqrt[2],-1/2,
-1/2},{-1/2-1/Sqrt[2],-1/2,1/2},
{-1/2-1/Sqrt[2],1/2,-1/2},{-1/2-1/Sqrt[2],1/2,
1/2},{1/2+1/Sqrt[2],-1/2,-1/2},
{1/2+1/Sqrt[2],-1/2,1/2},{1/2+1/Sqrt[2],1/2,-1/2},
{1/2+1/Sqrt[2],1/2,1/2}}

(* Small Triakis Octahedron *)

PolyhedronName[SmallTriakisOctahedron]:="small triakis octahedron"
NetFaces[SmallTriakisOctahedron]:={{24,25,26},{24,26,22},{24,22,23},{21,20,18},{21,18,23},{21,23,22},{16,18,20},
{16,20,13},{16,13,14},{17,15,13},{17,13,20},{17,20,19},{10,12,14},{10,14,7},{10,7,8},{11,9,7},{11,7,14},
{11,14,13},{6,7,9},{6,9,4},{6,4,5},{3,2,1},{3,1,5},{3,5,4}}
NetVertices[SmallTriakisOctahedron]:={{-3.21637012435,-0.573638749873},{-2.75634063762,0.314264888667},
{-2.71536906624,-0.270086958122},{-2.67055466396,0.313982746499},{-2.21637553277,-0.57692764001},
{-2.17156113049,0.00714206461077},{-2.13058955911,-0.577209782179},{-1.67641042792,-1.46812016869},
{-1.67056007237,0.310693856362},{-1.63159602564,-0.884050464067},{-1.62958850099,-0.273657990428},
{-1.59062445426,-1.46840231086},{-1.58477409872,0.310411714193},{-1.13059496752,-0.580498672316},
{-1.12474461198,1.19831535273},{-1.08578056524,0.00357103230539},{-1.0837730406,0.613963505945},
{-1.04480899387,-0.580780814484},{-1.03895863832,1.19803321057},{-0.58477950713,0.307122824057},
{-0.54380793575,-0.277229022733},{-0.498993533471,0.306840681888},{-0.044814402279,-0.584069704621},
{0.,0.},{0.0409715713793,-0.58435184679},{0.501001058115,0.303551791751}}
PolyhedronDual[SmallTriakisOctahedron]:=TruncatedCube
PolyhedronFaces[SmallTriakisOctahedron]:=
{{6,8,4},{3,8,5},{7,6,2},{1,5,7},{12,8,6},{5,8,11},{10,6,7},{7,5,9},{14,6,
    10},{9,5,14},{14,12,6},{5,11,14},{11,8,14},{14,8,12},{7,9,14},{14,10,
    7},{6,13,2},{1,13,5},{6,4,13},{13,3,5},{3,13,8},{8,13,4},{13,1,7},{7,2,
    13}}
PolyhedronVertices[SmallTriakisOctahedron]:=
{{-1,-1,-1},{-1,-1,1},{-1,1,-1},{-1,1,1},
{0,0,-1-Sqrt[2]},{0,0,1+Sqrt[2]},
{0,-1-Sqrt[2],0},{0,1+Sqrt[2],0},{1,-1,-1},
{1,-1,1},{1,1,-1},{1,1,1},{-1-Sqrt[2],0,0},
{1+Sqrt[2],0,0}}

(* Snub Cube *)

PolyhedronName[SnubCube]:=PolyhedronName[SnubCube,"laevo"]
PolyhedronDual[SnubCube]:=PolyhedronDual[SnubCube,"laevo"]
PolyhedronFaces[SnubCube]:=PolyhedronFaces[SnubCube,"laevo"]
PolyhedronVertices[SnubCube]:=PolyhedronVertices[SnubCube,"laevo"]

NetFaces[SnubCube]:={{2,1,4},{1,3,4},{4,3,7,8},{6,4,8},{3,5,7},{8,7,11},{7,10,11},{9,8,12},{8,11,12},{12,11,16,17},{14,12,17},{11,13,16},{17,16,23},{16,22,23},{18,17,24},{17,23,24},{24,23,31,32},{28,24,32},{23,27,31},{32,31,38},{31,37,38},{33,32,39},{32,38,39},{39,38,43,44},{42,39,44},{38,41,43},{44,43,46},{43,45,46},{25,20,29},{15,19,20},{20,19,28,29},{29,28,34},{19,24,28},{31,27,36},{21,26,27},{27,26,35,36},{36,35,40},{26,30,35}}
NetVertices[SnubCube]:={{-1/(2Sqrt[3]),-1/2},{-1/(2Sqrt[3]),1/2},{1/Sqrt[3],-1},{1/Sqrt[3],0},{(3+2Sqrt[3])/6,(-2-Sqrt[3])/2},{(3+2Sqrt[3])/6,Sqrt[3]/2},{(3+Sqrt[3])/3,-1},{(3+Sqrt[3])/3,0},{(3+Sqrt[3])/3,1},{(6+5Sqrt[3])/6,-3/2},{(6+5Sqrt[3])/6,-1/2},{(6+5Sqrt[3])/6,1/2},{(9+5Sqrt[3])/6,(-1-Sqrt[3])/2},{(9+5Sqrt[3])/6,(1+Sqrt[3])/2},{(9+5Sqrt[3])/6,(3+Sqrt[3])/2},{(12+5Sqrt[3])/6,-1/2},{(12+5Sqrt[3])/6,1/2},{(12+5Sqrt[3])/6,3/2},{(9+8Sqrt[3])/6,(2+Sqrt[3])/2},{(9+8Sqrt[3])/6,(4+Sqrt[3])/2},{(5(3+Sqrt[3]))/6,(-1-Sqrt[3])/2},{(2(3+2Sqrt[3]))/3,-1},{(2(3+2Sqrt[3]))/3,0},{(2(3+2Sqrt[3]))/3,1},{(2(3+2Sqrt[3]))/3,2+Sqrt[3]},{(15+8Sqrt[3])/6,(-2-Sqrt[3])/2},{(15+8Sqrt[3])/6,-Sqrt[3]/2},{(15+8Sqrt[3])/6,(2+Sqrt[3])/2},{(15+8Sqrt[3])/6,(4+Sqrt[3])/2},{(9+4Sqrt[3])/3,-1-Sqrt[3]},{(9+4Sqrt[3])/3,0},{(9+4Sqrt[3])/3,1},{(9+4Sqrt[3])/3,2},{(15+11Sqrt[3])/6,(3+Sqrt[3])/2},{(21+8Sqrt[3])/6,(-2-Sqrt[3])/2},{(21+8Sqrt[3])/6,-Sqrt[3]/2},{(18+11Sqrt[3])/6,-1/2},{(18+11Sqrt[3])/6,1/2},{(18+11Sqrt[3])/6,3/2},{(21+11Sqrt[3])/6,(-1-Sqrt[3])/2},{(21+11Sqrt[3])/6,(1-Sqrt[3])/2},{(21+11Sqrt[3])/6,(3+Sqrt[3])/2},{(24+11Sqrt[3])/6,1/2},{(24+11Sqrt[3])/6,3/2},{(12+7Sqrt[3])/3,0},{(12+7Sqrt[3])/3,1}}

PolyhedronDual[SnubCube,"laevo"]:=Sequence[PentagonalIcositetrahedron,"dextro"]
PolyhedronName[SnubCube,"laevo"]:="snub cube (laevo)"
PolyhedronFaces[SnubCube,"laevo"]:=
{{3,1,17},{3,17,9},{3,19,2},{3,9,19},{1,4,20},{1,20,11},{1,11,17},{2,19,12},{2,18,4},
{2,12,18},{4,18,10},{4,10,20},{17,11,13},{19,9,15},{18,12,14},{20,10,16},{9,21,15},
{11,23,13},{12,24,14},{10,22,16},{13,23,7},{13,7,21},{15,21,5},{15,5,24},{16,22,6},
{16,6,23},{14,24,8},{14,8,22},{21,7,5},{23,6,7},{24,5,8},{22,8,6},{1,3,2,4},{21,9,17,13},
{24,12,19,15},{10,18,14,22},{11,20,16,23},{8,5,7,6}}
PolyhedronVertices[SnubCube,"laevo"]:=
{{Root[-1-12#1^2-32#1^4+32#1^6&,1,0],Root[-1+12#1^2-32#1^4+32#1^6&,1,0],Root[-1+4#1^2-16#1^4+32#1^6&,1,0]},
{Root[-1-12#1^2-32#1^4+32#1^6&,1,0],Root[-1+12#1^2-32#1^4+32#1^6&,2,0],Root[-1+4#1^2-16#1^4+32#1^6&,2,0]},
{Root[-1-12#1^2-32#1^4+32#1^6&,1,0],Root[-1+4#1^2-16#1^4+32#1^6&,1,0],Root[-1+12#1^2-32#1^4+32#1^6&,2,0]},
{Root[-1-12#1^2-32#1^4+32#1^6&,1,0],Root[-1+4#1^2-16#1^4+32#1^6&,2,0],Root[-1+12#1^2-32#1^4+32#1^6&,1,0]},
{Root[-1-12#1^2-32#1^4+32#1^6&,2,0],Root[-1+12#1^2-32#1^4+32#1^6&,1,0],Root[-1+4#1^2-16#1^4+32#1^6&,2,0]},
{Root[-1-12#1^2-32#1^4+32#1^6&,2,0],Root[-1+12#1^2-32#1^4+32#1^6&,2,0],Root[-1+4#1^2-16#1^4+32#1^6&,1,0]},
{Root[-1-12#1^2-32#1^4+32#1^6&,2,0],Root[-1+4#1^2-16#1^4+32#1^6&,1,0],Root[-1+12#1^2-32#1^4+32#1^6&,1,0]},
{Root[-1-12#1^2-32#1^4+32#1^6&,2,0],Root[-1+4#1^2-16#1^4+32#1^6&,2,0],Root[-1+12#1^2-32#1^4+32#1^6&,2,0]},
{Root[-1+12#1^2-32#1^4+32#1^6&,1,0],Root[-1-12#1^2-32#1^4+32#1^6&,1,0],Root[-1+4#1^2-16#1^4+32#1^6&,2,0]},
{Root[-1+12#1^2-32#1^4+32#1^6&,1,0],Root[-1-12#1^2-32#1^4+32#1^6&,2,0],Root[-1+4#1^2-16#1^4+32#1^6&,1,0]},
{Root[-1+12#1^2-32#1^4+32#1^6&,1,0],Root[-1+4#1^2-16#1^4+32#1^6&,1,0],Root[-1-12#1^2-32#1^4+32#1^6&,1,0]},
{Root[-1+12#1^2-32#1^4+32#1^6&,1,0],Root[-1+4#1^2-16#1^4+32#1^6&,2,0],Root[-1-12#1^2-32#1^4+32#1^6&,2,0]},
{Root[-1+12#1^2-32#1^4+32#1^6&,2,0],Root[-1-12#1^2-32#1^4+32#1^6&,1,0],Root[-1+4#1^2-16#1^4+32#1^6&,1,0]},
{Root[-1+12#1^2-32#1^4+32#1^6&,2,0],Root[-1-12#1^2-32#1^4+32#1^6&,2,0],Root[-1+4#1^2-16#1^4+32#1^6&,2,0]},
{Root[-1+12#1^2-32#1^4+32#1^6&,2,0],Root[-1+4#1^2-16#1^4+32#1^6&,1,0],Root[-1-12#1^2-32#1^4+32#1^6&,2,0]},
{Root[-1+12#1^2-32#1^4+32#1^6&,2,0],Root[-1+4#1^2-16#1^4+32#1^6&,2,0],Root[-1-12#1^2-32#1^4+32#1^6&,1,0]},
{Root[-1+4#1^2-16#1^4+32#1^6&,1,0],Root[-1-12#1^2-32#1^4+32#1^6&,1,0],Root[-1+12#1^2-32#1^4+32#1^6&,1,0]},
{Root[-1+4#1^2-16#1^4+32#1^6&,1,0],Root[-1-12#1^2-32#1^4+32#1^6&,2,0],Root[-1+12#1^2-32#1^4+32#1^6&,2,0]},
{Root[-1+4#1^2-16#1^4+32#1^6&,1,0],Root[-1+12#1^2-32#1^4+32#1^6&,1,0],Root[-1-12#1^2-32#1^4+32#1^6&,2,0]},
{Root[-1+4#1^2-16#1^4+32#1^6&,1,0],Root[-1+12#1^2-32#1^4+32#1^6&,2,0],Root[-1-12#1^2-32#1^4+32#1^6&,1,0]},
{Root[-1+4#1^2-16#1^4+32#1^6&,2,0],Root[-1-12#1^2-32#1^4+32#1^6&,1,0],Root[-1+12#1^2-32#1^4+32#1^6&,2,0]},
{Root[-1+4#1^2-16#1^4+32#1^6&,2,0],Root[-1-12#1^2-32#1^4+32#1^6&,2,0],Root[-1+12#1^2-32#1^4+32#1^6&,1,0]},
{Root[-1+4#1^2-16#1^4+32#1^6&,2,0],Root[-1+12#1^2-32#1^4+32#1^6&,1,0],Root[-1-12#1^2-32#1^4+32#1^6&,1,0]},
{Root[-1+4#1^2-16#1^4+32#1^6&,2,0],Root[-1+12#1^2-32#1^4+32#1^6&,2,0],Root[-1-12#1^2-32#1^4+32#1^6&,2,0]}}

PolyehdronName[SnubCube,"dextro"]:="snub cube (dextro)"
PolyhedronDual[SnubCube,"dextro"]:=Sequence[PentagonalIcositetrahedron,"laevo"]
PolyhedronFaces[SnubCube,"dextro"]:=
{{3,2,19},{3,17,1},{3,19,9},{3,9,17},{1,17,11},{1,20,4},{1,11,20},{2,4,18},
{2,18,12},{2,12,19},{4,20,10},{4,10,18},{17,13,11},{19,15,9},{18,14,12},
{20,16,10},{9,15,21},{11,13,23},{12,14,24},{10,16,22},{13,21,7},{13,7,23},
{15,24,5},{15,5,21},{16,23,6},{16,6,22},{14,22,8},{14,8,24},{21,5,7},{23,
7,6},{24,8,5},{22,6,8},{2,3,1,4},{21,13,17,9},{15,19,12,24},{14,18,10,22},
{23,16,20,11},{8,6,7,5}}
PolyhedronVertices[SnubCube,"dextro"]:=
{{Root[-1-12#1^2-32#1^4+32#1^6&,1,0],Root[-1+12#1^2-32#1^4+32#1^6&,1,0],Root[-1+4#1^2-16#1^4+32#1^6&,2,0]},
{Root[-1-12#1^2-32#1^4+32#1^6&,1,0],Root[-1+12#1^2-32#1^4+32#1^6&,2,0],Root[-1+4#1^2-16#1^4+32#1^6&,1,0]},
{Root[-1-12#1^2-32#1^4+32#1^6&,1,0],Root[-1+4#1^2-16#1^4+32#1^6&,1,0],Root[-1+12#1^2-32#1^4+32#1^6&,1,0]},
{Root[-1-12#1^2-32#1^4+32#1^6&,1,0],Root[-1+4#1^2-16#1^4+32#1^6&,2,0],Root[-1+12#1^2-32#1^4+32#1^6&,2,0]},
{Root[-1-12#1^2-32#1^4+32#1^6&,2,0],Root[-1+12#1^2-32#1^4+32#1^6&,1,0],Root[-1+4#1^2-16#1^4+32#1^6&,1,0]},
{Root[-1-12#1^2-32#1^4+32#1^6&,2,0],Root[-1+12#1^2-32#1^4+32#1^6&,2,0],Root[-1+4#1^2-16#1^4+32#1^6&,2,0]},
{Root[-1-12#1^2-32#1^4+32#1^6&,2,0],Root[-1+4#1^2-16#1^4+32#1^6&,1,0],Root[-1+12#1^2-32#1^4+32#1^6&,2,0]},
{Root[-1-12#1^2-32#1^4+32#1^6&,2,0],Root[-1+4#1^2-16#1^4+32#1^6&,2,0],Root[-1+12#1^2-32#1^4+32#1^6&,1,0]},
{Root[-1+12#1^2-32#1^4+32#1^6&,1,0],Root[-1-12#1^2-32#1^4+32#1^6&,1,0],Root[-1+4#1^2-16#1^4+32#1^6&,1,0]},
{Root[-1+12#1^2-32#1^4+32#1^6&,1,0],Root[-1-12#1^2-32#1^4+32#1^6&,2,0],Root[-1+4#1^2-16#1^4+32#1^6&,2,0]},
{Root[-1+12#1^2-32#1^4+32#1^6&,1,0],Root[-1+4#1^2-16#1^4+32#1^6&,1,0],Root[-1-12#1^2-32#1^4+32#1^6&,2,0]},
{Root[-1+12#1^2-32#1^4+32#1^6&,1,0],Root[-1+4#1^2-16#1^4+32#1^6&,2,0],Root[-1-12#1^2-32#1^4+32#1^6&,1,0]},
{Root[-1+12#1^2-32#1^4+32#1^6&,2,0],Root[-1-12#1^2-32#1^4+32#1^6&,1,0],Root[-1+4#1^2-16#1^4+32#1^6&,2,0]},
{Root[-1+12#1^2-32#1^4+32#1^6&,2,0],Root[-1-12#1^2-32#1^4+32#1^6&,2,0],Root[-1+4#1^2-16#1^4+32#1^6&,1,0]},
{Root[-1+12#1^2-32#1^4+32#1^6&,2,0],Root[-1+4#1^2-16#1^4+32#1^6&,1,0],Root[-1-12#1^2-32#1^4+32#1^6&,1,0]},
{Root[-1+12#1^2-32#1^4+32#1^6&,2,0],Root[-1+4#1^2-16#1^4+32#1^6&,2,0],Root[-1-12#1^2-32#1^4+32#1^6&,2,0]},
{Root[-1+4#1^2-16#1^4+32#1^6&,1,0],Root[-1-12#1^2-32#1^4+32#1^6&,1,0],Root[-1+12#1^2-32#1^4+32#1^6&,2,0]},
{Root[-1+4#1^2-16#1^4+32#1^6&,1,0],Root[-1-12#1^2-32#1^4+32#1^6&,2,0],Root[-1+12#1^2-32#1^4+32#1^6&,1,0]},
{Root[-1+4#1^2-16#1^4+32#1^6&,1,0],Root[-1+12#1^2-32#1^4+32#1^6&,1,0],Root[-1-12#1^2-32#1^4+32#1^6&,1,0]},
{Root[-1+4#1^2-16#1^4+32#1^6&,1,0],Root[-1+12#1^2-32#1^4+32#1^6&,2,0],Root[-1-12#1^2-32#1^4+32#1^6&,2,0]},
{Root[-1+4#1^2-16#1^4+32#1^6&,2,0],Root[-1-12#1^2-32#1^4+32#1^6&,1,0],Root[-1+12#1^2-32#1^4+32#1^6&,1,0]},
{Root[-1+4#1^2-16#1^4+32#1^6&,2,0],Root[-1-12#1^2-32#1^4+32#1^6&,2,0],Root[-1+12#1^2-32#1^4+32#1^6&,2,0]},
{Root[-1+4#1^2-16#1^4+32#1^6&,2,0],Root[-1+12#1^2-32#1^4+32#1^6&,1,0],Root[-1-12#1^2-32#1^4+32#1^6&,2,0]},
{Root[-1+4#1^2-16#1^4+32#1^6&,2,0],Root[-1+12#1^2-32#1^4+32#1^6&,2,0],Root[-1-12#1^2-32#1^4+32#1^6&,1,0]}}

(* Snub Dodecahedron *)

PolyhedronName[SnubDodecahedron]:=PolyhedronName[SnubDodecahedron,"laevo"]
PolyhedronDual[SnubDodecahedron]:=PolyhedronDual[SnubDodecahedron,"laevo"]
PolyhedronFaces[SnubDodecahedron]:=PolyhedronFaces[SnubDodecahedron,"laevo"]
PolyhedronVertices[SnubDodecahedron]:=PolyhedronVertices[SnubDodecahedron,"laevo"]

NetFaces[SnubDodecahedron]:={{27,18,17,25,29},{11,10,17},{17,10,16},{17,16,25},{10,9,16},{10,6,9},{6,3,5},{6,5,9},{9,5,8},{36,35,46},{46,35,45},{46,45,58},{35,34,45},{35,25,34},{25,16,24},{25,24,34},{34,24,33},{16,12,13,19,24},{48,37,33,44,50},{24,23,33},{33,23,32},{33,32,44},{23,22,32},{23,15,22},{15,7,14},{15,14,22},{22,14,21},{57,56,68},{68,56,67},{68,67,80},{56,55,67},{56,44,55},{44,32,43},{44,43,55},{55,43,54},{32,26,28,38,43},{70,59,54,66,72},{43,42,54},{54,42,53},{54,53,66},{42,41,53},{42,31,41},{31,20,30},{31,30,41},{41,30,40},{79,78,89},{89,78,88},{89,88,99},{78,77,88},{78,66,77},{66,53,65},{66,65,77},{77,65,76},{53,47,49,60,65},{91,81,76,87,93},{65,64,76},{76,64,75},{76,75,87},{64,63,75},{64,52,63},{52,39,51},{52,51,63},{63,51,62},{98,97,105},{105,97,104},{105,104,112},{97,96,104},{97,87,96},{87,75,86},{87,86,96},{96,86,95},{75,69,71,82,86},{106,100,95,103,107},{86,85,95},{95,85,94},{95,94,103},{85,84,94},{85,74,84},{74,61,73},{74,73,84},{84,73,83},{111,110,114},{114,110,113},{114,113,116},{110,109,113},{110,103,109},{103,94,102},{103,102,109},{109,102,108},{94,90,92,101,102},{3,1,2,4,5},{117,115,114,116,118}}
NetVertices[SnubDodecahedron]:={{(-50Sqrt[3]-6Sqrt[15]-3Sqrt[2(5-Sqrt[5])]-3Sqrt[10(5-Sqrt[5])])/48,(-26+2Sqrt[5]-Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])])/16},{(-88Sqrt[3]-9Sqrt[2(5-Sqrt[5])]-3Sqrt[10(5-Sqrt[5])]-3Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/96,(-56-3Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])]-Sqrt[6(5+Sqrt[5])]-Sqrt[30(5+Sqrt[5])])/32},{-7/(2Sqrt[3]),-3/2},{(-76Sqrt[3]+12Sqrt[15]+3Sqrt[2(5-Sqrt[5])]-3Sqrt[10(5-Sqrt[5])]-3Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/96,(-60-4Sqrt[5]+Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])]-Sqrt[6(5+Sqrt[5])]-Sqrt[30(5+Sqrt[5])])/32},{-2/Sqrt[3],-2},{-2/Sqrt[3],-1},{-1/(2Sqrt[3]),-7/2},{-1/(2Sqrt[3]),-5/2},{-1/(2Sqrt[3]),-3/2},{-1/(2Sqrt[3]),-1/2},{-1/(2Sqrt[3]),1/2},{(22Sqrt[3]-6Sqrt[15]-3Sqrt[2(5-Sqrt[5])]-3Sqrt[10(5-Sqrt[5])])/48,(-18+2Sqrt[5]-Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])])/16},{(56Sqrt[3]-9Sqrt[2(5-Sqrt[5])]-3Sqrt[10(5-Sqrt[5])]-3Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/96,(-40-3Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])]-Sqrt[6(5+Sqrt[5])]-Sqrt[30(5+Sqrt[5])])/32},{1/Sqrt[3],-4},{1/Sqrt[3],-3},{1/Sqrt[3],-1},{1/Sqrt[3],0},{(11Sqrt[3]-3Sqrt[15]+3Sqrt[2(5+Sqrt[5])])/24,(-1+Sqrt[5]+Sqrt[6(5+Sqrt[5])])/8},{(68Sqrt[3]+12Sqrt[15]+3Sqrt[2(5-Sqrt[5])]-3Sqrt[10(5-Sqrt[5])]-3Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/96,(-44-4Sqrt[5]+Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])]-Sqrt[6(5+Sqrt[5])]-Sqrt[30(5+Sqrt[5])])/32},{5/(2Sqrt[3]),-11/2},{5/(2Sqrt[3]),-9/2},{5/(2Sqrt[3]),-7/2},{5/(2Sqrt[3]),-5/2},{5/(2Sqrt[3]),-3/2},{5/(2Sqrt[3]),-1/2},{(70Sqrt[3]-6Sqrt[15]-3Sqrt[2(5-Sqrt[5])]-3Sqrt[10(5-Sqrt[5])])/48,(-50+2Sqrt[5]-Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])])/16},{(28Sqrt[3]+3Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/48,(-4+Sqrt[6(5+Sqrt[5])]+Sqrt[30(5+Sqrt[5])])/16},{(152Sqrt[3]-9Sqrt[2(5-Sqrt[5])]-3Sqrt[10(5-Sqrt[5])]-3Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/96,(-104-3Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])]-Sqrt[6(5+Sqrt[5])]-Sqrt[30(5+Sqrt[5])])/32},{(17Sqrt[3]+3Sqrt[15]+3Sqrt[2(5+Sqrt[5])])/24,(-3-Sqrt[5]+Sqrt[6(5+Sqrt[5])])/8},{4/Sqrt[3],-6},{4/Sqrt[3],-5},{4/Sqrt[3],-3},{4/Sqrt[3],-2},{4/Sqrt[3],-1},{4/Sqrt[3],0},{4/Sqrt[3],1},{(35Sqrt[3]-3Sqrt[15]+3Sqrt[2(5+Sqrt[5])])/24,(-17+Sqrt[5]+Sqrt[6(5+Sqrt[5])])/8},{(164Sqrt[3]+12Sqrt[15]+3Sqrt[2(5-Sqrt[5])]-3Sqrt[10(5-Sqrt[5])]-3Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/96,(-108-4Sqrt[5]+Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])]-Sqrt[6(5+Sqrt[5])]-Sqrt[30(5+Sqrt[5])])/32},{11/(2Sqrt[3]),-15/2},{11/(2Sqrt[3]),-13/2},{11/(2Sqrt[3]),-11/2},{11/(2Sqrt[3]),-9/2},{11/(2Sqrt[3]),-7/2},{11/(2Sqrt[3]),-5/2},{11/(2Sqrt[3]),-1/2},{11/(2Sqrt[3]),1/2},{(118Sqrt[3]-6Sqrt[15]-3Sqrt[2(5-Sqrt[5])]-3Sqrt[10(5-Sqrt[5])])/48,(-82+2Sqrt[5]-Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])])/16},{(76Sqrt[3]+3Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/48,(-36+Sqrt[6(5+Sqrt[5])]+Sqrt[30(5+Sqrt[5])])/16},{(248Sqrt[3]-9Sqrt[2(5-Sqrt[5])]-3Sqrt[10(5-Sqrt[5])]-3Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/96,(-168-3Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])]-Sqrt[6(5+Sqrt[5])]-Sqrt[30(5+Sqrt[5])])/32},{(41Sqrt[3]+3Sqrt[15]+3Sqrt[2(5+Sqrt[5])])/24,(-19-Sqrt[5]+Sqrt[6(5+Sqrt[5])])/8},{7/Sqrt[3],-8},{7/Sqrt[3],-7},{7/Sqrt[3],-5},{7/Sqrt[3],-4},{7/Sqrt[3],-3},{7/Sqrt[3],-2},{7/Sqrt[3],-1},{7/Sqrt[3],0},{(59Sqrt[3]-3Sqrt[15]+3Sqrt[2(5+Sqrt[5])])/24,(-33+Sqrt[5]+Sqrt[6(5+Sqrt[5])])/8},{(260Sqrt[3]+12Sqrt[15]+3Sqrt[2(5-Sqrt[5])]-3Sqrt[10(5-Sqrt[5])]-3Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/96,(-172-4Sqrt[5]+Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])]-Sqrt[6(5+Sqrt[5])]-Sqrt[30(5+Sqrt[5])])/32},{17/(2Sqrt[3]),-19/2},{17/(2Sqrt[3]),-17/2},{17/(2Sqrt[3]),-15/2},{17/(2Sqrt[3]),-13/2},{17/(2Sqrt[3]),-11/2},{17/(2Sqrt[3]),-9/2},{17/(2Sqrt[3]),-5/2},{17/(2Sqrt[3]),-3/2},{(166Sqrt[3]-6Sqrt[15]-3Sqrt[2(5-Sqrt[5])]-3Sqrt[10(5-Sqrt[5])])/48,(-114+2Sqrt[5]-Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])])/16},{(124Sqrt[3]+3Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/48,(-68+Sqrt[6(5+Sqrt[5])]+Sqrt[30(5+Sqrt[5])])/16},{(344Sqrt[3]-9Sqrt[2(5-Sqrt[5])]-3Sqrt[10(5-Sqrt[5])]-3Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/96,(-232-3Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])]-Sqrt[6(5+Sqrt[5])]-Sqrt[30(5+Sqrt[5])])/32},{(65Sqrt[3]+3Sqrt[15]+3Sqrt[2(5+Sqrt[5])])/24,(-35-Sqrt[5]+Sqrt[6(5+Sqrt[5])])/8},{10/Sqrt[3],-10},{10/Sqrt[3],-9},{10/Sqrt[3],-7},{10/Sqrt[3],-6},{10/Sqrt[3],-5},{10/Sqrt[3],-4},{10/Sqrt[3],-3},{10/Sqrt[3],-2},{(83Sqrt[3]-3Sqrt[15]+3Sqrt[2(5+Sqrt[5])])/24,(-49+Sqrt[5]+Sqrt[6(5+Sqrt[5])])/8},{(356Sqrt[3]+12Sqrt[15]+3Sqrt[2(5-Sqrt[5])]-3Sqrt[10(5-Sqrt[5])]-3Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/96,(-236-4Sqrt[5]+Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])]-Sqrt[6(5+Sqrt[5])]-Sqrt[30(5+Sqrt[5])])/32},{23/(2Sqrt[3]),-21/2},{23/(2Sqrt[3]),-19/2},{23/(2Sqrt[3]),-17/2},{23/(2Sqrt[3]),-15/2},{23/(2Sqrt[3]),-13/2},{23/(2Sqrt[3]),-9/2},{23/(2Sqrt[3]),-7/2},{(214Sqrt[3]-6Sqrt[15]-3Sqrt[2(5-Sqrt[5])]-3Sqrt[10(5-Sqrt[5])])/48,(-146+2Sqrt[5]-Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])])/16},{(172Sqrt[3]+3Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/48,(-100+Sqrt[6(5+Sqrt[5])]+Sqrt[30(5+Sqrt[5])])/16},{(440Sqrt[3]-9Sqrt[2(5-Sqrt[5])]-3Sqrt[10(5-Sqrt[5])]-3Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/96,(-296-3Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])]-Sqrt[6(5+Sqrt[5])]-Sqrt[30(5+Sqrt[5])])/32},{(89Sqrt[3]+3Sqrt[15]+3Sqrt[2(5+Sqrt[5])])/24,(-51-Sqrt[5]+Sqrt[6(5+Sqrt[5])])/8},{13/Sqrt[3],-9},{13/Sqrt[3],-8},{13/Sqrt[3],-7},{13/Sqrt[3],-6},{13/Sqrt[3],-5},{13/Sqrt[3],-4},{(107Sqrt[3]-3Sqrt[15]+3Sqrt[2(5+Sqrt[5])])/24,(-65+Sqrt[5]+Sqrt[6(5+Sqrt[5])])/8},{(452Sqrt[3]+12Sqrt[15]+3Sqrt[2(5-Sqrt[5])]-3Sqrt[10(5-Sqrt[5])]-3Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/96,(-300-4Sqrt[5]+Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])]-Sqrt[6(5+Sqrt[5])]-Sqrt[30(5+Sqrt[5])])/32},{29/(2Sqrt[3]),-19/2},{29/(2Sqrt[3]),-17/2},{29/(2Sqrt[3]),-13/2},{29/(2Sqrt[3]),-11/2},{(220Sqrt[3]+3Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/48,(-132+Sqrt[6(5+Sqrt[5])]+Sqrt[30(5+Sqrt[5])])/16},{(113Sqrt[3]+3Sqrt[15]+3Sqrt[2(5+Sqrt[5])])/24,(-67-Sqrt[5]+Sqrt[6(5+Sqrt[5])])/8},{16/Sqrt[3],-10},{16/Sqrt[3],-9},{16/Sqrt[3],-8},{16/Sqrt[3],-7},{16/Sqrt[3],-6},{35/(2Sqrt[3]),-17/2},{35/(2Sqrt[3]),-15/2},{(143Sqrt[3]-3Sqrt[15]+3Sqrt[2(5+Sqrt[5])])/24,(-61+Sqrt[5]+Sqrt[6(5+Sqrt[5])])/8},{19/Sqrt[3],-8},{(292Sqrt[3]+3Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/48,(-124+Sqrt[6(5+Sqrt[5])]+Sqrt[30(5+Sqrt[5])])/16},{(149Sqrt[3]+3Sqrt[15]+3Sqrt[2(5+Sqrt[5])])/24,(-63-Sqrt[5]+Sqrt[6(5+Sqrt[5])])/8}}

PolyhedronName[SnubDodecahedron,"laevo"]:="snub dodecahedron (laevo)"
PolyhedronDual[SnubDodecahedron,"laevo"]:=Sequence[PentagonalHexecontahedron,"dextro"]
PolyhedronFaces[SnubDodecahedron,"laevo"]:=
{{5,1,9},{5,9,3},{5,29,15},{5,3,29},{1,27,19},{1,19,25},{1,25,9},{15,29,7},
{15,45,39},{15,7,45},{27,39,41},{27,41,37},{27,37,19},{9,25,17},{39,45,21},
{39,21,41},{29,3,13},{3,43,13},{19,37,35},{25,31,17},{45,7,11},{7,49,11},
{41,21,23},{37,33,35},{17,31,59},{17,59,55},{13,43,53},{13,53,51},{21,47,23},
{43,55,48},{43,48,53},{35,33,58},{35,58,57},{31,57,34},{31,34,59},{11,49,52},
{11,52,54},{55,59,24},{55,24,48},{49,51,50},{49,50,52},{23,47,56},{23,56,60},
{51,53,12},{51,12,50},{33,60,32},{33,32,58},{57,58,36},{57,36,34},{47,54,44},
{47,44,56},{48,24,22},{54,52,14},{54,14,44},{60,56,18},{60,18,32},{34,36,38},
{24,42,22},{50,12,8},{12,46,8},{32,18,26},{36,20,38},{44,14,4},{22,42,40},
{22,40,46},{14,30,4},{18,10,26},{38,20,28},{38,28,42},{42,28,40},{8,46,16},
{8,16,30},{46,40,16},{26,10,2},{26,2,20},{20,2,28},{4,30,6},{4,6,10},
{30,16,6},{10,6,2},{39,27,1,5,15},{3,9,17,55,43},{51,49,7,29,13},
{57,31,25,19,35},{47,21,45,11,54},{33,37,41,23,60},{42,24,59,34,38},
{46,12,53,48,22},{36,58,32,26,20},{14,52,50,8,30},{44,4,10,18,56},
{16,40,28,2,6}}
PolyhedronVertices[SnubDodecahedron,"laevo"]:=
{{Root[3721-65628#1^2+31104#1^4+1050624#1^6+2426112#1^8-13188096#1^10+2985984#1^12&,1,0],
Root[1-68#1^2+1216#1^4-5888#1^6+9472#1^8-5120#1^10+4096#1^12&,1,0],
Root[1-10584#1^2+441072#1^4-3430080#1^6+7962624#1^8-3234816#1^10+2985984#1^12&,6,0]},
{Root[3721-65628#1^2+31104#1^4+1050624#1^6+2426112#1^8-13188096#1^10+2985984#1^12&,8,0],
Root[1-68#1^2+1216#1^4-5888#1^6+9472#1^8-5120#1^10+4096#1^12&,1,0],
Root[1-10584#1^2+441072#1^4-3430080#1^6+7962624#1^8-3234816#1^10+2985984#1^12&,3,0]},
{Root[5041-289884#1^2+2745792#1^4-10091520#1^6+17190144#1^8-13188096#1^10+2985984#1^12&,1,0],
Root[1-68#1^2+1216#1^4-5888#1^6+9472#1^8-5120#1^10+4096#1^12&,8,0],
Root[361-292488#1^2+2858544#1^4-6492096#1^6+1078272#1^8-3234816#1^10+2985984#1^12&,8,0]},
{Root[5041-289884#1^2+2745792#1^4-10091520#1^6+17190144#1^8-13188096#1^10+2985984#1^12&,8,0],
Root[1-68#1^2+1216#1^4-5888#1^6+9472#1^8-5120#1^10+4096#1^12&,8,0],
Root[361-292488#1^2+2858544#1^4-6492096#1^6+1078272#1^8-3234816#1^10+2985984#1^12&,1,0]},
{Root[1-1272#1^2+158544#1^4+763776#1^6-787968#1^8-12939264#1^10+2985984#1^12&,1,0],
Root[1-56#1^2+912#1^4-6016#1^6+15872#1^8-12288#1^10+4096#1^12&,6,0],
Root[961-95784#1^2+1076976#1^4-3568320#1^6+1824768#1^8+1741824#1^10+2985984#1^12&,6,0]},
{Root[1-1272#1^2+158544#1^4+763776#1^6-787968#1^8-12939264#1^10+2985984#1^12&,8,0],
Root[1-56#1^2+912#1^4-6016#1^6+15872#1^8-12288#1^10+4096#1^12&,6,0],
Root[961-95784#1^2+1076976#1^4-3568320#1^6+1824768#1^8+1741824#1^10+2985984#1^12&,3,0]},
{Root[17161-386076#1^2+3170448#1^4-11460096#1^6+17750016#1^8-12192768#1^10+2985984#1^12&,1,0],
Root[1-100#1^2+1872#1^4-9472#1^6+9216#1^8-13312#1^10+4096#1^12&,8,0],
Root[961-95784#1^2+1076976#1^4-3568320#1^6+1824768#1^8+1741824#1^10+2985984#1^12&,3,0]},
{Root[17161-386076#1^2+3170448#1^4-11460096#1^6+17750016#1^8-12192768#1^10+2985984#1^12&,8,0],
Root[1-100#1^2+1872#1^4-9472#1^6+9216#1^8-13312#1^10+4096#1^12&,8,0],
Root[961-95784#1^2+1076976#1^4-3568320#1^6+1824768#1^8+1741824#1^10+2985984#1^12&,6,0]},
{Root[3481-83304#1^2+701136#1^4-2782080#1^6+7423488#1^8-11943936#1^10+2985984#1^12&,1,0],
Root[1-56#1^2+912#1^4-6016#1^6+15872#1^8-12288#1^10+4096#1^12&,3,0],
Root[22801-432792#1^2+2409264#1^4-3340224#1^6-3898368#1^8+746496#1^10+2985984#1^12&,8,0]},
{Root[3481-83304#1^2+701136#1^4-2782080#1^6+7423488#1^8-11943936#1^10+2985984#1^12&,8,0],
Root[1-56#1^2+912#1^4-6016#1^6+15872#1^8-12288#1^10+4096#1^12&,3,0],
Root[22801-432792#1^2+2409264#1^4-3340224#1^6-3898368#1^8+746496#1^10+2985984#1^12&,1,0]},
{Root[6241-211068#1^2+2283984#1^4-9542016#1^6+16941312#1^8-11943936#1^10+2985984#1^12&,1,0],
Root[1-36#1^2+272#1^4+384#1^6-768#1^8-12288#1^10+4096#1^12&,8,0],
Root[22801-432792#1^2+2409264#1^4-3340224#1^6-3898368#1^8+746496#1^10+2985984#1^12&,1,0]},
{Root[6241-211068#1^2+2283984#1^4-9542016#1^6+16941312#1^8-11943936#1^10+2985984#1^12&,8,0],
Root[1-36#1^2+272#1^4+384#1^6-768#1^8-12288#1^10+4096#1^12&,8,0],
Root[22801-432792#1^2+2409264#1^4-3340224#1^6-3898368#1^8+746496#1^10+2985984#1^12&,8,0]},
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PolyhedronName[SnubDodecahedron,"dextro"]:="snub dodecahedron (dextro)"
PolyhedronDual[SnubDodecahedron,"dextro"]:=Sequence[PentagonalHexecontahedron,"laevo"]
PolyhedronFaces[SnubDodecahedron,"dextro"]:=
{{5,15,29},{5,9,1},{5,29,3},{5,3,9},{1,9,25},{1,19,27},{1,25,19},{15,39,45},
{15,45,7},{15,7,29},{27,19,37},{27,41,39},{27,37,41},{9,17,25},{39,41,21},
{39,21,45},{29,13,3},{3,13,43},{19,35,37},{25,17,31},{45,11,7},{7,11,49},
{41,23,21},{37,35,33},{17,55,59},{17,59,31},{13,51,53},{13,53,43},{21,23,47},
{43,53,48},{43,48,55},{35,57,58},{35,58,33},{31,59,34},{31,34,57},{11,54,52},
{11,52,49},{55,48,24},{55,24,59},{49,52,50},{49,50,51},{23,60,56},{23,56,47},
{51,50,12},{51,12,53},{33,58,32},{33,32,60},{57,34,36},{57,36,58},{47,56,44},
{47,44,54},{48,22,24},{54,44,14},{54,14,52},{60,32,18},{60,18,56},{34,38,36},
{24,22,42},{50,8,12},{12,8,46},{32,26,18},{36,38,20},{44,4,14},{22,46,40},
{22,40,42},{14,4,30},{18,26,10},{38,42,28},{38,28,20},{42,40,28},{8,30,16},
{8,16,46},{46,16,40},{26,20,2},{26,2,10},{20,28,2},{4,10,6},{4,6,30},{30,6,16},
{10,2,6},{39,15,5,1,27},{9,3,43,55,17},{51,13,29,7,49},{57,35,19,25,31},
{11,45,21,47,54},{23,41,37,33,60},{42,38,34,59,24},{46,22,48,53,12},
{26,32,58,36,20},{8,50,52,14,30},{18,10,4,44,56},{28,40,16,6,2}}
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Root[961-90696#1^2+1566576#1^4-7750080#1^6+15925248#1^8-14183424#1^10+2985984#1^12&,8,0]},
{Root[11881-341376#1^2+3016368#1^4-8434368#1^6+8957952#1^8-5225472#1^10+2985984#1^12&,8,0],
Root[1-128#1^2+1392#1^4-4672#1^6+4096#1^8-1024#1^10+4096#1^12&,1,0],
Root[961-90696#1^2+1566576#1^4-7750080#1^6+15925248#1^8-14183424#1^10+2985984#1^12&,1,0]},
{Root[361-29856#1^2+588816#1^4-1926720#1^6-1907712#1^8-4230144#1^10+2985984#1^12&,1,0],
Root[1-416#1^2+6672#1^4-21696#1^6+28672#1^8-17408#1^10+4096#1^12&,1,0],
Root[361-292488#1^2+2858544#1^4-6492096#1^6+1078272#1^8-3234816#1^10+2985984#1^12&,1,0]},
{Root[361-29856#1^2+588816#1^4-1926720#1^6-1907712#1^8-4230144#1^10+2985984#1^12&,8,0],
Root[1-416#1^2+6672#1^4-21696#1^6+28672#1^8-17408#1^10+4096#1^12&,1,0],
Root[361-292488#1^2+2858544#1^4-6492096#1^6+1078272#1^8-3234816#1^10+2985984#1^12&,8,0]},
{Root[1-504#1^2+65664#1^4-2647296#1^6+8335872#1^8-3732480#1^10+2985984#1^12&,3,0],
Root[1-328#1^2+2944#1^4-8960#1^6+9728#1^8-3072#1^10+4096#1^12&,2,0],
Root[1-360#1^2+12528#1^4-478656#1^6+5723136#1^8-14183424#1^10+2985984#1^12&,1,0]},
{Root[1-504#1^2+65664#1^4-2647296#1^6+8335872#1^8-3732480#1^10+2985984#1^12&,6,0],
Root[1-328#1^2+2944#1^4-8960#1^6+9728#1^8-3072#1^10+4096#1^12&,2,0],
Root[1-360#1^2+12528#1^4-478656#1^6+5723136#1^8-14183424#1^10+2985984#1^12&,8,0]},
{Root[1-18792#1^2+524016#1^4-3443904#1^6+6718464#1^8-1244160#1^10+2985984#1^12&,2,0],
Root[1-40#1^2+624#1^4-4672#1^6+16384#1^8-21504#1^10+4096#1^12&,1,0],
Root[1-10584#1^2+441072#1^4-3430080#1^6+7962624#1^8-3234816#1^10+2985984#1^12&,3,0]},
{Root[1-18792#1^2+524016#1^4-3443904#1^6+6718464#1^8-1244160#1^10+2985984#1^12&,7,0],
Root[1-40#1^2+624#1^4-4672#1^6+16384#1^8-21504#1^10+4096#1^12&,1,0],
Root[1-10584#1^2+441072#1^4-3430080#1^6+7962624#1^8-3234816#1^10+2985984#1^12&,6,0]},
{Root[1-7104#1^2+351216#1^4-3300480#1^6+2757888#1^8+4230144#1^10+2985984#1^12&,2,0],
Root[1-48#1^2+752#1^4-4736#1^6+14592#1^8-19456#1^10+4096#1^12&,1,0],
Root[1-176136#1^2+2371248#1^4-9332928#1^6+14764032#1^8-10202112#1^10+2985984#1^12&,8,0]},
{Root[1-7104#1^2+351216#1^4-3300480#1^6+2757888#1^8+4230144#1^10+2985984#1^12&,7,0],
Root[1-48#1^2+752#1^4-4736#1^6+14592#1^8-19456#1^10+4096#1^12&,1,0],
Root[1-176136#1^2+2371248#1^4-9332928#1^6+14764032#1^8-10202112#1^10+2985984#1^12&,1,0]},
{Root[361-23352#1^2+432864#1^4-2666304#1^6+2115072#1^8+4478976#1^10+2985984#1^12&,4,0],
Root[1-488#1^2+5728#1^4-20160#1^6+29184#1^8-18432#1^10+4096#1^12&,1,0],
Root[1681-113304#1^2+1571184#1^4-7390656#1^6+11446272#1^8-11197440#1^10+2985984#1^12&,8,0]},
{Root[361-23352#1^2+432864#1^4-2666304#1^6+2115072#1^8+4478976#1^10+2985984#1^12&,5,0],
Root[1-488#1^2+5728#1^4-20160#1^6+29184#1^8-18432#1^10+4096#1^12&,1,0],
Root[1681-113304#1^2+1571184#1^4-7390656#1^6+11446272#1^8-11197440#1^10+2985984#1^12&,1,0]},
{Root[5041-143496#1^2+1133568#1^4-2958336#1^6+124416#1^8+5225472#1^10+2985984#1^12&,1,0],
Root[1-56#1^2+1024#1^4-7168#1^6+18944#1^8-15360#1^10+4096#1^12&,5,0],
Root[1-360#1^2+12528#1^4-478656#1^6+5723136#1^8-14183424#1^10+2985984#1^12&,8,0]},
{Root[5041-143496#1^2+1133568#1^4-2958336#1^6+124416#1^8+5225472#1^10+2985984#1^12&,8,0],
Root[1-56#1^2+1024#1^4-7168#1^6+18944#1^8-15360#1^10+4096#1^12&,5,0],
Root[1-360#1^2+12528#1^4-478656#1^6+5723136#1^8-14183424#1^10+2985984#1^12&,1,0]},
{Root[961-36984#1^2+466992#1^4-2185920#1^6+1741824#1^8+6718464#1^10+2985984#1^12&,4,0],
Root[1-40#1^2+624#1^4-4672#1^6+16384#1^8-21504#1^10+4096#1^12&,8,0],
Root[841-44232#1^2+705456#1^4-4688064#1^6+12939264#1^8-11197440#1^10+2985984#1^12&,8,0]},
{Root[961-36984#1^2+466992#1^4-2185920#1^6+1741824#1^8+6718464#1^10+2985984#1^12&,5,0],
Root[1-40#1^2+624#1^4-4672#1^6+16384#1^8-21504#1^10+4096#1^12&,8,0],
Root[841-44232#1^2+705456#1^4-4688064#1^6+12939264#1^8-11197440#1^10+2985984#1^12&,1,0]},
{Root[1-528#1^2+69264#1^4-1498176#1^6+2737152#1^8+6718464#1^10+2985984#1^12&,2,0],
Root[1-416#1^2+6672#1^4-21696#1^6+28672#1^8-17408#1^10+4096#1^12&,8,0],
Root[961-90696#1^2+1566576#1^4-7750080#1^6+15925248#1^8-14183424#1^10+2985984#1^12&,8,0]},
{Root[1-528#1^2+69264#1^4-1498176#1^6+2737152#1^8+6718464#1^10+2985984#1^12&,7,0],
Root[1-416#1^2+6672#1^4-21696#1^6+28672#1^8-17408#1^10+4096#1^12&,8,0],
Root[961-90696#1^2+1566576#1^4-7750080#1^6+15925248#1^8-14183424#1^10+2985984#1^12&,1,0]}}

(* Tetrakis Hexahedron *)

PolyhedronName[TetrakisHexahedron]:="tetrakis hexahedron"
NetFaces[TetrakisHexahedron]:={{11,15,16},{11,16,7},{11,7,6},{11,6,14},{13,18,17},{13,17,8},{13,8,7},{13,7,16},
{4,6,7},{4,7,2},{4,2,1},{4,1,5},{9,12,14},{9,14,6},{9,6,3},{9,3,10},{19,22,21},{19,21,16},{19,16,15},
{19,15,20},{23,26,24},{23,24,21},{23,21,22},{23,22,25}}
NetVertices[TetrakisHexahedron]:={{-1.81018796079,0.0420586415981},{-1.3766342059,0.943186347101},
{-1.09030548402,-1.26331159012},{-1.08966538187,0.250258577394},{-0.962812686481,-0.488935844712},
{-0.656751729135,-0.36218388462},{-0.432913652733,0.612442462014},{-0.429844986981,1.61243775365},
{-0.369782905099,-1.05511165433},{-0.242930209712,-1.79430607643},{0.,0.},
{0.0631307476338,-1.66755411634},{0.127635042462,1.11072467153},{0.286968824036,-0.692927769707},
{0.557480029443,-0.501713082122},{0.560548695194,0.49828220951200003},{0.564299286669,1.72049867706},
{0.782151570617,0.744516923968},{1.11802872464,-0.00343087261061},{1.55094237737,-0.615873334625},
{1.55469296884,0.606343132928},{1.77254525279,-0.369638620169},{2.209209497,0.24013538537},
{2.50042562384,0.931288890367},{2.76668952644,-0.261577696752},{2.92844072001,0.027517305589}}
PolyhedronDual[TetrakisHexahedron]:=TruncatedOctahedron
PolyhedronFaces[TetrakisHexahedron]:=
{{3,2,1},{1,2,4},{3,1,5},{6,2,3},{1,4,9},{10,4,2},{5,6,3},{1,7,5},{6,8,2},{1,
    9,7},{8,10,2},{4,10,9},{11,6,5},{5,7,13},{14,8,6},{7,9,13},{14,10,8},{9,
    10,12},{11,5,13},{14,6,11},{9,12,13},{14,12,10},{13,14,11},{12,14,13}}
PolyhedronVertices[TetrakisHexahedron]:=
{{-3/2,0,-3/(2Sqrt[2])},{-3/2,0,3/(2Sqrt[2])},
{-9/8,-9/8,0},{-9/8,9/8,0},{0,-3/2,-3/(2Sqrt[2])},
{0,-3/2,3/(2Sqrt[2])},{0,0,-9/(4Sqrt[2])},
{0,0,9/(4Sqrt[2])},{0,3/2,-3/(2Sqrt[2])},
{0,3/2,3/(2Sqrt[2])},{9/8,-9/8,0},{9/8,9/8,0},
{3/2,0,-3/(2Sqrt[2])},{3/2,0,3/(2Sqrt[2])}}

(* Triakis Icosahedron *)

PolyhedronName[TriakisIcosahedron]:="triakis icosahedron"
NetFaces[TriakisIcosahedron]:={{5,7,9},{5,9,1},{5,1,3},{17,19,21},{17,21,13},{17,13,15},{29,31,33},{29,33,25},
{29,25,27},{41,43,45},{41,45,37},{41,37,39},{53,55,57},{53,57,49},{53,49,51},{10,8,6},{10,6,14},{10,14,12},
{22,20,18},{22,18,26},{22,26,24},{34,32,30},{34,30,38},{34,38,36},{46,44,42},{46,42,50},{46,50,48},{58,56,54},
{58,54,62},{58,62,60},{4,2,1},{4,1,9},{4,9,6},{11,13,14},{11,14,6},{11,6,9},{16,14,13},{16,13,21},{16,21,18},
{23,25,26},{23,26,18},{23,18,21},{28,26,25},{28,25,33},{28,33,30},{35,37,38},{35,38,30},{35,30,33},{40,38,37},
{40,37,45},{40,45,42},{47,49,50},{47,50,42},{47,42,45},{52,50,49},{52,49,57},{52,57,54},{59,61,62},{59,62,54},
{59,54,57}}
NetVertices[TriakisIcosahedron]:={{-1.01320351487,-1.16977998261},{-0.528830295465,-0.294918516823},
{-0.526757023452,-2.04349035281},{-0.513552805645,-0.874896063864},{-0.512854927033,-1.46347820769},
{-0.499650709226,-0.294883918748},{-0.497577437213,-2.04345575473},{-0.0152774898199,0.579977547041},
{-0.0132042178064,-1.16859428894},{0.,0.},{0.000697878612331,-0.588582143827},
{0.0139020964188,0.580012145116},{0.0159753684323,-1.16855969087},{0.500348587839,-0.293698225079},
{0.502421859852,-2.04227006106},{0.515626077659,-0.873675772119},{0.516323956271,-1.46225791595},
{0.529528174077,-0.293663627004},{0.531601446091,-2.04223546299},{1.01390139348,0.581197838785},
{1.0159746655,-1.1673739972},{1.0291788833,0.00122029174424},{1.02987676192,-0.587361852082},
{1.04308097972,0.58123243686},{1.04515425174,-1.16733939912},{1.52952747114,-0.292477933334},
{1.53160074316,-2.04104976932},{1.54480496096,-0.872455480375},{1.54550283957,-1.4610376242},
{1.55870705738,-0.292443335259},{1.56078032939,-2.04101517124},{2.04308027679,0.582418130529},
{2.0451535488,-1.16615370545},{2.05835776661,0.00244058348848},{2.05905564522,-0.586141560338},
{2.07225986303,0.582452728604},{2.07433313504,-1.16611910738},{2.55870635445,-0.29125764159},
{2.56077962646,-2.03982947757},{2.57398384427,-0.871235188631},{2.57468172288,-1.45981733246},
{2.58788594068,-0.291223043515},{2.5899592127,-2.0397948795},{3.07225916009,0.583638422273},
{3.0743324321,-1.16493341371},{3.08753664991,0.00366087523273},{3.08823452852,-0.584921268594},
{3.10143874633,0.583673020348},{3.10351201834,-1.16489881563},{3.58788523775,-0.290037349846},
{3.58995850976,-2.03860918583},{3.60316272757,-0.870014896887},{3.60386060618,-1.45859704071},
{3.61706482399,-0.290002751771},{3.619138096,-2.03857458775},{4.10143804339,0.584858714018},
{4.10351131541,-1.16371312197},{4.11671553321,0.00488116697697},{4.11741341183,-0.58370097685},
{4.13061762963,0.584893312093},{4.13269090165,-1.16367852389},{4.61706412105,-0.288817058102}}
PolyhedronDual[TriakisIcosahedron]:=TruncatedDodecahedron
PolyhedronFaces[TriakisIcosahedron]:=
{{1,26,2},{5,31,6},{3,27,1},{7,32,5},{19,26,31},{32,27,20},{26,23,2},{3,24,
    27},{30,29,13},{29,26,21},{22,27,30},{16,26,29},{30,27,17},{31,29,6},{7,
    30,32},{31,23,19},{20,24,32},{18,24,23},{8,28,31},{9,32,28},{14,23,
    31},{32,24,15},{23,24,4},{31,28,14},{28,32,15},{21,25,29},{30,25,22},{28,
    23,14},{15,24,28},{18,23,28},{28,24,18},{29,25,13},{13,25,30},{2,23,1},{8,
    31,5},{1,24,3},{5,32,9},{19,23,26},{27,24,20},{11,26,1},{6,29,5},{1,27,
    12},{5,30,7},{1,25,11},{5,29,10},{12,25,1},{10,30,5},{1,23,4},{5,28,8},{4,
    24,1},{9,28,5},{10,29,30},{21,26,25},{22,25,27},{16,29,31},{32,30,17},{25,
    26,11},{27,25,12},{31,26,16},{17,27,32}}
PolyhedronVertices[TriakisIcosahedron]:=
{{0,0,Sqrt[25/4+(5Sqrt[5])/2]},{Sqrt[5(25+2Sqrt[5])]/22,(-5(3+2Sqrt[5]))/22,
Sqrt[1325+590Sqrt[5]]/22},{Sqrt[5(25+2Sqrt[5])]/22,(5(3+2Sqrt[5]))/22,Sqrt[1325+590Sqrt[5]]/22},
{Sqrt[(5(85+31Sqrt[5]))/2]/11,0,Sqrt[1325+590Sqrt[5]]/22},{0,0,-Sqrt[5(5+2Sqrt[5])]/2},
{-Sqrt[5(25+2Sqrt[5])]/22,(-5(3+2Sqrt[5]))/22,-Sqrt[1325+590Sqrt[5]]/22},
{-Sqrt[5(25+2Sqrt[5])]/22,(5(3+2Sqrt[5]))/22,-Sqrt[1325+590Sqrt[5]]/22},
{Sqrt[(5(205+89Sqrt[5]))/2]/22,(-5(7+Sqrt[5]))/44,-Sqrt[1325+590Sqrt[5]]/22},
{Sqrt[(5(205+89Sqrt[5]))/2]/22,(5(7+Sqrt[5]))/44,-Sqrt[1325+590Sqrt[5]]/22},
{Root[125-425#1^2+121#1^4&,1,0],0,-Sqrt[1325+590Sqrt[5]]/22},
{-Sqrt[(5(205+89Sqrt[5]))/2]/22,(-5(7+Sqrt[5]))/44,Sqrt[1325+590Sqrt[5]]/22},
{-Sqrt[(5(205+89Sqrt[5]))/2]/22,(5(7+Sqrt[5]))/44,Sqrt[1325+590Sqrt[5]]/22},
{Root[125-1025#1^2+121#1^4&,1,0],0,-Sqrt[5(25+2Sqrt[5])]/22},
{Sqrt[1325+590Sqrt[5]]/22,(-5(3+2Sqrt[5]))/22,-Sqrt[5(25+2Sqrt[5])]/22},
{Sqrt[1325+590Sqrt[5]]/22,(5(3+2Sqrt[5]))/22,-Sqrt[5(25+2Sqrt[5])]/22},
{-Sqrt[(5(85+31Sqrt[5]))/2]/22,(-5(13+5Sqrt[5]))/44,-Sqrt[5(25+2Sqrt[5])]/22},
{-Sqrt[(5(85+31Sqrt[5]))/2]/22,(5(13+5Sqrt[5]))/44,-Sqrt[5(25+2Sqrt[5])]/22},
{Sqrt[(5(205+89Sqrt[5]))/2]/11,0,Sqrt[5(25+2Sqrt[5])]/22},
{Sqrt[(5(85+31Sqrt[5]))/2]/22,(-5(13+5Sqrt[5]))/44,Sqrt[5(25+2Sqrt[5])]/22},
{Sqrt[(5(85+31Sqrt[5]))/2]/22,(5(13+5Sqrt[5]))/44,Sqrt[5(25+2Sqrt[5])]/22},
{-Sqrt[1325+590Sqrt[5]]/22,(-5(3+2Sqrt[5]))/22,Sqrt[5(25+2Sqrt[5])]/22},
{-Sqrt[1325+590Sqrt[5]]/22,(5(3+2Sqrt[5]))/22,Sqrt[5(25+2Sqrt[5])]/22},
{Sqrt[25/8+(11Sqrt[5])/8],(-5-Sqrt[5])/4,Sqrt[5+2Sqrt[5]]/2},
{Sqrt[25/8+(11Sqrt[5])/8],(5+Sqrt[5])/4,Sqrt[5+2Sqrt[5]]/2},
{-Sqrt[5+2Sqrt[5]],0,Sqrt[5+2Sqrt[5]]/2},{-Sqrt[(5+Sqrt[5])/2]/2,(-5-3Sqrt[5])/4,
Sqrt[5+2Sqrt[5]]/2},{-Sqrt[(5+Sqrt[5])/2]/2,(5+3Sqrt[5])/4,Sqrt[5+2Sqrt[5]]/2},
{Sqrt[5+2Sqrt[5]],0,-Sqrt[5+2Sqrt[5]]/2},{Root[5-100#1^2+16#1^4&,1,0],(-5-Sqrt[5])/4,
-Sqrt[5+2Sqrt[5]]/2},{Root[5-100#1^2+16#1^4&,1,0],(5+Sqrt[5])/4,-Sqrt[5+2Sqrt[5]]/2},
{Sqrt[5/8+Sqrt[5]/8],(-5-3Sqrt[5])/4,-Sqrt[5+2Sqrt[5]]/2},{Sqrt[5/8+Sqrt[5]/8],(5+3Sqrt[5])/4,
-Sqrt[5+2Sqrt[5]]/2}}

(* Triakis Tetrahedron *)

PolyhedronName[TriakisTetrahedron]:="triakis tetrahedron"
NetFaces[TriakisTetrahedron]:={{8,10,11},{8,11,4},{8,4,6},{12,11,10},{12,10,14},{12,14,13},{7,4,11},{7,11,9},{7,9,3},{2,6,4},{2,4,1},{2,1,5}}
NetVertices[TriakisTetrahedron]:={{-1.19569314749,-0.390563847883},{-0.609881600786,-0.260853278728},
{-0.588955493862,0.533606938224},{-0.50156635741,0.329288914351},{-0.262565641171,-0.750109473576},
{-0.108315243376,-0.590142193079},{-0.00314394716,0.66331750738},{0.,0.},
{0.105171296216,1.25345970046},{0.113904482771,-0.589088931151},{0.49842241025,0.334028593029},
{0.612326893021,-0.255060338122},{0.720642136397,0.335081854957},{1.11389325043,-0.584349252474}}
PolyhedronDual[TriakisTetrahedron]:=TruncatedTetrahedron
PolyhedronFaces[TriakisTetrahedron]:=
{{6,4,1},{1,5,7},{6,3,4},{7,5,3},{6,2,3},{7,3,2},{7,2,6},{7,6,8},{4,3,1},{3,5,
    1},{6,1,8},{1,7,8}}
PolyhedronVertices[TriakisTetrahedron]:=
{{0, 0, (-3Sqrt[3/2])/2}, {0, 0, (9Sqrt[3/2])/10}, 
 {-Sqrt[3], 0, Sqrt[3/2]/2}, {(-3Sqrt[3])/10, -9/10, 
  (-3Sqrt[3/2])/10}, {(-3Sqrt[3])/10, 9/10, 
  (-3Sqrt[3/2])/10}, {Sqrt[3]/2, -3/2, Sqrt[3/2]/2}, 
 {Sqrt[3]/2, 3/2, Sqrt[3/2]/2}, {(3Sqrt[3])/5, 0, 
  (-3Sqrt[3/2])/10}}

(* Truncated Cube *)

PolyhedronName[TruncatedCube]:="truncated cube"
NetFaces[TruncatedCube]:={{4,2,1,3,5,11,12,6},{17,12,11,16,20,25,26,21},{32,26,25,31,33,35,36,34},{40,36,35,39,41,43,44,42},{18,14,13,17,21,27,28,22},{16,10,9,15,19,23,24,20},{42,44,46},{41,45,43},{34,36,38},{33,37,35},{21,26,30},{20,29,25},{8,12,17},{7,16,11}}
NetVertices[TruncatedCube]:={{(-1-Sqrt[2])/2,-1/2},{(-1-Sqrt[2])/2,1/2},{-1/2,(-1-Sqrt[2])/2},{-1/2,(1+Sqrt[2])/2},{1/2,(-1-Sqrt[2])/2},{1/2,(1+Sqrt[2])/2},{(2+3Sqrt[2]-Sqrt[6])/4,(-2-Sqrt[2]-Sqrt[6])/4},{(2+3Sqrt[2]-Sqrt[6])/4,(2+Sqrt[2]+Sqrt[6])/4},{(1+Sqrt[2])/2,(-3-2Sqrt[2])/2},{(1+Sqrt[2])/2,(-1-2Sqrt[2])/2},{(1+Sqrt[2])/2,-1/2},{(1+Sqrt[2])/2,1/2},{(1+Sqrt[2])/2,(1+2Sqrt[2])/2},{(1+Sqrt[2])/2,(3+2Sqrt[2])/2},{(1+2Sqrt[2])/2,(-3(1+Sqrt[2]))/2},{(1+2Sqrt[2])/2,(-1-Sqrt[2])/2},{(1+2Sqrt[2])/2,(1+Sqrt[2])/2},{(1+2Sqrt[2])/2,(3(1+Sqrt[2]))/2},{(3+2Sqrt[2])/2,(-3(1+Sqrt[2]))/2},{(3+2Sqrt[2])/2,(-1-Sqrt[2])/2},{(3+2Sqrt[2])/2,(1+Sqrt[2])/2},{(3+2Sqrt[2])/2,(3(1+Sqrt[2]))/2},{(3(1+Sqrt[2]))/2,(-3-2Sqrt[2])/2},{(3(1+Sqrt[2]))/2,(-1-2Sqrt[2])/2},{(3(1+Sqrt[2]))/2,-1/2},{(3(1+Sqrt[2]))/2,1/2},{(3(1+Sqrt[2]))/2,(1+2Sqrt[2])/2},{(3(1+Sqrt[2]))/2,(3+2Sqrt[2])/2},{(6+5Sqrt[2]+Sqrt[6])/4,(-2-Sqrt[2]-Sqrt[6])/4},{(6+5Sqrt[2]+Sqrt[6])/4,(2+Sqrt[2]+Sqrt[6])/4},{(3+4Sqrt[2])/2,(-1-Sqrt[2])/2},{(3+4Sqrt[2])/2,(1+Sqrt[2])/2},{(5+4Sqrt[2])/2,(-1-Sqrt[2])/2},{(5+4Sqrt[2])/2,(1+Sqrt[2])/2},{(5(1+Sqrt[2]))/2,-1/2},{(5(1+Sqrt[2]))/2,1/2},{(10+9Sqrt[2]+Sqrt[6])/4,(-2-Sqrt[2]-Sqrt[6])/4},{(10+9Sqrt[2]+Sqrt[6])/4,(2+Sqrt[2]+Sqrt[6])/4},{(5+6Sqrt[2])/2,(-1-Sqrt[2])/2},{(5+6Sqrt[2])/2,(1+Sqrt[2])/2},{(7+6Sqrt[2])/2,(-1-Sqrt[2])/2},{(7+6Sqrt[2])/2,(1+Sqrt[2])/2},{(7(1+Sqrt[2]))/2,-1/2},{(7(1+Sqrt[2]))/2,1/2},{(14+13Sqrt[2]+Sqrt[6])/4,(-2-Sqrt[2]-Sqrt[6])/4},{(14+13Sqrt[2]+Sqrt[6])/4,(2+Sqrt[2]+Sqrt[6])/4}}
PolyhedronDual[TruncatedCube]:=SmallTriakisOctahedron
PolyhedronFaces[TruncatedCube]:=
{{6,12,10,8,4,18,20,2},{1,19,17,3,7,9,11,5},{3,24,23,4,8,15,16,7},{5,14,13,6,
    2,21,22,1},{9,16,15,10,12,13,14,11},{19,22,21,20,18,23,24,17},{16,9,7},{5,
    11,14},{3,17,24},{22,19,1},{8,10,15},{13,12,6},{23,18,4},{2,20,21}}
PolyhedronVertices[TruncatedCube]:=
{{-1/2,1/2+1/Sqrt[2],1/2+1/Sqrt[2]},
{-1/2,1/2+1/Sqrt[2],(2-2Sqrt[2])^(-1)},
{-1/2,(2-2Sqrt[2])^(-1),1/2+1/Sqrt[2]},
{-1/2,(2-2Sqrt[2])^(-1),(2-2Sqrt[2])^(-1)},
{1/2,1/2+1/Sqrt[2],1/2+1/Sqrt[2]},
{1/2,1/2+1/Sqrt[2],(2-2Sqrt[2])^(-1)},
{1/2,(2-2Sqrt[2])^(-1),1/2+1/Sqrt[2]},
{1/2,(2-2Sqrt[2])^(-1),(2-2Sqrt[2])^(-1)},
{1/2+1/Sqrt[2],-1/2,1/2+1/Sqrt[2]},
{1/2+1/Sqrt[2],-1/2,(2-2Sqrt[2])^(-1)},
{1/2+1/Sqrt[2],1/2,1/2+1/Sqrt[2]},
{1/2+1/Sqrt[2],1/2,(2-2Sqrt[2])^(-1)},
{1/2+1/Sqrt[2],1/2+1/Sqrt[2],-1/2},
{1/2+1/Sqrt[2],1/2+1/Sqrt[2],1/2},
{1/2+1/Sqrt[2],(2-2Sqrt[2])^(-1),-1/2},
{1/2+1/Sqrt[2],(2-2Sqrt[2])^(-1),1/2},
{(2-2Sqrt[2])^(-1),-1/2,1/2+1/Sqrt[2]},
{(2-2Sqrt[2])^(-1),-1/2,(2-2Sqrt[2])^(-1)},
{(2-2Sqrt[2])^(-1),1/2,1/2+1/Sqrt[2]},
{(2-2Sqrt[2])^(-1),1/2,(2-2Sqrt[2])^(-1)},
{(2-2Sqrt[2])^(-1),1/2+1/Sqrt[2],-1/2},
{(2-2Sqrt[2])^(-1),1/2+1/Sqrt[2],1/2},
{(2-2Sqrt[2])^(-1),(2-2Sqrt[2])^(-1),-1/2},
{(2-2Sqrt[2])^(-1),(2-2Sqrt[2])^(-1),1/2}}

(* Truncated Dodecahedron *)

PolyhedronName[TruncatedDodecahedron]:="truncated dodecahedron"
NetFaces[TruncatedDodecahedron]:={{80,90,94,97,98,95,91,81,75,74},{67,74,75},{53,60,73,79,80,74,61,54,49,48},{42,49,54},{83,90,80},{99,107,108,100,94,90,86,85,89,93},{78,89,85},{103,97,94},{118,116,112,106,98,97,105,111,115,117},{114,115,111},{104,95,98},{96,92,88,87,91,95,101,109,110,102},{113,110,109},{84,81,91},{51,50,55,62,75,81,82,76,63,56},{72,63,76},{28,38,44,45,39,29,25,22,21,24},{15,24,21},{23,27,31,32,28,24,18,10,9,17},{6,9,10},{35,38,28},{68,69,64,57,44,38,37,43,56,63},{47,56,43},{52,45,44},{66,59,46,40,39,45,58,65,70,71},{77,70,65},{36,29,39},{20,12,11,19,25,29,33,34,30,26},{41,30,34},{16,22,25},{1,3,7,13,21,22,14,8,4,2},{5,4,8}}
NetVertices[TruncatedDodecahedron]:={{(2Sqrt[3]-15Sqrt[2(5+Sqrt[5])]-6Sqrt[10(5+Sqrt[5])])/12,(13+5Sqrt[5])/4},{(2Sqrt[3]-15Sqrt[2(5+Sqrt[5])]-6Sqrt[10(5+Sqrt[5])])/12,(17+5Sqrt[5])/4},{(4Sqrt[3]-33Sqrt[2(5+Sqrt[5])]-9Sqrt[10(5+Sqrt[5])])/24,3+Sqrt[5]},{(4Sqrt[3]-33Sqrt[2(5+Sqrt[5])]-9Sqrt[10(5+Sqrt[5])])/24,(3(3+Sqrt[5]))/2},{(7Sqrt[3]-3Sqrt[15]-30Sqrt[2(5+Sqrt[5])]-9Sqrt[10(5+Sqrt[5])])/24,(35+13Sqrt[5]+Sqrt[6(5+Sqrt[5])])/8},{(-4Sqrt[3]-15Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/12,(5+Sqrt[5])/2},{(4Sqrt[3]-27Sqrt[2(5+Sqrt[5])]-9Sqrt[10(5+Sqrt[5])])/24,(13+3Sqrt[5])/4},{(4Sqrt[3]-27Sqrt[2(5+Sqrt[5])]-9Sqrt[10(5+Sqrt[5])])/24,(17+7Sqrt[5])/4},{(2Sqrt[3]-15Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/12,(4+Sqrt[5])/2},{(2Sqrt[3]-15Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/12,(6+Sqrt[5])/2},{(2Sqrt[3]-15Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/12,(9+4Sqrt[5])/2},{(2Sqrt[3]-15Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/12,(11+4Sqrt[5])/2},{(4Sqrt[3]-21Sqrt[2(5+Sqrt[5])]-9Sqrt[10(5+Sqrt[5])])/24,3+Sqrt[5]},{(4Sqrt[3]-21Sqrt[2(5+Sqrt[5])]-9Sqrt[10(5+Sqrt[5])])/24,(3(3+Sqrt[5]))/2},{(2Sqrt[3]-6Sqrt[15]-51Sqrt[2(5+Sqrt[5])]-9Sqrt[10(5+Sqrt[5])])/48,(50+18Sqrt[5]+Sqrt[6(5+Sqrt[5])]-Sqrt[30(5+Sqrt[5])])/16},{(2Sqrt[3]-6Sqrt[15]-51Sqrt[2(5+Sqrt[5])]-9Sqrt[10(5+Sqrt[5])])/48,(70+22Sqrt[5]-Sqrt[6(5+Sqrt[5])]+Sqrt[30(5+Sqrt[5])])/16},{(4Sqrt[3]-33Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/24,(7+Sqrt[5])/4},{(4Sqrt[3]-33Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/24,(13+3Sqrt[5])/4},{(4Sqrt[3]-33Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/24,(17+7Sqrt[5])/4},{(4Sqrt[3]-33Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/24,(23+9Sqrt[5])/4},{(2Sqrt[3]-12Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/12,(13+5Sqrt[5])/4},{(2Sqrt[3]-12Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/12,(17+5Sqrt[5])/4},{(4Sqrt[3]-27Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/24,2},{(4Sqrt[3]-27Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/24,3+Sqrt[5]},{(4Sqrt[3]-27Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/24,(3(3+Sqrt[5]))/2},{(4Sqrt[3]-27Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/24,(11+5Sqrt[5])/2},{(4Sqrt[3]-21Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/24,(7+Sqrt[5])/4},{(4Sqrt[3]-21Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/24,(13+3Sqrt[5])/4},{(4Sqrt[3]-21Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/24,(17+7Sqrt[5])/4},{(4Sqrt[3]-21Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/24,(23+9Sqrt[5])/4},{(Sqrt[3]-6Sqrt[2(5+Sqrt[5])])/6,(4+Sqrt[5])/2},{(Sqrt[3]-6Sqrt[2(5+Sqrt[5])])/6,(6+Sqrt[5])/2},{(Sqrt[3]-6Sqrt[2(5+Sqrt[5])])/6,(9+4Sqrt[5])/2},{(Sqrt[3]-6Sqrt[2(5+Sqrt[5])])/6,(11+4Sqrt[5])/2},{(Sqrt[3]+3Sqrt[15]-18Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/24,(25+7Sqrt[5]-Sqrt[6(5+Sqrt[5])])/8},{(Sqrt[3]+3Sqrt[15]-18Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/24,(35+13Sqrt[5]+Sqrt[6(5+Sqrt[5])])/8},{(4Sqrt[3]-15Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/24,2+Sqrt[5]},{(4Sqrt[3]-15Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/24,3+Sqrt[5]},{(4Sqrt[3]-15Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/24,(3(3+Sqrt[5]))/2},{(4Sqrt[3]-15Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/24,(11+3Sqrt[5])/2},{(14Sqrt[3]+6Sqrt[15]-45Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/48,(90+34Sqrt[5]-Sqrt[6(5+Sqrt[5])]+Sqrt[30(5+Sqrt[5])])/16},{(2Sqrt[3]-6Sqrt[15]-21Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/48,(-10-2Sqrt[5]-Sqrt[6(5+Sqrt[5])]+Sqrt[30(5+Sqrt[5])])/16},{(2Sqrt[3]-9Sqrt[2(5+Sqrt[5])])/12,(7+3Sqrt[5])/4},{(2Sqrt[3]-9Sqrt[2(5+Sqrt[5])])/12,(13+5Sqrt[5])/4},{(2Sqrt[3]-9Sqrt[2(5+Sqrt[5])])/12,(17+5Sqrt[5])/4},{(2Sqrt[3]-9Sqrt[2(5+Sqrt[5])])/12,(23+7Sqrt[5])/4},{(7Sqrt[3]-3Sqrt[15]-15Sqrt[2(5+Sqrt[5])])/24,(15+5Sqrt[5]-Sqrt[6(5+Sqrt[5])])/8},{(4Sqrt[3]-9Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/24,(-7-Sqrt[5])/4},{(4Sqrt[3]-9Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/24,(-3-Sqrt[5])/4},{(4Sqrt[3]-9Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/24,(3+Sqrt[5])/4},{(4Sqrt[3]-9Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/24,(7+Sqrt[5])/4},{(8Sqrt[3]-9Sqrt[2(5+Sqrt[5])])/12,(5(3+Sqrt[5]))/4},{(Sqrt[3]-3Sqrt[2(5+Sqrt[5])])/6,(-4-Sqrt[5])/2},{(Sqrt[3]-3Sqrt[2(5+Sqrt[5])])/6,-1/2},{(Sqrt[3]-3Sqrt[2(5+Sqrt[5])])/6,1/2},{(Sqrt[3]-3Sqrt[2(5+Sqrt[5])])/6,(4+Sqrt[5])/2},{(Sqrt[3]-3Sqrt[2(5+Sqrt[5])])/6,(3(2+Sqrt[5]))/2},{(Sqrt[3]-3Sqrt[2(5+Sqrt[5])])/6,(9+2Sqrt[5])/2},{(Sqrt[3]-3Sqrt[2(5+Sqrt[5])])/6,(11+4Sqrt[5])/2},{(2Sqrt[3]-3Sqrt[2(5+Sqrt[5])])/12,(-7-3Sqrt[5])/4},{(2Sqrt[3]-3Sqrt[2(5+Sqrt[5])])/12,(-3+Sqrt[5])/4},{(2Sqrt[3]-3Sqrt[2(5+Sqrt[5])])/12,(3-Sqrt[5])/4},{(2Sqrt[3]-3Sqrt[2(5+Sqrt[5])])/12,(7+3Sqrt[5])/4},{(2Sqrt[3]-3Sqrt[2(5+Sqrt[5])])/12,(13+5Sqrt[5])/4},{(2Sqrt[3]-3Sqrt[2(5+Sqrt[5])])/12,(17+5Sqrt[5])/4},{(2Sqrt[3]-3Sqrt[2(5+Sqrt[5])])/12,(23+7Sqrt[5])/4},{-(1/Sqrt[3]),0},{(4Sqrt[3]-9Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/24,2+Sqrt[5]},{(4Sqrt[3]-9Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/24,3+Sqrt[5]},{(4Sqrt[3]-9Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/24,(3(3+Sqrt[5]))/2},{(4Sqrt[3]-9Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/24,(11+3Sqrt[5])/2},{(Sqrt[3]+3Sqrt[15]-3Sqrt[2(5+Sqrt[5])])/24,(15+5Sqrt[5]+Sqrt[6(5+Sqrt[5])])/8},{1/(2Sqrt[3]),(-4-Sqrt[5])/2},{1/(2Sqrt[3]),-1/2},{1/(2Sqrt[3]),1/2},{1/(2Sqrt[3]),(4+Sqrt[5])/2},{(14Sqrt[3]+6Sqrt[15]-15Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/48,(70+22Sqrt[5]+Sqrt[6(5+Sqrt[5])]-Sqrt[30(5+Sqrt[5])])/16},{(2Sqrt[3]-6Sqrt[15]+9Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/48,(-30-14Sqrt[5]+Sqrt[6(5+Sqrt[5])]-Sqrt[30(5+Sqrt[5])])/16},{(4Sqrt[3]-3Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/24,(-7-Sqrt[5])/4},{(4Sqrt[3]-3Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/24,(-3-Sqrt[5])/4},{(4Sqrt[3]-3Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/24,(3+Sqrt[5])/4},{(4Sqrt[3]-3Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/24,(7+Sqrt[5])/4},{(7Sqrt[3]-3Sqrt[15]+3Sqrt[10(5+Sqrt[5])])/24,(-5-3Sqrt[5]-Sqrt[6(5+Sqrt[5])])/8},{(7Sqrt[3]-3Sqrt[15]+3Sqrt[10(5+Sqrt[5])])/24,(5+3Sqrt[5]+Sqrt[6(5+Sqrt[5])])/8},{(2Sqrt[3]+3Sqrt[2(5+Sqrt[5])])/12,(-7-3Sqrt[5])/4},{(2Sqrt[3]+3Sqrt[2(5+Sqrt[5])])/12,(-3(1+Sqrt[5]))/4},{(2Sqrt[3]+3Sqrt[2(5+Sqrt[5])])/12,(3(1+Sqrt[5]))/4},{(2Sqrt[3]+3Sqrt[2(5+Sqrt[5])])/12,(7+3Sqrt[5])/4},{(4Sqrt[3]+3Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/24,-2-Sqrt[5]},{(4Sqrt[3]+3Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/24,(-1-Sqrt[5])/2},{(4Sqrt[3]+3Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/24,(1+Sqrt[5])/2},{(4Sqrt[3]+3Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/24,2+Sqrt[5]},{(4Sqrt[3]+9Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/24,(-7-5Sqrt[5])/4},{(4Sqrt[3]+9Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/24,(-3-Sqrt[5])/4},{(4Sqrt[3]+9Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/24,(3+Sqrt[5])/4},{(4Sqrt[3]+9Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/24,(7+5Sqrt[5])/4},{(2Sqrt[3]+3Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/12,-1/2},{(2Sqrt[3]+3Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/12,1/2},{(4Sqrt[3]+15Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/24,-2-Sqrt[5]},{(4Sqrt[3]+15Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/24,(-1-Sqrt[5])/2},{(4Sqrt[3]+15Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/24,(1+Sqrt[5])/2},{(4Sqrt[3]+15Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/24,2+Sqrt[5]},{(14Sqrt[3]+6Sqrt[15]+15Sqrt[2(5+Sqrt[5])]+9Sqrt[10(5+Sqrt[5])])/48,(-10-2Sqrt[5]+Sqrt[6(5+Sqrt[5])]-Sqrt[30(5+Sqrt[5])])/16},{(14Sqrt[3]+6Sqrt[15]+15Sqrt[2(5+Sqrt[5])]+9Sqrt[10(5+Sqrt[5])])/48,(10+2Sqrt[5]-Sqrt[6(5+Sqrt[5])]+Sqrt[30(5+Sqrt[5])])/16},{(4Sqrt[3]+3Sqrt[2(5+Sqrt[5])]+9Sqrt[10(5+Sqrt[5])])/24,(-3-Sqrt[5])/4},{(4Sqrt[3]+3Sqrt[2(5+Sqrt[5])]+9Sqrt[10(5+Sqrt[5])])/24,(3+Sqrt[5])/4},{(2Sqrt[3]+6Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/12,(-7-3Sqrt[5])/4},{(2Sqrt[3]+6Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/12,(-3(1+Sqrt[5]))/4},{(2Sqrt[3]+6Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/12,(3(1+Sqrt[5]))/4},{(2Sqrt[3]+6Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/12,(7+3Sqrt[5])/4},{(4Sqrt[3]+9Sqrt[2(5+Sqrt[5])]+9Sqrt[10(5+Sqrt[5])])/24,(-1-Sqrt[5])/2},{(4Sqrt[3]+9Sqrt[2(5+Sqrt[5])]+9Sqrt[10(5+Sqrt[5])])/24,(1+Sqrt[5])/2},{(8Sqrt[3]+6Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/12,(5+3Sqrt[5])/4},{(Sqrt[3]+3Sqrt[15]+12Sqrt[2(5+Sqrt[5])]+9Sqrt[10(5+Sqrt[5])])/24,(-5-3Sqrt[5]-Sqrt[6(5+Sqrt[5])])/8},{(4Sqrt[3]+15Sqrt[2(5+Sqrt[5])]+9Sqrt[10(5+Sqrt[5])])/24,(-3-Sqrt[5])/4},{(4Sqrt[3]+15Sqrt[2(5+Sqrt[5])]+9Sqrt[10(5+Sqrt[5])])/24,(3+Sqrt[5])/4},{(Sqrt[3]+3Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/6,-1/2},{(Sqrt[3]+3Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/6,1/2}}
PolyhedronDual[TruncatedDodecahedron]:=TriakisIcosahedron
PolyhedronFaces[TruncatedDodecahedron]:=
{{3,42,46,44,40,1,34,48,50,36},{47,43,4,37,51,49,35,2,41,45},{2,35,19,24,21,
    16,5,59,55,14},{49,51,20,25,29,31,30,28,24,19},{37,4,15,56,60,6,17,22,25,
    20},{43,47,52,9,33,27,11,13,56,15},{45,41,14,55,12,10,26,32,9,52},{6,60,
    13,11,54,58,42,3,8,39},{27,33,32,26,53,57,44,46,58,54},{10,12,59,5,38,7,1,
    40,57,53},{16,21,28,30,18,23,48,34,7,38},{31,29,22,17,39,8,36,50,23,
    18},{9,32,33},{18,30,31},{47,45,52},{50,48,23},{10,53,26},{27,54,11},{21,
    24,28},{29,25,22},{40,44,57},{58,46,42},{35,49,19},{20,51,37},{12,55,
    59},{60,56,13},{41,2,14},{15,4,43},{34,1,7},{8,3,36},{38,5,16},{17,6,39}}
PolyhedronVertices[TruncatedDodecahedron]:=
{{0,(-1-Sqrt[5])/2,Sqrt[25/8+(11Sqrt[5])/8]},
{0,(-1-Sqrt[5])/2,Root[5-100#1^2+16#1^4&,1,
0]},{0,(1+Sqrt[5])/2,Sqrt[25/8+(11Sqrt[5])/8]},
{0,(1+Sqrt[5])/2,Root[5-100#1^2+16#1^4&,1,
0]},{Sqrt[1/8+1/(8Sqrt[5])],(-5-3Sqrt[5])/4,
Root[1-20#1^2+80#1^4&,3,0]},
{Sqrt[1/8+1/(8Sqrt[5])],(5+3Sqrt[5])/4,
Root[1-20#1^2+80#1^4&,3,0]},
{Sqrt[1/4+1/(2Sqrt[5])],-1-Sqrt[5]/2,
Sqrt[17/8+31/(8Sqrt[5])]},{Sqrt[1/4+1/(2Sqrt[5])],
(2+Sqrt[5])/2,Sqrt[17/8+31/(8Sqrt[5])]},
{-2Sqrt[1+2/Sqrt[5]],0,Root[1-100#1^2+80#1^4&,
1,0]},{(-3Sqrt[1+2/Sqrt[5]])/2,-1-Sqrt[5]/2,
Root[1-20#1^2+80#1^4&,3,0]},
{(-3Sqrt[1+2/Sqrt[5]])/2,(2+Sqrt[5])/2,
Root[1-20#1^2+80#1^4&,3,0]},
{-Sqrt[1+2/Sqrt[5]],(-3-Sqrt[5])/2,
Root[1-20#1^2+80#1^4&,2,0]},
{-Sqrt[1+2/Sqrt[5]],(3+Sqrt[5])/2,
Root[1-20#1^2+80#1^4&,2,0]},
{-Sqrt[1+2/Sqrt[5]]/2,-1-Sqrt[5]/2,
-Sqrt[17/8+31/(8Sqrt[5])]},{-Sqrt[1+2/Sqrt[5]]/2,
(2+Sqrt[5])/2,-Sqrt[17/8+31/(8Sqrt[5])]},
{Sqrt[1+2/Sqrt[5]],(-3-Sqrt[5])/2,
Root[1-20#1^2+80#1^4&,3,0]},
{Sqrt[1+2/Sqrt[5]],(3+Sqrt[5])/2,
Root[1-20#1^2+80#1^4&,3,0]},
{2Sqrt[1+2/Sqrt[5]],0,Sqrt[5/8+11/(8Sqrt[5])]},
{Sqrt[13/8+29/(8Sqrt[5])],(-3-Sqrt[5])/4,
-Sqrt[17/8+31/(8Sqrt[5])]},
{Sqrt[13/8+29/(8Sqrt[5])],(3+Sqrt[5])/4,
-Sqrt[17/8+31/(8Sqrt[5])]},{Sqrt[9/4+9/(2Sqrt[5])],
-1-Sqrt[5]/2,Root[1-20#1^2+80#1^4&,2,0]},
{Sqrt[9/4+9/(2Sqrt[5])],(2+Sqrt[5])/2,
Root[1-20#1^2+80#1^4&,2,0]},
{Sqrt[5/2+11/(2Sqrt[5])],0,
Sqrt[17/8+31/(8Sqrt[5])]},{Sqrt[5/2+11/(2Sqrt[5])],
(-1-Sqrt[5])/2,Root[1-100#1^2+80#1^4&,1,0]},
{Sqrt[5/2+11/(2Sqrt[5])],(1+Sqrt[5])/2,
Root[1-100#1^2+80#1^4&,1,0]},
{-Sqrt[29/8+61/(8Sqrt[5])],(-3-Sqrt[5])/4,
Root[1-20#1^2+80#1^4&,3,0]},
{-Sqrt[29/8+61/(8Sqrt[5])],(3+Sqrt[5])/4,
Root[1-20#1^2+80#1^4&,3,0]},
{Sqrt[29/8+61/(8Sqrt[5])],(-3-Sqrt[5])/4,
Root[1-20#1^2+80#1^4&,2,0]},
{Sqrt[29/8+61/(8Sqrt[5])],(3+Sqrt[5])/4,
Root[1-20#1^2+80#1^4&,2,0]},
{Sqrt[17/4+19/(2Sqrt[5])],-1/2,
Root[1-20#1^2+80#1^4&,3,0]},
{Sqrt[17/4+19/(2Sqrt[5])],1/2,
Root[1-20#1^2+80#1^4&,3,0]},
{-Sqrt[17+38/Sqrt[5]]/2,-1/2,
Root[1-20#1^2+80#1^4&,2,0]},
{-Sqrt[17+38/Sqrt[5]]/2,1/2,
Root[1-20#1^2+80#1^4&,2,0]},
{Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,
Sqrt[25/8+(11Sqrt[5])/8]},{Sqrt[5/8+Sqrt[5]/8],
(-3-Sqrt[5])/4,Root[5-100#1^2+16#1^4&,1,0]},
{Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,
Sqrt[25/8+(11Sqrt[5])/8]},{Sqrt[5/8+Sqrt[5]/8],
(3+Sqrt[5])/4,Root[5-100#1^2+16#1^4&,1,0]},
{Sqrt[(5+Sqrt[5])/10],(-3-Sqrt[5])/2,
Sqrt[5/8+11/(8Sqrt[5])]},{Sqrt[(5+Sqrt[5])/10],
(3+Sqrt[5])/2,Sqrt[5/8+11/(8Sqrt[5])]},
{-Sqrt[(5+Sqrt[5])/2]/2,(-3-Sqrt[5])/4,
Sqrt[25/8+(11Sqrt[5])/8]},{-Sqrt[(5+Sqrt[5])/2]/2,
(-3-Sqrt[5])/4,Root[5-100#1^2+16#1^4&,1,0]},
{-Sqrt[(5+Sqrt[5])/2]/2,(3+Sqrt[5])/4,
Sqrt[25/8+(11Sqrt[5])/8]},{-Sqrt[(5+Sqrt[5])/2]/2,
(3+Sqrt[5])/4,Root[5-100#1^2+16#1^4&,1,0]},
{-Sqrt[5+2Sqrt[5]]/2,-1/2,
Sqrt[25/8+(11Sqrt[5])/8]},{-Sqrt[5+2Sqrt[5]]/2,
-1/2,Root[5-100#1^2+16#1^4&,1,0]},
{-Sqrt[5+2Sqrt[5]]/2,1/2,Sqrt[25/8+(11Sqrt[5])/8]},
{-Sqrt[5+2Sqrt[5]]/2,1/2,
Root[5-100#1^2+16#1^4&,1,0]},
{Sqrt[5+2Sqrt[5]]/2,-1/2,Sqrt[25/8+(11Sqrt[5])/8]},
{Sqrt[5+2Sqrt[5]]/2,-1/2,
Root[5-100#1^2+16#1^4&,1,0]},
{Sqrt[5+2Sqrt[5]]/2,1/2,Sqrt[25/8+(11Sqrt[5])/8]},
{Sqrt[5+2Sqrt[5]]/2,1/2,
Root[5-100#1^2+16#1^4&,1,0]},
{Root[1-25#1^2+5#1^4&,1,0],0,
-Sqrt[17/8+31/(8Sqrt[5])]},
{Root[1-25#1^2+5#1^4&,1,0],(-1-Sqrt[5])/2,
Sqrt[5/8+11/(8Sqrt[5])]},
{Root[1-25#1^2+5#1^4&,1,0],(1+Sqrt[5])/2,
Sqrt[5/8+11/(8Sqrt[5])]},
{Root[1-5#1^2+5#1^4&,1,0],(-3-Sqrt[5])/2,
Root[1-100#1^2+80#1^4&,1,0]},
{Root[1-5#1^2+5#1^4&,1,0],(3+Sqrt[5])/2,
Root[1-100#1^2+80#1^4&,1,0]},
{Root[1-260#1^2+80#1^4&,1,0],(-3-Sqrt[5])/4,
Sqrt[17/8+31/(8Sqrt[5])]},
{Root[1-260#1^2+80#1^4&,1,0],(3+Sqrt[5])/4,
Sqrt[17/8+31/(8Sqrt[5])]},
{Root[1-20#1^2+80#1^4&,1,0],(-5-3Sqrt[5])/4,
Root[1-20#1^2+80#1^4&,2,0]},
{Root[1-20#1^2+80#1^4&,1,0],(5+3Sqrt[5])/4,
Root[1-20#1^2+80#1^4&,2,0]}}

(* Truncated Icosahedron *)

PolyhedronName[TruncatedIcosahedron]:="truncated icosahedron"
NetFaces[TruncatedIcosahedron]:={{24,29,49,50,30},{49,60,71,72,61,50},{71,96,106,107,97,72},{4,14,39,40,15,5},{39,50,61,62,51,40},{61,81,91,82,62},{25,31,51,52,32},{51,62,73,74,63,52},{73,98,108,109,99,74},{6,16,41,42,17,7},{41,52,63,64,53,42},{63,83,92,84,64},{26,33,53,54,34},{53,64,75,76,65,54},{75,100,110,111,101,76},{8,18,43,44,19,9},{43,54,65,66,55,44},{65,85,93,86,66},{27,35,55,56,36},{55,66,77,78,67,56},{77,102,112,113,103,78},{10,20,45,46,21,11},{45,56,67,68,57,46},{67,87,94,88,68},{28,37,57,58,38},{57,68,79,80,69,58},{79,104,114,115,105,80},{12,22,47,48,23,13},{47,58,69,70,59,48},{69,89,95,90,70},{1,2,4,5,3},{108,116,118,117,109}}
NetVertices[TruncatedIcosahedron]:={{(-15Sqrt[3]-Sqrt[10(5+Sqrt[5])])/10,3/2},{(-60Sqrt[3]-5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,(5-Sqrt[5])/4},{(-60Sqrt[3]-5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,(7+Sqrt[5])/4},{(-60Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,1},{(-60Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,2},{(-60Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,4},{(-60Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,5},{(-60Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,7},{(-60Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,8},{(-60Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,10},{(-60Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,11},{(-60Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,13},{(-60Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,14},{(-40Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,1/2},{(-40Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,5/2},{(-40Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,7/2},{(-40Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,11/2},{(-40Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,13/2},{(-40Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,17/2},{(-40Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,19/2},{(-40Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,23/2},{(-40Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,25/2},{(-40Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,29/2},{-Sqrt[(5+Sqrt[5])/10],0},{-Sqrt[(5+Sqrt[5])/10],3},{-Sqrt[(5+Sqrt[5])/10],6},{-Sqrt[(5+Sqrt[5])/10],9},{-Sqrt[(5+Sqrt[5])/10],12},{((-5+Sqrt[5])Sqrt[(5+Sqrt[5])/2])/20,(-1-Sqrt[5])/4},{((-5+Sqrt[5])Sqrt[(5+Sqrt[5])/2])/20,(1+Sqrt[5])/4},{((-5+Sqrt[5])Sqrt[(5+Sqrt[5])/2])/20,(11-Sqrt[5])/4},{((-5+Sqrt[5])Sqrt[(5+Sqrt[5])/2])/20,(13+Sqrt[5])/4},{((-5+Sqrt[5])Sqrt[(5+Sqrt[5])/2])/20,(23-Sqrt[5])/4},{((-5+Sqrt[5])Sqrt[(5+Sqrt[5])/2])/20,(25+Sqrt[5])/4},{((-5+Sqrt[5])Sqrt[(5+Sqrt[5])/2])/20,(35-Sqrt[5])/4},{((-5+Sqrt[5])Sqrt[(5+Sqrt[5])/2])/20,(37+Sqrt[5])/4},{((-5+Sqrt[5])Sqrt[(5+Sqrt[5])/2])/20,(47-Sqrt[5])/4},{((-5+Sqrt[5])Sqrt[(5+Sqrt[5])/2])/20,(49+Sqrt[5])/4},{(-20Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,1},{(-20Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,2},{(-20Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,4},{(-20Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,5},{(-20Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,7},{(-20Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,8},{(-20Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,10},{(-20Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,11},{(-20Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,13},{(-20Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,14},{(5+Sqrt[5])^(3/2)/(20Sqrt[2]),-1/2},{(5+Sqrt[5])^(3/2)/(20Sqrt[2]),1/2},{(5+Sqrt[5])^(3/2)/(20Sqrt[2]),5/2},{(5+Sqrt[5])^(3/2)/(20Sqrt[2]),7/2},{(5+Sqrt[5])^(3/2)/(20Sqrt[2]),11/2},{(5+Sqrt[5])^(3/2)/(20Sqrt[2]),13/2},{(5+Sqrt[5])^(3/2)/(20Sqrt[2]),17/2},{(5+Sqrt[5])^(3/2)/(20Sqrt[2]),19/2},{(5+Sqrt[5])^(3/2)/(20Sqrt[2]),23/2},{(5+Sqrt[5])^(3/2)/(20Sqrt[2]),25/2},{(5+Sqrt[5])^(3/2)/(20Sqrt[2]),29/2},{(20Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,-1},{(20Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,1},{(20Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,2},{(20Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,4},{(20Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,5},{(20Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,7},{(20Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,8},{(20Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,10},{(20Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,11},{(20Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,13},{(20Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,14},{(40Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,-1/2},{(40Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,1/2},{(40Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,5/2},{(40Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,7/2},{(40Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,11/2},{(40Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,13/2},{(40Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,17/2},{(40Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,19/2},{(40Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,23/2},{(40Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,25/2},{(20Sqrt[3]+5Sqrt[2(5-Sqrt[5])]+5Sqrt[10(5-Sqrt[5])]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,(5-Sqrt[5])/4},{(40Sqrt[3]-5Sqrt[2(5-Sqrt[5])]+5Sqrt[10(5-Sqrt[5])]+15Sqrt[2(5+Sqrt[5])]+7Sqrt[10(5+Sqrt[5])])/80,(7+Sqrt[5])/4},{(20Sqrt[3]+5Sqrt[2(5-Sqrt[5])]+5Sqrt[10(5-Sqrt[5])]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,(17-Sqrt[5])/4},{(40Sqrt[3]-5Sqrt[2(5-Sqrt[5])]+5Sqrt[10(5-Sqrt[5])]+15Sqrt[2(5+Sqrt[5])]+7Sqrt[10(5+Sqrt[5])])/80,(19+Sqrt[5])/4},{(20Sqrt[3]+5Sqrt[2(5-Sqrt[5])]+5Sqrt[10(5-Sqrt[5])]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,(29-Sqrt[5])/4},{(40Sqrt[3]-5Sqrt[2(5-Sqrt[5])]+5Sqrt[10(5-Sqrt[5])]+15Sqrt[2(5+Sqrt[5])]+7Sqrt[10(5+Sqrt[5])])/80,(31+Sqrt[5])/4},{(20Sqrt[3]+5Sqrt[2(5-Sqrt[5])]+5Sqrt[10(5-Sqrt[5])]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,(41-Sqrt[5])/4},{(40Sqrt[3]-5Sqrt[2(5-Sqrt[5])]+5Sqrt[10(5-Sqrt[5])]+15Sqrt[2(5+Sqrt[5])]+7Sqrt[10(5+Sqrt[5])])/80,(43+Sqrt[5])/4},{(20Sqrt[3]+5Sqrt[2(5-Sqrt[5])]+5Sqrt[10(5-Sqrt[5])]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,(53-Sqrt[5])/4},{(40Sqrt[3]-5Sqrt[2(5-Sqrt[5])]+5Sqrt[10(5-Sqrt[5])]+15Sqrt[2(5+Sqrt[5])]+7Sqrt[10(5+Sqrt[5])])/80,(55+Sqrt[5])/4},{(40Sqrt[3]+15Sqrt[2(5-Sqrt[5])]+5Sqrt[10(5-Sqrt[5])]+15Sqrt[2(5+Sqrt[5])]+7Sqrt[10(5+Sqrt[5])])/80,3/2},{(40Sqrt[3]+15Sqrt[2(5-Sqrt[5])]+5Sqrt[10(5-Sqrt[5])]+15Sqrt[2(5+Sqrt[5])]+7Sqrt[10(5+Sqrt[5])])/80,9/2},{(40Sqrt[3]+15Sqrt[2(5-Sqrt[5])]+5Sqrt[10(5-Sqrt[5])]+15Sqrt[2(5+Sqrt[5])]+7Sqrt[10(5+Sqrt[5])])/80,15/2},{(40Sqrt[3]+15Sqrt[2(5-Sqrt[5])]+5Sqrt[10(5-Sqrt[5])]+15Sqrt[2(5+Sqrt[5])]+7Sqrt[10(5+Sqrt[5])])/80,21/2},{(40Sqrt[3]+15Sqrt[2(5-Sqrt[5])]+5Sqrt[10(5-Sqrt[5])]+15Sqrt[2(5+Sqrt[5])]+7Sqrt[10(5+Sqrt[5])])/80,27/2},{(60Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,-1},{(60Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,1},{(60Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,2},{(60Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,4},{(60Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,5},{(60Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,7},{(60Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,8},{(60Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,10},{(60Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,11},{(60Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,13},{(80Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,-1/2},{(80Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,1/2},{(80Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,5/2},{(80Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,7/2},{(80Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,11/2},{(80Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,13/2},{(80Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,17/2},{(80Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,19/2},{(80Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,23/2},{(80Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,25/2},{(80Sqrt[3]+5Sqrt[2(5-Sqrt[5])]+5Sqrt[10(5-Sqrt[5])]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,(11-Sqrt[5])/4},{(160Sqrt[3]-5Sqrt[2(5-Sqrt[5])]+5Sqrt[10(5-Sqrt[5])]+15Sqrt[2(5+Sqrt[5])]+7Sqrt[10(5+Sqrt[5])])/80,(13+Sqrt[5])/4},{(160Sqrt[3]+15Sqrt[2(5-Sqrt[5])]+5Sqrt[10(5-Sqrt[5])]+15Sqrt[2(5+Sqrt[5])]+7Sqrt[10(5+Sqrt[5])])/80,3}}
PolyhedronDual[TruncatedIcosahedron]:=PentakisDodecahedron
PolyhedronFaces[TruncatedIcosahedron]:=
{{53,11,24,23,9},{51,39,40,52,30},{60,28,16,12,2},{20,42,48,55,18},{19,17,54,
    47,41},{1,10,15,27,59},{36,26,44,50,38},{4,58,22,32,8},{34,29,33,45,
    46},{21,57,3,6,31},{37,49,43,25,35},{13,5,56,7,14},{9,59,27,51,30,53},{53,
    30,52,28,60,11},{11,60,2,42,20,24},{24,20,18,17,19,23},{23,19,41,1,59,
    9},{13,25,43,3,57,5},{5,57,21,33,29,56},{56,29,34,22,58,7},{7,58,4,44,26,
    14},{14,26,36,35,25,13},{40,38,50,16,28,52},{16,50,44,4,8,12},{12,8,32,48,
    42,2},{48,32,22,34,46,55},{55,46,45,54,17,18},{54,45,33,21,31,47},{47,31,
    6,10,1,41},{10,6,3,43,49,15},{15,49,37,39,51,27},{39,37,35,36,38,40}}
PolyhedronVertices[TruncatedIcosahedron]:=
{{-Sqrt[1-2/Sqrt[5]]/2,-1-Sqrt[5]/2,Sqrt[9/8+9/(8Sqrt[5])]},
{-Sqrt[1-2/Sqrt[5]]/2,(2+Sqrt[5])/2,Sqrt[9/8+9/(8Sqrt[5])]},
{Sqrt[1-2/Sqrt[5]]/2,-1-Sqrt[5]/2,(-3Sqrt[(5+Sqrt[5])/10])/2},
{Sqrt[1-2/Sqrt[5]]/2,(2+Sqrt[5])/2,(-3Sqrt[(5+Sqrt[5])/10])/2},
{-Sqrt[2-2/Sqrt[5]]/4,(1-Sqrt[5])^(-1),-Sqrt[25/8+41/(8Sqrt[5])]},
{-Sqrt[2-2/Sqrt[5]]/4,(-3(1+Sqrt[5]))/4,Root[1-20#1^2+80#1^4&,1,0]},
{-Sqrt[2-2/Sqrt[5]]/4,(1+Sqrt[5])/4,-Sqrt[25/8+41/(8Sqrt[5])]},
{-Sqrt[2-2/Sqrt[5]]/4,(3(1+Sqrt[5]))/4,Root[1-20#1^2+80#1^4&,1,0]},
{Sqrt[2-2/Sqrt[5]]/4,(1-Sqrt[5])^(-1),Sqrt[25/8+41/(8Sqrt[5])]},
{Sqrt[2-2/Sqrt[5]]/4,(-3(1+Sqrt[5]))/4,Sqrt[1/8+1/(8Sqrt[5])]},
{Sqrt[2-2/Sqrt[5]]/4,(1+Sqrt[5])/4,Sqrt[25/8+41/(8Sqrt[5])]},
{Sqrt[2-2/Sqrt[5]]/4,(3(1+Sqrt[5]))/4,Sqrt[1/8+1/(8Sqrt[5])]},
{Sqrt[1/4+1/(2Sqrt[5])],-1/2,-Sqrt[25/8+41/(8Sqrt[5])]},
{Sqrt[1/4+1/(2Sqrt[5])],1/2,-Sqrt[25/8+41/(8Sqrt[5])]},
{Sqrt[5/4+1/(2Sqrt[5])],-1-Sqrt[5]/2,Sqrt[1/8+1/(8Sqrt[5])]},
{Sqrt[5/4+1/(2Sqrt[5])],(2+Sqrt[5])/2,Sqrt[1/8+1/(8Sqrt[5])]},
{(-3Sqrt[1+2/Sqrt[5]])/2,-1/2,Sqrt[9/8+9/(8Sqrt[5])]},
{(-3Sqrt[1+2/Sqrt[5]])/2,1/2,Sqrt[9/8+9/(8Sqrt[5])]},
{-Sqrt[1+2/Sqrt[5]],-1,Sqrt[26+58/Sqrt[5]]/4},
{-Sqrt[1+2/Sqrt[5]],1,Sqrt[26+58/Sqrt[5]]/4},
{-Sqrt[1+2/Sqrt[5]],-2/(-1+Sqrt[5]),(-3Sqrt[(5+Sqrt[5])/10])/2},
{-Sqrt[1+2/Sqrt[5]],(1+Sqrt[5])/2,(-3Sqrt[(5+Sqrt[5])/10])/2},
{-Sqrt[1+2/Sqrt[5]]/2,-1/2,Sqrt[25/8+41/(8Sqrt[5])]},
{-Sqrt[1+2/Sqrt[5]]/2,1/2,Sqrt[25/8+41/(8Sqrt[5])]},
{Sqrt[1+2/Sqrt[5]],-1,Root[1-260#1^2+80#1^4&,1,0]},
{Sqrt[1+2/Sqrt[5]],1,Root[1-260#1^2+80#1^4&,1,0]},
{Sqrt[1+2/Sqrt[5]],-2/(-1+Sqrt[5]),Sqrt[9/8+9/(8Sqrt[5])]},
{Sqrt[1+2/Sqrt[5]],(1+Sqrt[5])/2,Sqrt[9/8+9/(8Sqrt[5])]},
{-Sqrt[2+2/Sqrt[5]],0,Root[1-260#1^2+80#1^4&,1,0]},
{Sqrt[2+2/Sqrt[5]],0,Sqrt[26+58/Sqrt[5]]/4},
{-Sqrt[5+2/Sqrt[5]]/2,-1-Sqrt[5]/2,Root[1-20#1^2+80#1^4&,1,0]},
{-Sqrt[5+2/Sqrt[5]]/2,(2+Sqrt[5])/2,Root[1-20#1^2+80#1^4&,1,0]},
{-Sqrt[17/8+31/(8Sqrt[5])],(1-Sqrt[5])^(-1),(-3Sqrt[(5+Sqrt[5])/10])/2},
{-Sqrt[17/8+31/(8Sqrt[5])],(1+Sqrt[5])/4,(-3Sqrt[(5+Sqrt[5])/10])/2},
{Sqrt[9/4+9/(2Sqrt[5])],-1/2,(-3Sqrt[(5+Sqrt[5])/10])/2},
{Sqrt[9/4+9/(2Sqrt[5])],1/2,(-3Sqrt[(5+Sqrt[5])/10])/2},
{Sqrt[5/2+11/(2Sqrt[5])],-1,Root[1-20#1^2+80#1^4&,1,0]},
{Sqrt[5/2+11/(2Sqrt[5])],1,Root[1-20#1^2+80#1^4&,1,0]},
{Sqrt[13/4+11/(2Sqrt[5])],-1/2,Sqrt[1/8+1/(8Sqrt[5])]},
{Sqrt[13/4+11/(2Sqrt[5])],1/2,Sqrt[1/8+1/(8Sqrt[5])]},
{-Sqrt[10+22/Sqrt[5]]/4,(-5-Sqrt[5])/4,Sqrt[9/8+9/(8Sqrt[5])]},
{-Sqrt[10+22/Sqrt[5]]/4,(5+Sqrt[5])/4,Sqrt[9/8+9/(8Sqrt[5])]},
{Sqrt[10+22/Sqrt[5]]/4,(-5-Sqrt[5])/4,(-3Sqrt[(5+Sqrt[5])/10])/2},
{Sqrt[10+22/Sqrt[5]]/4,(5+Sqrt[5])/4,(-3Sqrt[(5+Sqrt[5])/10])/2},
{-Sqrt[13+22/Sqrt[5]]/2,-1/2,Root[1-20#1^2+80#1^4&,1,0]},
{-Sqrt[13+22/Sqrt[5]]/2,1/2,Root[1-20#1^2+80#1^4&,1,0]},
{-Sqrt[26+38/Sqrt[5]]/4,(-5-Sqrt[5])/4,Sqrt[1/8+1/(8Sqrt[5])]},
{-Sqrt[26+38/Sqrt[5]]/4,(5+Sqrt[5])/4,Sqrt[1/8+1/(8Sqrt[5])]},
{Sqrt[26+38/Sqrt[5]]/4,(-5-Sqrt[5])/4,Root[1-20#1^2+80#1^4&,1,0]},
{Sqrt[26+38/Sqrt[5]]/4,(5+Sqrt[5])/4,Root[1-20#1^2+80#1^4&,1,0]},
{Sqrt[34+62/Sqrt[5]]/4,(1-Sqrt[5])^(-1),Sqrt[9/8+9/(8Sqrt[5])]},
{Sqrt[34+62/Sqrt[5]]/4,(1+Sqrt[5])/4,Sqrt[9/8+9/(8Sqrt[5])]},
{Sqrt[(5+Sqrt[5])/10],0,Sqrt[25/8+41/(8Sqrt[5])]},
{Root[1-25#1^2+5#1^4&,1,0],-1,Sqrt[1/8+1/(8Sqrt[5])]},
{Root[1-25#1^2+5#1^4&,1,0],1,Sqrt[1/8+1/(8Sqrt[5])]},
{Root[1-5#1^2+5#1^4&,1,0],0,-Sqrt[25/8+41/(8Sqrt[5])]},
{Root[1-5#1^2+5#1^4&,2,0],-2/(-1+Sqrt[5]),Root[1-260#1^2+80#1^4&,1,0]},
{Root[1-5#1^2+5#1^4&,2,0],(1+Sqrt[5])/2,Root[1-260#1^2+80#1^4&,1,0]},
{Root[1-5#1^2+5#1^4&,3,0],-2/(-1+Sqrt[5]),Sqrt[26+58/Sqrt[5]]/4},
{Root[1-5#1^2+5#1^4&,3,0],(1+Sqrt[5])/2,Sqrt[26+58/Sqrt[5]]/4}}

(* Truncated Octahedron *)

PolyhedronName[TruncatedOctahedron]:="truncated octahedron"
PolyhedronDual[TruncatedOctahedron]:=TetrakisHexahedron
NetFaces[TruncatedOctahedron]:={{4,3,6,7},{3,1,2,5,9,6},{10,6,9,13,17,14},{9,8,12,13},{18,17,21,22},{17,13,16,20,26,21},{27,21,26,30,34,31},{26,25,29,30},{35,34,38,39},{34,30,33,37,41,38},{42,38,41,44,46,45},{41,40,43,44},{28,23,27,31,36,32},{16,11,15,19,24,20}}
NetVertices[TruncatedOctahedron]:={{-1,(-1-Sqrt[3])/2},{-1/2,(-1-2Sqrt[3])/2},{-1/2,-1/2},{-1/2,1/2},{1/2,(-1-2Sqrt[3])/2},{1/2,-1/2},{1/2,1/2},{1,(-3-Sqrt[3])/2},{1,(-1-Sqrt[3])/2},{1,(-1+Sqrt[3])/2},{2,(-1-3Sqrt[3])/2},{2,(-3-Sqrt[3])/2},{2,(-1-Sqrt[3])/2},{2,(-1+Sqrt[3])/2},{5/2,(-1-4Sqrt[3])/2},{5/2,(-1-2Sqrt[3])/2},{5/2,-1/2},{5/2,1/2},{7/2,(-1-4Sqrt[3])/2},{7/2,(-1-2Sqrt[3])/2},{7/2,-1/2},{7/2,1/2},{7/2,(-1+2Sqrt[3])/2},{4,(-1-3Sqrt[3])/2},{4,(-3-Sqrt[3])/2},{4,(-1-Sqrt[3])/2},{4,(-1+Sqrt[3])/2},{4,(-1+3Sqrt[3])/2},{5,(-3-Sqrt[3])/2},{5,(-1-Sqrt[3])/2},{5,(-1+Sqrt[3])/2},{5,(-1+3Sqrt[3])/2},{11/2,(-1-2Sqrt[3])/2},{11/2,-1/2},{11/2,1/2},{11/2,(-1+2Sqrt[3])/2},{13/2,(-1-2Sqrt[3])/2},{13/2,-1/2},{13/2,1/2},{7,(-3-Sqrt[3])/2},{7,(-1-Sqrt[3])/2},{7,(-1+Sqrt[3])/2},{8,(-3-Sqrt[3])/2},{8,(-1-Sqrt[3])/2},{8,(-1+Sqrt[3])/2},{17/2,-1/2}}
PolyhedronVertices[TruncatedOctahedron]:=TetrakisHexahedron
PolyhedronFaces[TruncatedOctahedron]:=
{{17,11,9,15},{14,8,10,16},{22,24,21,18},{12,5,2,6},{13,19,23,20},{4,1,3,
    7},{19,13,7,3,8,14},{15,9,4,7,13,20},{16,10,5,12,18,21},{22,18,12,6,11,
    17},{20,23,24,22,17,15},{14,16,21,24,23,19},{9,11,6,2,1,4},{3,1,2,5,10,8}}
PolyhedronVertices[TruncatedOctahedron]:=
{{-3/2,-1/2,0},{-3/2,1/2,0},{-1,-1,-(1/Sqrt[2])},
{-1,-1,1/Sqrt[2]},{-1,1,-(1/Sqrt[2])},
{-1,1,1/Sqrt[2]},{-1/2,-3/2,0},
{-1/2,-1/2,-Sqrt[2]},{-1/2,-1/2,Sqrt[2]},
{-1/2,1/2,-Sqrt[2]},{-1/2,1/2,Sqrt[2]},
{-1/2,3/2,0},{1/2,-3/2,0},{1/2,-1/2,-Sqrt[2]},
{1/2,-1/2,Sqrt[2]},{1/2,1/2,-Sqrt[2]},
{1/2,1/2,Sqrt[2]},{1/2,3/2,0},{1,-1,-(1/Sqrt[2])},
{1,-1,1/Sqrt[2]},{1,1,-(1/Sqrt[2])},
{1,1,1/Sqrt[2]},{3/2,-1/2,0},{3/2,1/2,0}}

(* Truncated Tetrahedron *)

PolyhedronName[TruncatedTetrahedron]:="truncated tetrahedron"
NetFaces[TruncatedTetrahedron]:={{3,1,2,5,8,6},{2,4,5},{9,8,11},{8,5,7,10,12,11},{13,11,12,15,18,16},{12,14,15},{19,18,21},{18,15,17,20,22,21}}
NetVertices[TruncatedTetrahedron]:={{-1,0},{-1/2,-Sqrt[3]/2},{-1/2,Sqrt[3]/2},{0,-Sqrt[3]},{1/2,-Sqrt[3]/2},{1/2,Sqrt[3]/2},{1,-Sqrt[3]},{1,0},{3/2,Sqrt[3]/2},{2,-Sqrt[3]},{2,0},{5/2,-Sqrt[3]/2},{5/2,Sqrt[3]/2},{3,-Sqrt[3]},{7/2,-Sqrt[3]/2},{7/2,Sqrt[3]/2},{4,-Sqrt[3]},{4,0},{9/2,Sqrt[3]/2},{5,-Sqrt[3]},{5,0},{11/2,-Sqrt[3]/2}}
PolyhedronDual[TruncatedTetrahedron]:=TriakisTetrahedron
PolyhedronFaces[TruncatedTetrahedron]:=
{{11,12,8},{3,9,1},{2,10,4},{6,5,7},{11,8,7,5,3,1},{2,4,6,7,8,12},{9,3,5,6,4,
    10},{2,12,11,1,9,10}}
PolyhedronVertices[TruncatedTetrahedron]:=
{{0,-1,-Sqrt[3/2]/2},{0,1,-Sqrt[3/2]/2},
{-(1/Sqrt[3]),-1,1/(2Sqrt[6])},
{-(1/Sqrt[3]),1,1/(2Sqrt[6])},
{-1/(2Sqrt[3]),-1/2,5/(2Sqrt[6])},
{-1/(2Sqrt[3]),1/2,5/(2Sqrt[6])},
{1/Sqrt[3],0,5/(2Sqrt[6])},
{2/Sqrt[3],0,1/(2Sqrt[6])},
{-Sqrt[3]/2,-1/2,-Sqrt[3/2]/2},
{-Sqrt[3]/2,1/2,-Sqrt[3/2]/2},
{Sqrt[3]/2,-1/2,-Sqrt[3/2]/2},
{Sqrt[3]/2,1/2,-Sqrt[3/2]/2}}
  
End[]

(* Protect[  ] *)

EndPackage[]

Prisms

(*:Mathematica Version: 5.2 *)

(*:Package Version: 1.08 *)

(*:Name: MathWorld`Prisms` *)

(*:Author: Eric W. Weisstein *)

(*:URL:
  http://mathworld.wolfram.com/packages/Prisms.m
*)

(*:Summary:
*)

(*:History:
  v1.00 (2003-01-12): Broken off from MiscPolyhedra.m.
                      PentagonalPrism6Compound
  v1.01 (2003-08-17): Rhombohedron
  v1.02 (2003-10-05): Renamed Wedge to SolidWegde
  v1.03 (2003-10-19): updated contexts
  v1.04 (2003-11-21): Fixed Antiprism
  v1.05 (2003-12-25): Converted contructors to use PolyhedronFaces[]
                      and PolyehdronVertices[].  Moved Prism-type solids
                      here.
  v1.06 (2004-02-10): Fixed Frustum
  v1.07 (2004-09-14): Fixed Net[Antiprism[]], added names for antiprisms
  v1.08 (2005-02-15): PolyhedronName for solids with polygonal bases
  
  (c) 2002-2005 Eric W. Weisstein
*)

(*:Keywords:
  
*)

(*:Requirements:

  Requires
  
    http://mathworld.wolfram.com/packages/Polygons.m
    http://mathworld.wolfram.com/packages/PolyhedronOperations.m

*)

(*:Discussion:
*)

(*:References: *)

(*:Limitations:

Should rewrite constructors to generate faces and vertices directly instead of
generating solids and then extracting the face and vertex information.

Ditto for nets.
*)

BeginPackage["MathWorld`Prisms`",
	{
	"MathWorld`PolyhedronOperations`",
	"MathWorld`Platonic`", (* AugmentedDodecahedron *)
	"MathWorld`Polygons`"
	}
]


Antiprism::usage =
"Polyhedron[Antiprism[n]] generates an n-sided regular antiprism of side and \
lateral edge length 1.  Polyhedron[Antiprism[n,h]] generates an n-sided regular \
antiprism of side length 1 and height h.  Polyhedron[Antiprism[n,h,s]] generates \
an n-sided regular antiprism of side length s and height h.  \
Net[Antiprism[n]] gives the net of the n-gonal antiprism of height 1.  \
Net[Antiprism[n,h]] gives the net of the n-gonal antiprism of height h."

AugmentedDodecahedron::usage =
"Polyhedron[AugmentedDodecahedron] gives an augemented dodecahedron.  \
Polyhedron[AugmentedDodecahedron[l]] gives a dodecehedron with faces \
augmented."

AugmentedPrism::usage =
"Polyhedron[AugmentedPrism[n]] gives an n-gonal prism with square pyramid \
attached to one lateral face."

Cupola::usage =
"Polyhedron[Cupola[n]] gives the n-gonal cupola of unit edge lengths (which only \
exists for n=3, 4, 5.  Polyhedron[Cupola[n,h]] gives the n-gonal cupola of height h."

Dipyramid::usage =
"Polyhedron[Dipyramid[{v1,...,vn},{x,y,h}]] gives the dipyramid with base v and vertex \
{x,y,n}.  Polyhedron[Dipyramid[{v1,...,vn},h]] gives the dipyramid with base v and \
vertex {0,0,h}."

ElongatedCupola::usage =
"Polyhedron[ElongatedCupola[n]] gives a list of polygons representing the n-gonal \
elongated cupola."

ElongatedDipyramid::usage =
"Polyhedron[ElongatedDipyramid[n]] gives a list of polygons representing the n-gonal \
elongated dipyramid."

ElongatedGyrobicupola::usage =
"Polyhedron[ElongatedGyrobicupola[n]] gives an elongated gyrobicupola."

ElongatedGyrobirotunda::usage =
"Polyhedron[ElongatedGyrobirotunda[n]] gives an elongated gyrobirotunda \
for values of n sufficiently close to 5."

ElongatedGyrocupolarotunda::usage =
"Polyhedron[ElongatedGyrocupolarotunda[n]] gives the elongated orthocupolarotunda \
for values of n sufficiently close to 5."

ElongatedOrthobicupola::usage =
"Polyhedron[ElongatedOrthobicupola[n]] gives the elongated orthobicupola."

ElongatedOrthobirotunda::usage =
"Polyhedron[ElongatedOrthobirotunda[n]] gives the elongated orthobirotunda \
for values of n sufficiently close to 5."

ElongatedOrthocupolarotunda::usage =
"Polyhedron[ElongatedOrthocupolarotunda[n]] gives the elongated orthocupolarotunda \
for values of n sufficiently close to 5."

ElongatedPyramid::usage =
"Polyhedron[ElongatedPyramid[n]] gives a list of polygons representing the n-gonal \
elongated pyramid."

ElongatedRotunda::usage =
"Polyhedron[ElongatedRotunda[5]] gives a list of polygons representing the 5-gonal \
elongated rotunda."

Frustum::usage =
"Polyhedron[Frustum[{v1,...,vn},{x,y,h},s]] gives the frustun with base v, top center \
{x,y,h}, and top scaled by a factor s.  Polyhedron[Frustum[{v1,...,vn},h,s]] gives the \
frustum with base v, top centered at {0,0,h}, and top scaled by a factor s."

GoldenRhombohedron::usage =
"GoldenRhombohedron gives a golden rhombohedron."

Gyrobicupola::usage =
"Polyhedron[Gyrobicupola[n]] gives the n-gonal gyrobicupola."

Gyrobifastigium::usage =
"Polyhedron[Gyrobifastigium] gives the gyrobifastigium."

Gyrocupolarotunda::usage =
"Polyhedron[Gyrocupolarotunda[n]] gives the n-gonal gyrocupolarotunda \
for values of n sufficiently close to 5."

GyroelongatedBicupola::usage =
"Polyhedron[GyroelongatedBicupola[n]] gives the gyroelongated bicupola \
of n sides."

GyroelongatedBirotunda::usage =
"Polyhedron[GyroelongatedBirotunda[n]] gives the gyroelongated birotunda \
of n sides for values of n sufficiently close to 5."

GyroelongatedCupola::usage =
"Polyhedron[GyroelongatedCupola[n]] gives the gyroelongated cupola on \
a 2n-gonal base."

GyroelongatedCupolarotunda::usage =
"Polyhedron[GyroelongatedCupolarotunda[n]] gives the n-gonal \
gyroelongatedcupolarotunda for values of n sufficiently close to 5."

GyroelongatedPyramid::usage =
"Polyhedron[GyroelongatedPyramid[n]] gives the gyroelongated pyramid on \
an n-gonal base."

GyroelongatedRotunda::usage =
"Polyhedron[GyroelongatedRotunda[n]] gives the gyroelongated pentagonal rotunda \
for values of n sufficiently close to 5."

Orthobicupola::usage =
"Polyhedron[Orthobicupola[n]] gives the orthobicupola on n vertices."

Orthobirotunda::usage =
"Polyhedron[Orthobirotunda[n]] gives the pentagonal orthobirotunda for values of \
n sufficiently close to 5."

Orthocupolarotunda::usage =
"Polyhedron[Orthocupolarotunda[n]] gives the orthocupolarotunda for values of n \
sufficiently close to 5."

PentagonalPrism6Compound::usage =
"Polyhedron[PentagonalPrism6Compound] gives the analytic vertices of \
a compound of six pentagonal pyramids."

Prism::usage =
"Polyhedron[Prism[n]] generates an n-sided regualar prism.  Prism[{v1,...,vn},h,{x,y}] \
generates a prism having top and bottom of shape v, height h, and top-bottom offset of \
{x,y}.  Net[Prism[n]] gives the net of the n-gonal prism of height 1.  Net[Prism[n,h]] \
gives the net of the n-gonal prism of height h."

Pyramid::usage =
"Polyhedron[Pyramid[n,h]] gives a right regular n-sided pyramid of height h.  \
Polyhedron[Pyramid[{v1,...,vn},h]] gives the pyramid with base v and vertex {0,0,h}.  \
Polyhedron[Pyramid[{v1,...,vn},{x,y,h}]] gives the pyramid with base v and vertex \
{x,y,n}."

PyramidalFrustum::usage =
"Polyhedron[PyramidalFrustum[n,h,s]] gives a pyramidal frustum of n sides, \
height h, and top scaled by s."

Rhombohedron::usage =
"Polyhedron[Rhombohedron[{vx,vy}]] gives a rhombohedron using {1,0,0} and \
{vx,vy,0} as basis vectors in the plane."

Rotunda::usage =
"Polyhedron[Rotunda[n]] gives the n-rotunda, with the only true member corresponding \
to n=5."

SolidWedge::usage =
"Polyhedron[SolidWedge[{a,b},h,x]] gives a wedge with base rectangle a x b, height h, \
and indents x on the top.  Polyhedron[SolidWedge[{a,b},h]] gives \
Polyhedron[SolidWedge[{a,b},h,0]]."



Begin["`Private`"]

addn[n_,k_,m_]:=Mod[n+k-1,m]+1

(* Antiprism *)

antiprism[n_Integer?(#>2&)]:=antiprism[n,Sqrt[1-Sec[Pi/2/n]^2/4]]

antiprism[n_Integer?(#>2&),h_]:=antiprism[n,h,1]

antiprism[n_Integer?(#>2&),h_,s_]:=Module[
  {i,R=s Csc[Pi/n]/2,z=h/2},
  Orient[Flatten[{
    Polygon[Table[rim2[-z,R,n][[i]],{i,n}]],
    Polygon[Table[rim1[z,R,n][[i]],{i,n}]],
    Table[Polygon[{rim1[z,R,n][[i]],rim1[z,R,n][[i+1]],
    	rim2[-z,R,n][[i]]}],{i,n}],
    Table[Polygon[{rim2[-z,R,n][[i]],rim2[-z,R,n][[i+1]],
    	rim1[z,R,n][[i+1]]}],{i,n}]
   }]]
]

PolyhedronName[Antiprism[n_Integer,args___]]:=PolygonalName[n]<>" antiprism" /;
	n>2 && Head[PolygonalName[n]]=!=PolygonalName
PolyhedronName[Antiprism[args__]]:="antiprism"
PolyhedronFaces[Antiprism[args__]]:=PolyhedronFaces[antiprism[args]]
PolyhedronVertices[Antiprism[args__]]:=PolyhedronVertices[antiprism[args]]

antiprismNet[n_,h_:1]:=antiprismNet[n,h,1]

antiprismNet[n_,h_,s_]:=Module[
	{half=Floor[n/2],r=s Cot[Pi/n]/2,R=s Csc[Pi/n]/2,
	bot,top,ribbon1,ribbon2,
	i,x,dx,ddx,dy},
    dy=If[OddQ[n],R,r];
    dx=If[OddQ[n],1/2,0];
    ddx=If[OddQ[n],0,1];
    bot=Table[{s(dx+half+1/2)+R Sin[2Pi(i-3/2)/n],dy+R Cos[2Pi(i-3/2)/n]},
    	{i,0,n}];
    top=Table[{s(dx+half)+R Sin[2Pi (i-3/2)/n],dy+h-R Cos[2Pi(i-3/2)/n]+2r},
    	{i,0,n}];
    ribbon1=Table[{{s x,r+dy},{s(x+1/2),r+h+dy},{s(x+1),r+dy},{s x,r+dy}},
    	{x,dx,dx+n-1}];
    ribbon2=Table[{{s(x-1/2),r+h+dy},{s x,r+dy},{s(x+1/2),r+h+dy},{s(x-1/2),r+h+dy}},
    	{x,ddx+dx,ddx+dx+n-1}];
    Flatten[{Line/@{bot,top},Line/@ribbon1,Line/@ribbon2}]
]

NetFaces[Antiprism[args__]]:=NetFaces[antiprismNet[args]]
NetVertices[Antiprism[args__]]:=NetVertices[antiprismNet[args]]

(* Augmented Dodecahedron *)

augmentedDodecahedron[l_List:{1}]:=Module[
	{d=Flatten[Polyhedron[Dodecahedron]],h=Sqrt[1-Csc[Pi/5]^2/4]},
    {drop[d,l],CumulateFace[d[[#]],h]&/@l}
]

PolyhedronName[AugmentedDodecahedron[args___]]:="augmented dodecahedron"
PolyhedronFaces[AugmentedDodecahedron[args___]]:=
	PolyhedronFaces[augmentedDodecahedron[args]]
PolyhedronVertices[AugmentedDodecahedron[args___]]:=
	PolyhedronVertices[augmentedDodecahedron[args]]

(* Augmented Prism *)

augmentedPrism[n_Integer?(3<=#<=6&),l0_List:{1}]:=Module[
	{p=Polyhedron[Prism[n]],h=Sqrt[1-Csc[Pi/4]^2/4],l=l0+2},
    {drop[p,l],CumulateFace[p[[#]],h]&/@l}
]

PolyhedronName[AugmentedPrism[args__]]:="augmented prism"
PolyhedronFaces[AugmentedPrism[args__]]:=PolyhedronFaces[augmentedPrism[args]]
PolyhedronVertices[AugmentedPrism[args__]]:=PolyhedronVertices[augmentedPrism[args]]

(* Cupola *)

cupola[n_,h_:0]:=Module[{
      i,top,bot,
      r=Csc[Pi/n]/2,
      R=Csc[Pi/2/n]/2
	},
    z=If[h==0,Sqrt[1-Csc[Pi/n]^2/4],h];
    bot=Table[{R Cos[#],R Sin[#],0}&[Pi(2i+1)/2/n],{i,2n}];
    top=Table[{r Cos[#],r Sin[#],z}&[2Pi i/n],{i,n}];
	{
      Polygon/@{top,Reverse[bot]},
      Polygon/@Table[{top[[addn[i,1,n]]],top[[i]],bot[[2i]],bot[[addn[2i,1,2n]]]},
        {i,n}],
      Polygon/@Table[Reverse[{top[[i]],bot[[2i]],bot[[addn[2i,-1,2n]]]}],{i,n}]
	}
]

PolyhedronName[Cupola[args__]]:="cupola"
PolyhedronFaces[Cupola[args__]]:=PolyhedronFaces[cupola[args]]
PolyhedronVertices[Cupola[args__]]:=PolyhedronVertices[cupola[args]]

(* Dipyramid *)

dipyramid[v_List,{x_,y_,h_},opts___]:=Module[
	{base=Map[Append[#,0]&,Partition[v,2,1,1],{2}]},
	Map[Polygon,{Append[#,{x,y,h}]&/@base,Reverse/@(Append[#,{x,y,-h}]&/@base)},{2}]
]

dipyramid[v_List,h_:1,opts___]:=dipyramid[v,{0,0,h},opts]

dipyramid[n_Integer /; n>=3,{x_,y_,h_},opts___]:=Module[{i,v,R=Csc[Pi/n]/2},
	v=Table[R{Cos[2Pi i/n],Sin[2Pi i/n]},{i,n}];
	dipyramid[v,{x,y,h},opts]
]

dipyramid[n_Integer /; n>=3,h_,opts___]:=dipyramid[n,{0,0,h},opts]

dipyramid[n_Integer /; n>=3,opts___]:=Module[{R=Csc[Pi/n]/2},
	dipyramid[n,{0,0,Sqrt[1-R^2]},opts]
]

PolyhedronName[Dipyramid[n_Integer,args___]]:=PolygonalName[n]<>" dipyramid" /;
	n>2 && Head[PolygonalName[n]]=!=PolygonalName
PolyhedronName[Dipyramid[args__]]:="dipyramid"
PolyhedronFaces[Dipyramid[args__]]:=PolyhedronFaces[dipyramid[args]]
PolyhedronVertices[Dipyramid[args__]]:=PolyhedronVertices[dipyramid[args]]


drop[l_List,items_List]:=Part[l,Complement[Range[Length[l]],items]]


(* Elongated Cupola *)

elongatedCupola[n_Integer?(3<=#<=5&)]:=Module[{c=cupola[n],bot,botbot},
    bot=c[[1,2]]/.Polygon:>Identity;
    botbot={0,0,-1}+#&/@bot;
    {Rest[c],c[[1,1]],Polygon[botbot],ribbon[2n,1,-1/2]}
]

PolyhedronName[ElongatedCupola[args_]]:="elongated cupola"
PolyhedronFaces[ElongatedCupola[args_]]:=PolyhedronFaces[elongatedCupola[args]]
PolyhedronVertices[ElongatedCupola[args_]]:=PolyhedronVertices[elongatedCupola[args]]

(* Elongated Dipyramid *)

elongatedDipyramid[v_List,{x_,y_,h_},opts___]:=Module[
	{p=Partition[v,2,1,1],base1,base2},
	base1=Map[Append[#,1/2]&,p,{2}];
	base2=Map[Append[#,-1/2]&,p,{2}];
	{Map[Polygon,{
		Append[#,{x,y,h+1/2}]&/@base1,
		Reverse/@(Append[#,{x,y,-(h+1/2)}]&/@base2)},
	{2}],
	Polygon/@Reverse/@(Join[#[[1]],Reverse[#[[2]]]]&/@Transpose[{base1,base2}])
	}
]

elongatedDipyramid[v_List,h_:1,opts___]:=elongatedDipyramid[v,{0,0,h},opts]

elongatedDipyramid[n_Integer /; n>=3,{x_,y_,h_},opts___]:=Module[{i,v,R=Csc[Pi/n]/2},
	v=Table[R{Cos[2Pi i/n],Sin[2Pi i/n]},{i,n}];
	elongatedDipyramid[v,{x,y,h},opts]
]

elongatedDipyramid[n_Integer /; n>=3,h_,opts___]:=dipyramid[n,{0,0,h},opts]

elongatedDipyramid[n_Integer /; n>=3,opts___]:=Module[{R=Csc[Pi/n]/2},
	elongatedDipyramid[n,{0,0,Sqrt[1-R^2]},opts]
]

PolyhedronName[ElongatedDipyramid[args_]]:="elongated dipyramid"
PolyhedronFaces[ElongatedDipyramid[args_]]:=PolyhedronFaces[elongatedDipyramid[args]]
PolyhedronVertices[ElongatedDipyramid[args_]]:=PolyhedronVertices[elongatedDipyramid[args]]

(* Elongated Gyrobicupola*)

elongatedGyrobicupola[n_Integer?(3<=#<=5&)]:=Module[
	{
		cup={#[[1,1]],Rest[#]}&@cupola[n],top,bot,l,
		c=Cos[2Pi/2/n],s=Sin[2Pi/2/n]
	},
    top=cup/.Polygon[l_List]:>Polygon[{0,0,1/2}+#&/@l];
    bot=cup/.Polygon[l_List]:>Polygon[{0,0,-1/2}+{1,1,-1}#&/@l];
    {
    	top,
    	bot/.Polygon[l_List]:>Polygon[Reverse[{{c,s,0},{-s,c,0},{0,0,1}}.#&/@l]],
		ribbon[2n,1]
    }
]

PolyhedronName[ElongatedGyrobicupola[args_]]:="elongated gyrobicupola"
PolyhedronFaces[ElongatedGyrobicupola[args_]]:=PolyhedronFaces[elongatedGyrobicupola[args]]
PolyhedronVertices[ElongatedGyrobicupola[args_]]:=PolyhedronVertices[elongatedGyrobicupola[args]]

(* Elongated Gyrobirotunda *)

elongatedGyrobirotunda[n_Integer?(#==5&)]:=Module[
	{
		rot=First[rotunda[n]],l,
		c=Cos[2Pi/2/n],s=Sin[2Pi/2/n]
	},
	{
    	rot/.Polygon[l_List]:>Polygon[{0,0,1/2}+#&/@l],
    	rot/.Polygon[l_List]:>
    		Polygon[Reverse[{0,0,-1/2}+{{c,s,0},{-s,c,0},{0,0,-1}}.#&/@l]],
    	ribbon[10,1]
    }
]

PolyhedronName[ElongatedGyrobirotunda[args_]]:="elongated gyrobirotunda"
PolyhedronFaces[ElongatedGyrobirotunda[args_]]:=PolyhedronFaces[elongatedGyrobirotunda[args]]
PolyhedronVertices[ElongatedGyrobirotunda[args_]]:=PolyhedronVertices[elongatedGyrobirotunda[args]]

(* Elongated Gyrocupolarotunda *)

elongatedGyrocupolarotunda[n_Integer?(#==5&)]:=Module[
	{
		cup={#[[1,1]],Rest[#]}&@cupola[n],
		rot=First[rotunda[n]],
		l
	},
	{
		rot/.Polygon[l_List]:>Polygon[{0,0,1/2}+#&/@l],
		cup/.Polygon[l_List]:>Polygon[Reverse[({0,0,-1/2}+{1,1,-1}#)&/@l]],
    	ribbon[2n,1]
	}
]

PolyhedronName[ElongatedGyrocupolarotunda[args_]]:="elongated gyrocupolarotunda"
PolyhedronFaces[ElongatedGyrocupolarotunda[args_]]:=PolyhedronFaces[elongatedGyrocupolarotunda[args]]
PolyhedronVertices[ElongatedGyrocupolarotunda[args_]]:=PolyhedronVertices[elongatedGyrocupolarotunda[args]]

(* Elongated Orthobicupola *)

elongatedOrthobicupola[n_Integer?(3<=#<=5&)]:=Module[{c=cupola[n],top,bot},
    top=c/.Polygon[l_List]:>Polygon[{0,0,1/2}+#&/@l];
    bot=c/.Polygon[l_List]:>Polygon[Reverse[{0,0,-1/2}+{1,1,-1}#&/@l]];
    {Rest[top],top[[1,1]],Rest[bot],bot[[1,1]],ribbon[2n,1]}
]

PolyhedronName[ElongatedOrthobicupola[args_]]:="elongated orthobicupola"
PolyhedronFaces[ElongatedOrthobicupola[args_]]:=
	PolyhedronFaces[elongatedOrthobicupola[args]]
PolyhedronVertices[ElongatedOrthobicupola[args_]]:=
	PolyhedronVertices[elongatedOrthobicupola[args]]

(* Elongated Orthobirotunda *)

elongatedOrthobirotunda[n_Integer?(#==5&)]:=Module[{c=First[rotunda[n]],l},
    {
    	c/.Polygon[l_List]:>Polygon[{0,0,1/2}+#&/@l],
    	c/.Polygon[l_List]:>Polygon[Reverse[{0,0,-1/2}+{1,1,-1}#&/@l]],
    	ribbon[10,1]
    }
]

PolyhedronName[ElongatedOrthobirotunda[args_]]:="elongated orthobitotunda"
PolyhedronFaces[ElongatedOrthobirotunda[args_]]:=PolyhedronFaces[elongatedOrthobirotunda[args]]
PolyhedronVertices[ElongatedOrthobirotunda[args_]]:=PolyhedronVertices[elongatedOrthobirotunda[args]]

(* Elongated Orthocupolarotunda *)

elongatedOrthocupolarotunda[n_Integer?(#==5&)]:=Module[
	{
		cup={#[[1,1]],Rest[#]}&@cupola[n],
		rot=First[rotunda[n]],
		s=Sin[2Pi/2/n],c=Cos[2Pi/2/n]
	},
	{
		rot/.Polygon[l_List]:>Polygon[{0,0,1/2}+#&/@l],
		cup/.Polygon[l_List]:>
			Polygon[Reverse[{0,0,-1/2}+{{c,s,0},{-s,c,0},{0,0,-1}}.#&/@l]],
		ribbon[2n,1]
	}
]

PolyhedronName[ElongatedOrthocupolarotunda[args_]]:="elongated orthocupolarotunda"
PolyhedronFaces[ElongatedOrthocupolarotunda[args_]]:=PolyhedronFaces[elongatedOrthocupolarotunda[args]]
PolyhedronVertices[ElongatedOrthocupolarotunda[args_]]:=PolyhedronVertices[elongatedOrthocupolarotunda[args]]

(* Elongated Pyramid *)

elongatedPyramid[v_List,{x_,y_,h_},opts___]:=Module[
	{p=Partition[v,2,1,1],base1,base2},
	base1=Map[Append[#,1]&,p,{2}];
	base2=Map[Append[#,0]&,p,{2}];
	{Polygon/@(Append[#,{x,y,h+1}]&/@base1),
	Polygon/@Reverse/@(Join[#[[1]],Reverse[#[[2]]]]&/@Transpose[{base1,base2}]),
	Polygon[Append[#,0]&/@Reverse[v]]}
]

elongatedPyramid[v_List,h_:1,opts___]:=elongatedPyramid[v,{0,0,h},opts]

elongatedPyramid[n_Integer /; n>=3,{x_,y_,h_},opts___]:=Module[{i,v,R=Csc[Pi/n]/2},
	v=Table[R{Cos[2Pi i/n],Sin[2Pi i/n]},{i,n}];
	elongatedPyramid[v,{x,y,h},opts]
]

elongatedPyramid[n_Integer /; n>=3,h_,opts___]:=pyramid[n,{0,0,h},opts]

elongatedPyramid[n_Integer /; n>=3,opts___]:=Module[{R=Csc[Pi/n]/2},
	elongatedPyramid[n,{0,0,Sqrt[1-R^2]},opts]
]

PolyhedronName[ElongatedPyramid[args__]]:="elongated pyramid"
PolyhedronFaces[ElongatedPyramid[args__]]:=PolyhedronFaces[elongatedPyramid[args]]
PolyhedronVertices[ElongatedPyramid[args__]]:=PolyhedronVertices[elongatedPyramid[args]]

(* Elongated Rotunda *)

elongatedRotunda[n_Integer?(#==5&)]:=Module[{r=rotunda[n],bot,botbot},
	bot=Last[r]/.Polygon:>Identity;
    botbot={0,0,-1}+#&/@bot;
    {Drop[r,-1],Polygon[botbot],ribbon[2n,1,-1/2]}
]

PolyhedronName[ElongatedRotunda[args_]]:="elongated rotunda"
PolyhedronFaces[ElongatedRotunda[args_]]:=PolyhedronFaces[elongatedRotunda[args]]
PolyhedronVertices[ElongatedRotunda[args_]]:=PolyhedronVertices[elongatedRotunda[args]]

(* Frustum *)

frustum[v_List,{x_,y_,h_},s_]:=Module[{i,n=Length[v],g,bot,top},
	g=Append[Plus@@v/n,0];
	bot=Table[{v[[i,1]],v[[i,2]],0}-g,{i,n}];
	top=Table[g+s bot[[i]]+{x,y,h},{i,n}];
	{
	Polygon/@{top,bot},
	Polygon/@Table[{bot[[i]],bot[[addn[i,1,n]]],top[[addn[i,1,n]]],top[[i]]},{i,n}]
	}
]

frustum[v_List,h_,s_]:=frustum[v,{0,0,h},s]

PolyhedronName[Frustum[args__]]:="frustum"
PolyhedronFaces[Frustum[args__]]:=PolyhedronFaces[frustum[args]]
PolyhedronVertices[Frustum[args__]]:=PolyhedronVertices[frustum[args]]

(* Golden Rhombohedron *)

PolyhedronName[GoldenRhombohedron]:="golden rhombohedron"
GoldenRhombohedron:=Rhombohedron[{1,2}/Sqrt[5]]

(* Gyrobicupola *)

gyrobicupola[n_Integer?(3<=#<=5&)]:=Module[{cup={#[[1,1]],Rest[#]}&@cupola[n],
	c=Cos[2Pi/n/2],s=Sin[2Pi/n/2]},
    {cup,cup/.Polygon[l_List]:>Polygon[Reverse[{{c,s,0},{-s,c,0},{0,0,-1}}.#&/@l]]}
]

PolyhedronName[Gyrobicupola[args_]]:="gyrobicupola"
PolyhedronFaces[Gyrobicupola[args_]]:=
	PolyhedronFaces[gyrobicupola[args]]
PolyhedronVertices[Gyrobicupola[args_]]:=
	PolyhedronVertices[gyrobicupola[args]]

(* Gyrobifastigium *)

gyrobifastigium:=Module[{i,h=Sqrt[3],
      v1={1,-1,0},v2={1,1,0},v3={-1,-1,0},
      v4={-1,1,0},v5,v6,
      c=Cos[2Pi/4],s=Sin[2Pi/4],
      l},
    v5={1,0,h};
    v6={-1,0,h};
    l={{v1,v2,v5},{v6,v4,v3},Reverse[{v1,v3,v6,v5}],Reverse[{v4,v2,v5,v6}]}/2;
    {Polygon/@l,
    Polygon/@Reverse/@Map[{{c,s,0},{-s,c,0},{0,0,-1}}.#&,l,{2}]}
]

PolyhedronName[Gyrobifastigium]:="gyrobifastigium"
PolyhedronFaces[Gyrobifastigium]:=PolyhedronFaces[gyrobifastigium]
PolyhedronVertices[Gyrobifastigium]:=PolyhedronVertices[gyrobifastigium]

(* Gyrocupolarotunda *)

gyrocupolarotunda[n_Integer?(#==5&)]:=
  Module[{cup={#[[1,1]],Rest[#]}&@cupola[n],rot=First[rotunda[n]],
      s=Sin[2Pi/2/n],c=Cos[2Pi/2/n]},
    {rot,cup/.Polygon[l_List]:>Polygon[Reverse[{1,1,-1}#&/@l]]}
]

PolyhedronName[Gyrocupolarotunda[args_]]:="gyrocupolarotunda"
PolyhedronFaces[Gyrocupolarotunda[args_]]:=
	PolyhedronFaces[gyrocupolarotunda[args]]
PolyhedronVertices[Gyrocupolarotunda[args_]]:=
	PolyhedronVertices[gyrocupolarotunda[args]]

(* Gyroelongated Bicupola *)

gyroelongatedBicupola[n_Integer?(3<=#<=5&)]:=Module[
	{
		a=Drop[Polyhedron[Antiprism[2n]],{1,2}],
		cup={#[[1,1]],Rest[#]}&@cupola[n],
		h=Sqrt[1-Sec[Pi/2/n]^2/4],
		s=Sin[2Pi/4/n],
		c=Cos[2Pi/4/n]
    },
    {a,
    cup/.Polygon[l_List]:>
    	Polygon[{0,0,h/2}+{{c,s,0},{-s,c,0},{0,0,1}}.#&/@l],
	cup/.Polygon[l_List]:>Polygon[Reverse[{0,0,-h/2}-#&/@l]]
	}
]

PolyhedronName[GyroelongatedBicupola[args_]]:="gyroelongated bicupola"
PolyhedronFaces[GyroelongatedBicupola[args_]]:=
	PolyhedronFaces[gyroelongatedBicupola[args]]
PolyhedronVertices[GyroelongatedBicupola[args_]]:=
	PolyhedronVertices[gyroelongatedBicupola[args]]

(* Gyroelongated Birotunda *)

gyroelongatedBirotunda[n_Integer?(#==5&)]:=
  Module[{a=Drop[Polyhedron[Antiprism[2n]],{1,2}],
      rot=First[rotunda[n]],
      h=Sqrt[1-Sec[Pi/2/n]^2/4],s=Sin[2Pi/4/n],c=Cos[2Pi/4/n]},
      {a,
      rot/.Polygon[l_List]:>
          Polygon[{0,0,h/2}+{{c,s,0},{-s,c,0},{0,0,1}}.#&/@l],
      rot/.Polygon[l_List]:>Polygon[Reverse[{0,0,-h/2}+{1,1,-1}#&/@l]]}
]

PolyhedronName[GyroelongatedBirotunda[args_]]:="gyroelongated birotunda"
PolyhedronFaces[GyroelongatedBirotunda[args_]]:=
	PolyhedronFaces[gyroelongatedBirotunda[args]]
PolyhedronVertices[GyroelongatedBirotunda[args_]]:=
	PolyhedronVertices[gyroelongatedBirotunda[args]]

(* Gyroelongated Cupola *)

gyroelongatedCupola[n_Integer?(3<=#<=5&)]:=Module[
	{a=Polyhedron[Antiprism[2n]],c=cupola[n],l},
      {Rest[a] /. Polygon[l_]:>Polygon[Reverse[{0,0,a[[1,1,1,-1]]}-#&/@l]],
      c[[1,1]],Rest[c]}
]

PolyhedronName[GyroelongatedCupola[args_]]:="gyroelongated cupola"
PolyhedronFaces[GyroelongatedCupola[args_]]:=
	PolyhedronFaces[gyroelongatedCupola[args]]
PolyhedronVertices[GyroelongatedCupola[args_]]:=
	PolyhedronVertices[gyroelongatedCupola[args]]

(* Gyroelongated Cupolarotunda *)

gyroelongatedCupolarotunda[n_Integer?(#==5&)]:=
  Module[{a=Drop[Polyhedron[Antiprism[2n]],{1,2}],cup={#[[1,1]],Rest[#]}&@cupola[n],
      rot=First[rotunda[n]],
      h=Sqrt[1-Sec[Pi/2/n]^2/4],s=Sin[2Pi/4/n],c=Cos[2Pi/4/n]},
      {a,
      cup/.Polygon[l_List]:>Polygon[Reverse[{0,0,-h/2}+{1,1,-1}#&/@l]],
      rot/.Polygon[l_List]:>
          Polygon[{0,0,h/2}+{{c,s,0},{-s,c,0},{0,0,1}}.#&/@l]
      }
]

PolyhedronName[GyroelongatedCupolarotunda[args_]]:="gyroelongated cupolarotunda"
PolyhedronFaces[GyroelongatedCupolarotunda[args_]]:=
	PolyhedronFaces[gyroelongatedCupolarotunda[args]]
PolyhedronVertices[GyroelongatedCupolarotunda[args_]]:=
	PolyhedronVertices[gyroelongatedCupolarotunda[args]]

(* Gyroelongated Pyramid *)

gyroelongatedPyramid[n_Integer?(3<=#<=5&)]:=
	Module[{a=Polyhedron[Antiprism[n]],p=Polyhedron[Pyramid[n]],l},
	{Drop[a,{2}],
      Most[p]/.Polygon[l_List]:>Polygon[{0,0,a[[2,1,1,-1]]}+#&/@l]
	}
]

PolyhedronName[GyroelongatedPyramid[args_]]:="gyroelongated pyramid"
PolyhedronFaces[GyroelongatedPyramid[args_]]:=
	PolyhedronFaces[gyroelongatedPyramid[args]]
PolyhedronVertices[GyroelongatedPyramid[args_]]:=
	PolyhedronVertices[gyroelongatedPyramid[args]]

(* Gyroelongated Rotunda *)

gyroelongatedRotunda[n_Integer?(#==5&)]:=
  Module[{a=Polyhedron[Antiprism[2n]],r=rotunda[n],l},
      {Drop[a,{1}] /. Polygon[l_]:>Polygon[Reverse[{0,0,a[[1,1,1,-1]]}-#&/@l]],
      First[r]}
]

PolyhedronName[GyroelongatedRotunda[args_]]:="gyroelongated rotunda"
PolyhedronFaces[GyroelongatedRotunda[args_]]:=
	PolyhedronFaces[gyroelongatedRotunda[args]]
PolyhedronVertices[GyroelongatedRotunda[args_]]:=
	PolyhedronVertices[gyroelongatedRotunda[args]]

(* Orthobicupola *)

orthobicupola[n_Integer?(3<=#<=5&)]:=Module[
	{cup={#[[1,1]],Rest[#]}&@cupola[n],l},
    {cup,cup/.Polygon[l_List]:>Polygon[Reverse[{1,1,-1}#&/@l]]}
]

PolyhedronName[Orthobicupola[args_]]:="orthobicupola"
PolyhedronFaces[Orthobicupola[args_]]:=
	PolyhedronFaces[orthobicupola[args]]
PolyhedronVertices[Orthobicupola[args_]]:=
	PolyhedronVertices[orthobicupola[args]]

(* Orthobirotunda *)

orthobirotunda[n_Integer?(#==5&)]:=Module[{r=First[rotunda[n]]},
    {r,r/.Polygon[l_List]:>Polygon[Reverse[{1,1,-1}#&/@l]]}
]

PolyhedronName[Orthobirotunda[args_]]:="orthobirotunda"
PolyhedronFaces[Orthobirotunda[args_]]:=
	PolyhedronFaces[orthobirotunda[args]]
PolyhedronVertices[Orthobirotunda[args_]]:=
	PolyhedronVertices[orthobirotunda[args]]

(* Orthocupolarotunda *)
		
orthocupolarotunda[n_]:=
  Module[{cup={#[[1,1]],Rest[#]}&@cupola[n],rot=First[rotunda[n]],
      s=Sin[2Pi/2/n],c=Cos[2Pi/2/n]},
    {rot,
    	cup/.Polygon[l_List]:>Polygon[Reverse[{{c,s,0},{-s,c,0},{0,0,-1}}.#&/@l]]}
]

PolyhedronName[Orthocupolarotunda[args_]]:="orthocupolarotunda"
PolyhedronFaces[Orthocupolarotunda[args_]]:=
	PolyhedronFaces[orthocupolarotunda[args]]
PolyhedronVertices[Orthocupolarotunda[args_]]:=
	PolyhedronVertices[orthocupolarotunda[args]]

(* Pentagonal Prism 6-compound *)

PolyhedronName[PentagonalPrism6Compound]:="pentagonal prism 6-compound"
NetPieces[PentagonalPrism6Compound]:={{60,{1,2,3}}}
NetFaces[PentagonalPrism6Compound]:={{11,3,9,8,12,2,1},{1,13,10,4,11},{2,5,7,6,1}}
NetVertices[PentagonalPrism6Compound]:=
{{0, 0}, {0, (3 - Sqrt[5])/2}, {(5 - 3*Sqrt[5])/10, (5 - 2*Sqrt[5])/5}, 
 {(1 - Sqrt[5])/4, -Sqrt[5/8 - 11/(8*Sqrt[5])]}, 
 {(3 - Sqrt[5])/5, (3 - Sqrt[5])/10}, {(5 - Sqrt[5])/10, 0}, 
 {(5 - Sqrt[5])/10, (5 - Sqrt[5])/20}, {(-5 + Sqrt[5])/10, (5 + Sqrt[5])/20}, 
 {(-5 + Sqrt[5])/10, (-5 + 3*Sqrt[5])/10}, 
 {(-3 + Sqrt[5])/4, -(1/Sqrt[2*(65 + 29*Sqrt[5])])}, {(-3 + Sqrt[5])/2, 0}, 
 {(-5 + 2*Sqrt[5])/5, 1/Sqrt[5]}, {(-5 + 3*Sqrt[5])/20, 
  -Sqrt[1/8 - 1/(8*Sqrt[5])]}}

PolyhedronPolyhedra[PentagonalPrism6Compound]:=Partition[Range[42],7]
PolyhedronFaces[PentagonalPrism6Compound]:=
{{50,7,40,1,9},{10,2,41,8,51},{50,7,8,51},{7,40,41,8},{40,1,2,41},{1,9,10,
    2},{9,50,51,10},{49,54,22,19,42},{29,33,3,43,58},{49,54,43,58},{54,22,3,
    43},{22,19,33,3},{19,42,29,33},{42,49,58,29},{27,21,37,35,6},{44,30,16,11,
    56},{27,21,11,56},{21,37,16,11},{37,35,30,16},{35,6,44,30},{6,27,56,
    44},{31,20,14,39,15},{25,57,24,28,48},{31,20,28,48},{20,14,24,28},{14,39,
    57,24},{39,15,25,57},{15,31,48,25},{60,26,4,34,47},{5,13,17,52,53},{60,26,
    52,53},{26,4,17,52},{4,34,13,17},{34,47,5,13},{47,60,53,5},{55,12,46,32,
    59},{45,38,36,18,23},{55,12,18,23},{12,46,36,18},{46,32,38,36},{32,59,45,
    38},{59,55,23,45}}
PolyhedronVertices[PentagonalPrism6Compound]:=
{{-1/2, -Sqrt[1/4 + 1/(2*Sqrt[5])], -1/2}, {-1/2, -Sqrt[1/4 + 1/(2*Sqrt[5])], 1/2}, {-1/2, Sqrt[1/4 + 1/(2*Sqrt[5])], -1/2}, 
 {-1/2, Sqrt[1/4 + 1/(2*Sqrt[5])], 1/2}, {0, -Sqrt[1/2 + 1/(2*Sqrt[5])], -1/2}, {0, -Sqrt[1/2 + 1/(2*Sqrt[5])], 1/2}, 
 {0, Sqrt[1/2 + 1/(2*Sqrt[5])], -1/2}, {0, Sqrt[1/2 + 1/(2*Sqrt[5])], 1/2}, {1/2, -Sqrt[1/4 + 1/(2*Sqrt[5])], -1/2}, 
 {1/2, -Sqrt[1/4 + 1/(2*Sqrt[5])], 1/2}, {1/2, Sqrt[1/4 + 1/(2*Sqrt[5])], -1/2}, {1/2, Sqrt[1/4 + 1/(2*Sqrt[5])], 1/2}, 
 {-3/(2*Sqrt[5]), -Sqrt[1/4 + 1/(2*Sqrt[5])], 1/(2*Sqrt[5])}, {-3/(2*Sqrt[5]), Sqrt[1/4 + 1/(2*Sqrt[5])], 1/(2*Sqrt[5])}, 
 {-(1/Sqrt[5]), -Sqrt[1/2 + 1/(2*Sqrt[5])], -1/(2*Sqrt[5])}, {-(1/Sqrt[5]), Sqrt[1/2 + 1/(2*Sqrt[5])], -1/(2*Sqrt[5])}, 
 {-1/(2*Sqrt[5]), -Sqrt[1/4 - 1/(2*Sqrt[5])], (5 + 2*Sqrt[5])/10}, {-1/(2*Sqrt[5]), Sqrt[1/4 - 1/(2*Sqrt[5])], 
  (5 + 2*Sqrt[5])/10}, {-1/(2*Sqrt[5]), -Sqrt[1/4 + 1/(2*Sqrt[5])], -3/(2*Sqrt[5])}, 
 {-1/(2*Sqrt[5]), Sqrt[1/4 + 1/(2*Sqrt[5])], -3/(2*Sqrt[5])}, {1/(2*Sqrt[5]), -Sqrt[1/4 - 1/(2*Sqrt[5])], (-5 - 2*Sqrt[5])/10}, 
 {1/(2*Sqrt[5]), Sqrt[1/4 - 1/(2*Sqrt[5])], (-5 - 2*Sqrt[5])/10}, {1/(2*Sqrt[5]), -Sqrt[1/4 + 1/(2*Sqrt[5])], 3/(2*Sqrt[5])}, 
 {1/(2*Sqrt[5]), Sqrt[1/4 + 1/(2*Sqrt[5])], 3/(2*Sqrt[5])}, {1/Sqrt[5], -Sqrt[1/2 + 1/(2*Sqrt[5])], 1/(2*Sqrt[5])}, 
 {1/Sqrt[5], Sqrt[1/2 + 1/(2*Sqrt[5])], 1/(2*Sqrt[5])}, {3/(2*Sqrt[5]), -Sqrt[1/4 + 1/(2*Sqrt[5])], -1/(2*Sqrt[5])}, 
 {3/(2*Sqrt[5]), Sqrt[1/4 + 1/(2*Sqrt[5])], -1/(2*Sqrt[5])}, {(-5 - 3*Sqrt[5])/20, -Sqrt[1/8 + 1/(8*Sqrt[5])], 3/(2*Sqrt[5])}, 
 {(-5 - 3*Sqrt[5])/20, Sqrt[1/8 + 1/(8*Sqrt[5])], 3/(2*Sqrt[5])}, {(5 - 3*Sqrt[5])/20, -Sqrt[1/8 - 1/(8*Sqrt[5])], 
  (-5 - 2*Sqrt[5])/10}, {(5 - 3*Sqrt[5])/20, Sqrt[1/8 - 1/(8*Sqrt[5])], (-5 - 2*Sqrt[5])/10}, 
 {(-5 - 2*Sqrt[5])/10, -Sqrt[1/4 - 1/(2*Sqrt[5])], -1/(2*Sqrt[5])}, {(-5 - 2*Sqrt[5])/10, Sqrt[1/4 - 1/(2*Sqrt[5])], 
  -1/(2*Sqrt[5])}, {(-15 - Sqrt[5])/20, -Sqrt[1/8 + 1/(8*Sqrt[5])], 1/(2*Sqrt[5])}, 
 {(-15 - Sqrt[5])/20, Sqrt[1/8 + 1/(8*Sqrt[5])], 1/(2*Sqrt[5])}, {(-5 - Sqrt[5])/10, 0, -3/(2*Sqrt[5])}, 
 {(-1 - Sqrt[5])/4, -Sqrt[1/8 - 1/(8*Sqrt[5])], -1/2}, {(-1 - Sqrt[5])/4, -Sqrt[1/8 - 1/(8*Sqrt[5])], 1/2}, 
 {(-1 - Sqrt[5])/4, Sqrt[1/8 - 1/(8*Sqrt[5])], -1/2}, {(-1 - Sqrt[5])/4, Sqrt[1/8 - 1/(8*Sqrt[5])], 1/2}, 
 {(5 - Sqrt[5])/20, -Sqrt[5/8 + Sqrt[5]/8], 1/(2*Sqrt[5])}, {(5 - Sqrt[5])/20, Sqrt[5/8 + Sqrt[5]/8], 1/(2*Sqrt[5])}, 
 {(5 - Sqrt[5])/10, 0, (5 + 2*Sqrt[5])/10}, {(-5 + Sqrt[5])/20, -Sqrt[5/8 + Sqrt[5]/8], -1/(2*Sqrt[5])}, 
 {(-5 + Sqrt[5])/20, Sqrt[5/8 + Sqrt[5]/8], -1/(2*Sqrt[5])}, {(-5 + Sqrt[5])/10, 0, (-5 - 2*Sqrt[5])/10}, 
 {(1 + Sqrt[5])/4, -Sqrt[1/8 - 1/(8*Sqrt[5])], -1/2}, {(1 + Sqrt[5])/4, -Sqrt[1/8 - 1/(8*Sqrt[5])], 1/2}, 
 {(1 + Sqrt[5])/4, Sqrt[1/8 - 1/(8*Sqrt[5])], -1/2}, {(1 + Sqrt[5])/4, Sqrt[1/8 - 1/(8*Sqrt[5])], 1/2}, 
 {(5 + Sqrt[5])/10, 0, 3/(2*Sqrt[5])}, {(15 + Sqrt[5])/20, -Sqrt[1/8 + 1/(8*Sqrt[5])], -1/(2*Sqrt[5])}, 
 {(15 + Sqrt[5])/20, Sqrt[1/8 + 1/(8*Sqrt[5])], -1/(2*Sqrt[5])}, {(5 + 2*Sqrt[5])/10, -Sqrt[1/4 - 1/(2*Sqrt[5])], 1/(2*Sqrt[5])}, 
 {(5 + 2*Sqrt[5])/10, Sqrt[1/4 - 1/(2*Sqrt[5])], 1/(2*Sqrt[5])}, {(-5 + 3*Sqrt[5])/20, -Sqrt[1/8 - 1/(8*Sqrt[5])], 
  (5 + 2*Sqrt[5])/10}, {(-5 + 3*Sqrt[5])/20, Sqrt[1/8 - 1/(8*Sqrt[5])], (5 + 2*Sqrt[5])/10}, 
 {(5 + 3*Sqrt[5])/20, -Sqrt[1/8 + 1/(8*Sqrt[5])], -3/(2*Sqrt[5])}, {(5 + 3*Sqrt[5])/20, Sqrt[1/8 + 1/(8*Sqrt[5])], 
  -3/(2*Sqrt[5])}}

(* Prism *)

PolyhedronName[Prism[n_Integer,args___]]:=PolygonalName[n]<>" prism" /;
	n>2 && Head[PolygonalName[n]]=!=PolygonalName
PolyhedronName[Prism[args__]]:="prism"

PolyhedronVertices[Prism[n_Integer?Positive,h_?Positive,s_?Positive]]:=Module[
	{i,z,R=s Csc[Pi/n]/2},
	Flatten[
		Table[{R Cos[2Pi i/n],R Sin[2Pi i/n],z h/2},{z,-1,1,2},{i,0,n-1}],
	1]
] /; n>2

PolyhedronVertices[Prism[n_Integer?Positive]]:=
	PolyhedronVertices[Prism[n,1,1]] /; n>2
PolyhedronVertices[Prism[n_Integer?Positive,h_:1]]:=
	PolyhedronVertices[Prism[n,h,1]] /; n>2

PolyhedronFaces[Prism[n_Integer?Positive,h_?Positive,s_?Positive]]:=Module[
    {r1=Range[n],r2=Range[n+1,2n]},
    Join[{Reverse[r1],r2},
      Flatten[{#1,Reverse[#2]}]&@@@Transpose[Partition[#,2,1,1]&/@{r1,r2}]
      ]
    ]
    
PolyhedronFaces[Prism[n_Integer?Positive]]:=
	PolyhedronFaces[Prism[n,1,1]] /; n>2
PolyhedronFaces[Prism[n_Integer?Positive,h_:1]]:=
	PolyhedronFaces[Prism[n,h,1]] /; n>2


Prism[v_List,h_:1]:=Prism[v,h,{0,0}]

Prism[v_List,h_,off_List]:=Module[{i,n=Length[v],bot,top,sides},
	bot=Table[Append[v[[i]],0],{i,n}];
	top=Table[Append[v[[i]]+off,h],{i,n}];
	sides=Table[{bot[[i]],bot[[addn[i,1,n]]],top[[addn[i,1,n]]],top[[i]]},{i,n}];
	Flatten[{Polygon[bot],Polygon[top],Polygon/@sides}]
]

prismNet[n_,h_:1]:=prismNet[n,h,1]

prismNet[n_,h_,s_]:=
  Module[{half=Floor[n/2],r=s Cot[Pi/n]/2,R=s Csc[Pi/n]/2,bot,top,ribbon,dy,i},
    dy=If[OddQ[n],R,r];
    bot=Table[{half s+s/2+R Sin[2Pi(i-3/2)/n],dy+R Cos[2Pi(i-3/2)/n]},{i,0,n}];
    top=Table[{half s+s/2+R Sin[2Pi(i-3/2)/n],dy+h-R Cos[2Pi(i-3/2)/n]+2r},{i,0,n}];
    ribbon=
      Table[{{s i,r+dy},{s(i+1),r+dy},{s(i+1),r+h+dy},{s i,r+h+dy},{s i,r+dy}},
      {i,0,n-1}];
    {Line/@{bot,top},Line/@ribbon}
]

NetFaces[Prism[args__]]:=NetFaces[prismNet[args]]
NetVertices[Prism[args__]]:=NetVertices[prismNet[args]]

(* Pyramid *)

pyramid[v_List,{x_,y_,h_}]:=Module[
	{base=Map[Append[#,0]&,Partition[v,2,1,1],{2}]},
	{Polygon/@(Append[#,{x,y,h}]&/@base),Polygon[Append[#,0]&/@Reverse[v]]}
]

pyramid[v_List,h_:1]:=pyramid[v,{0,0,h}]

pyramid[n_Integer /; n>=3,{x_,y_,h_}]:=Module[
	{i,v,R=Csc[Pi/n]/2},
	v=Table[R{Cos[2Pi i/n],Sin[2Pi i/n]},{i,n}];
	pyramid[v,{x,y,h}]
]

pyramid[n_Integer /; n>=3,h_]:=pyramid[n,{0,0,h}]

pyramid[n_Integer /; 3<=n<=5]:=Module[{R=Csc[Pi/n]/2},
	pyramid[n,{0,0,Sqrt[1-R^2]}]
]

pyramid[n_Integer /; n>=6]:=pyramid[n,1]

PolyhedronName[Pyramid[n_Integer,args___]]:=PolygonalName[n]<>" pyramid" /;
	n>2 && Head[PolygonalName[n]]=!=PolygonalName
PolyhedronName[Pyramid[args__]]:="pyramid"
PolyhedronFaces[Pyramid[args__]]:=PolyhedronFaces[pyramid[args]]
PolyhedronVertices[Pyramid[args__]]:=PolyhedronVertices[pyramid[args]]

(* Pyramidal Frsutum *)

pyramidalFrustum[n_Integer /; n>=3,h_:1,s_]:=Module[{i,v},
	v=Table[{Cos[2Pi i/n],Sin[2Pi i/n]},{i,n}];
	Frustum[v,h,s]
]

PolyhedronName[PyramidalFrustrum[args__]]:="pyramidal frustrum"
PolyhedronFaces[PyramidalFrustrum[args__]]:=PolyhedronFaces[pyramidalFrustrum[args]]
PolyhedronVertices[PyramidalFrustrum[args__]]:=PolyhedronVertices[pyramidalFrustrum[args]]

(* Rhombohedron *)

rhombohedron[v_]:=Module[
	{
		o={0,0,0},
		v0=Norm[v]{1,0,0},
		v1=Append[v,0],
		v2,
		theta=ArcTan@@v
	},
    v2={Cos[theta/2]Cos[theta],Sin[theta/2]Cos[theta],Sin[theta]}Norm[v];
	Polygon/@{
		{o,v1,v0+v1,v0},{o,v0,v0+v2,v2},{o,v2,v2+v1,v1},
		{v2,v0+v2,v0+v1+v2,v1+v2},{v1,v1+v2,v0+v1+v2,v0+v1},{v0,v0+v1,v0+v1+v2,v0+v2}
	}
]

PolyhedronName[Rhombohedron[args_]]:="rhombohedron"
PolyhedronFaces[Rhombohedron[args_]]:=PolyhedronFaces[rhombohedron[args]]
PolyhedronVertices[Rhombohedron[args_]]:=PolyhedronVertices[rhombohedron[args]]

(* ribbon *)

ribbon[n_,rot_:0,z_:0]:=Module[{c,i,top,bot,R=Csc[Pi/n]/2},
    c=Table[R{Cos[#],Sin[#]}&[2Pi(i+rot/2)/n],{i,n}];
    top=Append[#,z+1/2]&/@c;
    bot=Append[#,z-1/2]&/@c;
    Polygon/@Reverse/@
      Join@@@Transpose[{Partition[top,2,1,1],Reverse/@Partition[bot,2,1,1]}]
]

(* Rim stuff---rewrite *)

rim1[z_,R_,n_]:=Module[{i},
	Table[{R Cos[2 i Pi/n],R Sin[2 i Pi/n],z},{i,0,n}]
]

rim2[z_,R_,n_]:=Module[{i},
	Table[{R Cos[(2i+1) Pi/n],R Sin[(2i+1) Pi/n],z},{i,0,n}]
]

rim[z_,R_,n_]:=Module[{i},
	Table[{R Cos[2 i Pi/n],R Sin[2 i Pi/n],z},{i,0,n}]
]

(* Rotunda *)

rotunda[5]:=Module[
	{p,rim,ctr,i,h=Sqrt[(5+2Sqrt[5])/5],x2,golden=(Sqrt[5]+1)/2,
      z2=Sqrt[(5+Sqrt[5])/10],R=Sqrt[50+10Sqrt[5]]/10,half},
      x2=h;
    p=Table[{R Cos[2Pi(i+1/2)/5],R Sin[2Pi(i+1/2)/5],h},{i,6}];
    rim=Table[{x2 Cos[2Pi i/5],x2 Sin[2Pi i/5],z2},{i,6}];
    ctr=Table[golden{Cos[2Pi(i+1/2)/10],Sin[2Pi(i+1/2)/10],0},{i,11}];
    half=Flatten[{{Take[p,5]},Table[Reverse[{p[[i]],p[[i+1]],rim[[i+1]]}],{i,5}],
          Table[{ctr[[2i-1]],ctr[[2i]],rim[[i]]},{i,5}],
          Table[{ctr[[2i]],ctr[[2i+1]],rim[[i+1]],p[[i]],rim[[i]]},{i,5}]},
    1]; 
	{Polygon/@half,Polygon[Reverse[Drop[ctr,-1]]]}
]

PolyhedronName[Rotunda[5]]:="rotunda"
PolyhedronFaces[Rotunda[5]]:=PolyhedronFaces[rotunda[5]]
PolyhedronVertices[Rotunda[5]]:=PolyhedronVertices[rotunda[5]]

(* Wedge *)

solidWedge[{a_,b_},h_,x_:0]:=Orient[Polygon/@{
	{{-a,-b,0},{a,-b,0},{a,b,0},{-a,b,0}},
	{{-a,-b,0},{a,-b,0},{a-x,0,h},{-a+x,0,h}},
	{{-a,b,0},{a,b,0},{a-x,0,h},{-a+x,0,h}},
	{{-a,-b,0},{-a+x,0,h},{-a,b,0}},
	{{a,-b,0},{a-x,0,h},{a,b,0}}
}]

PolyhedronName[SolidWedge[args__]]:="wedge"
PolyhedronFaces[SolidWedge[args__]]:=PolyhedronFaces[solidWedge[args]]
PolyhedronVertices[SolidWedge[args__]]:=PolyhedronVertices[solidWedge[args]]

End[]

EndPackage[]

(* Protect[  ] *)

Miscellanius

(*:Mathematica Version: 5.2 *)

(*:Package Version: 1.97 *)

(*:Name: MathWorld`MiscPolyhedra` *)

(*:Author:

Eric W. Weisstein (eww@wolfram.com)

Additional authors:

David Eppstein (eppstein@ics.uci.edu):
Zonohedron (http://www.ics.uci.edu/~eppstein/junkyard/ukraine/ukraine.ma)

Ed Pegg Jr. (edp@wolfram.com):
2002-09-25 Hexahedron

Russell Towle (rustybel@foothill.net):
Trapezohedron

*)

(*:Summary:

Adds functionality to construct and perform operations on 
a number of polyhedra.  It is interlinked with other packages
by the author for Platonic solids, Johnson solids, and R. Maeder's
package for uniform polyhedra.

*)

(*:URL:
	Current version available from
	
	http://mathworld.wolfram.com/packages/MiscPolyhedra.m
*)


(*:History:
  v1.00 (1998-11-09): Written
  v1.01 (1998-11-15): cuboid, Jack, RhombicDodecahedronConstruction,
                      RhombicDodecahedron vertex fixed
  v1.02 (1998-12-11): Renamed from PolyhedronCompounds` to Polyhedra`
  v1.03 (1998-12-12): SquashedDodecahedron
  v1.04 (1999-01-08): Cube4Compound
  v1.05 (1999-01-27): RhombicDodecahedronStellation[n]
  v1.06 (1999-01-28): Named Archimdean solid duals added
  v1.07 (1999-06-18): CuboctahedronRhombicDodecahedron,
                      CuboctahedronRhombicDodecahedronSolid,
                      GreatRhombicTriacontahedron
  v1.10 (1999-07-25): Cumulate, Orient, PolyhedronEdgeLengths,
                      PolyhedronFaceCenters
  v1.11 (1999-07-26): TetrahedronExact
  v1.12 (1999-07-27): CubeExact, CumulateWireframe, DodecaedrhonExact, IcosahedronExact, 
                      OctahedronExact
  v1.20 (1999-07-27): Uniform polyhedron information added to this package
  v1.21 (1999-07-28): Changed DualArchimedeanPolyhedron[...] to 
                      Dual[...,Method->Tangential]
  v1.22 (1999-07-28): CircumsphereRadius, InsphereRadius, MidsphereRadius
  v1.23 (1999-07-29): Added Wireframe, wrapped Orient around CubeExact.
                      Broke out LabelVertices and LabelPolyhedronVertices
  v1.23 (1999-08-01): PlanarPolygon, PolyhedronFaces, PyramidVolume, PolyhedronVolume,
                      PolyhedronSurfaceArea, SmallTriambicIcosahedronExact.
  v1.24 (1999-08-02): Renamed ArchimedeanExact` to Archimedean`
  v1.25 (1999-08-02): UniformPolyhedronHullName
  v1.26 (1999-08-03): Cleaned up Orient a bit.
  v1.30 (1999-08-03): SnubDisphenoid, Deltahedron, Antiprism etc. moved
                      here.  Changed default for Antiprism to unit edge
                      length.  
  v1.31 (1999-08-04): GreatDodecahedronExact, GyroelongatedSquareDipyramid,
                      GreatStellatedDodecahedronExact, SmallStellatedDodecahedronExact
  v1.32 (1999-08-04): PolyhedronSymmetricSurfaceArea, PolyhedronSymmetricVolume
  v1.33 (1999-08-08): GreatIcosahedronExact
  v1.34 (1999-08-13): Corrected GreatIcosahedronExact pyramid height
  v1.35 (1999-08-14): ToExact, DualModel
  v1.36 (1999-08-16): AntiprismNet, PrismNet
  v1.37 (1999-08-17): <<Archimdean` removed
  v1.38 (1999-08-18): Modified ToExact with fix for Select[] bug under 
                      Linux
  v1.39 (1999-08-18): PointPlaneDistance, Polygon2D, Polygon3D
  v1.40 (1999-08-26): Arbitrary side length for Antiprism and Prism and Nets.
  v1.41 (1999-08-26): Fixed rotm to handle x==y==0
  v1.42 (1999-08-27): MidpointCumulate
  v1.43 (1999-08-29): Replaced SteveC's HiddenWireframe to now work rigorously.
  v1.44 (1999-08-29): PolyhedronCompoundSurfaceArea, PolyhedronCompoundVolume
                      Removed simplify from PolyhedronSurfaceArea
  v1.45 (1999-08-31): Replaced Union[] with Union[...,SameTest->SameQ] to workaround
                      Mathematica bug
  v1.46 (1999-09-02): Moved Cube-n-Compounds to Platonic`
  v1.47 (1999-09-13): ConnectivityTable,
                      Csaszar, FromFaces, Szilassi
  v1.48 (1999-09-17): Renamed cuboid to UnitCuboid to avoid conflict
                      with SolidGeometry cuboid.  Removed Tringulated
                      option from Pyramid.  Removed Graphics3D from
                      PyramidalFrustum
  v1.49 (1999-09-20): RectifiedPair, Rectify
  v1.50 (1999-10-07): ToExactN
  v1.51 (1999-10-29): RhombicIcosahedronExact
  v1.52 (1999-11-22): Moved SnubDisphenoid and Gyroelongated ...
                      to Johnson.m.  Dipyramid added,
                      and pyramid changed to default to unit edge lengths.
                      ElongatedDipyramid, ElongatedPyramid.
  v1.53 (2000-01-01): URL updated. Bipyramid, Cupola.
  v1.54 (2000-01-04): centroid fixed to Centroid in SquashedDodecahedron.
                      RhombicDodecahedron moved to Archimdean`.
  v1.55 (2000-01-14): DuerersSolid
  v1.56 (2000-01-28): PolyhedronFaceList change from {n, #} to {#, n}
  v1.57 (2000-02-02): Fixed deltahedra syntax
  v1.58 (2000-02-03): Skeleton
  v1.59 (2000-02-06): ElongatedCupola, ElongatedGyrobicupola,
                      ElongatedOrthobicupola, ElongatedOrthocupolarotunda,
                      ElongatedRotunda, 
                      Gyrobicupola, Gyrocupolarotunda, GyroelongatedCupola,
                      GyroelongatedPyramid, GyroelongatedRotunda, 
                      Orthobicupola, Orthobirotunda, Orthocupolarotunda, Rotunda
  v1.60 (2000-02-07): Made UniformPolyhedron evaluate explicitly.
  v1.61 (2000-02-09): GyroelongatedBicupola, GyroelongatedCupolarotunda
  v1.62 (2000-02-10): Bulletproofed and added messages to ToExact and ToExactN.
                      AugmentedDodecahedron, AugmentedPrism, Gyrobifastigium.
  v1.63 (2000-02-11): SymbolicUnion
  v1.64 (2000-02-15): Skeleton moved to Graphs`
  v1.65 (2000-02-17): Corrected base of pyramid to be oriented with outward normal
  v1.66 (2000-02-18): Corrected orientations for cupola and rotunda.
                      May break other routines relying on thier orderings?
  v1.67 (2000-02-22): Worked around Recognize bug in ToExactN.  NetVertices.
  v1.68 (2000-02-23): Corrected sign of Cumulate.
  v1.69 (2000-02-24): Added message for when VertexFigures fails
  v1.70 (2000-02-25): Removed outdated code which was shadowing some Archimedean duals
  v1.71 (2000-04-16): TriangleArea renamed TriangleArea3D
  v1.72 (2000-08-19): Continuation characters added to usage messages
  v1.73 (2000-08-22): PolyhedronFaceLists, PolyhedronRotationCompound
  v1.74 (2000-08-23): EschersSolid
  v1.75 (2000-11-02): NetConnectivity
  v1.76 (2000-11-11): Duals` no longer needed here, but now needed in Archimedean`
  v1.80 (2000-11-13): Code for general operations broken off into
                      PolyhedronOperations.  This package now only contains
                      code for constructing and describing specific polyhedra.
                      Package now called MiscPolyhedra`.
  v1.81 (2002-01-25): Analytic form of Zome added
  v1.82 (2002-09-25): Hexahedron
  v1.83 (2002-10-15): Zonohedron
  v1.84 (2002-11-09): DodecahedronSmallTriambicIcosahedronCompound
  v1.85 (2003-01-12): Pyramid and Prism-related code broken off into Prisms.m
  v1.86 (2003-08-13): Stripped some occurrences of Exact, did RhombicIcosahedron correctly
  v1.87 (2003-10-19): updated context
  v1.88 (2003-11-21): Boat.  fixed augmented solids and Deltahedra.  Threw out
                      some old unused code
  v1.89 (2003-12-01): Fixed UniformPolyhedron
  v1.90 (2003-12-07): RhombicEnneacontahedron, GreatRhombicTriacontahedron moved here from
                      Archimedean`
  v1.91 (2003-12-08): Removed undecipherable Towle code
  v1.92 (2003-12-25): Converted Johnson constructors to PolyhedronVertex syntax.
                      Moved Prism-type solids to Prisms`
  v1.93 (2004-02-02): Redid DuerersSolid
  v1.94 (2004-05-19): Fixed EschersSolid
  v1.95 (2004-09-20): Added Spiky
  v1.96 (2005-02-20): reimplemented JessensOrthogonalPolyhedron
  
  (c) 1998-2005 Eric W. Weisstein
*)

(*:Keywords:
*)

(*:Requirements:

  Requires a number of packages, e.g.
  
  	Archimedean.m
  	Convexify.m
  	JohnsonSolids.m
   	LiveGraphics3D.m
   	Platonic.m
   	UniformPolyhedra.m

  available from

    http://mathworld.wolfram.com/packages/

*)

(*:Discussion:

*)

(*:References: *)

(*:Limitations:
	
	This package is still under construction and is not intended to
	be a finished product.
	
	Documentation not complete.
	
	Dodecadodecahedron and Tetrahemihexahedron need analytic vertices
	
	Need to finish converting contents to PolyhedronVertices[...] syntax
	
	Zonohedron code is less than beautiful and would be nice to rewrite.  See
	Eppstein's paper for an explanation of what it's actually doing.  Insufficient
	checking is done in these routines so they can give error messages and wrong
	results for inexact or some exact inputs.
	
	Names for uniform truncations approximate for many solids.
*)

BeginPackage["MathWorld`MiscPolyhedra`",
	{
	"DiscreteMath`Combinatorica`",
	"Geometry`Rotations`",
	"Graphics`Shapes`",
	"NumberTheory`Recognize`",
	"MathWorld`Archimedean`",
	"MathWorld`ConvexHull`",
	"MathWorld`JohnsonSolids`",
	"Utilities`LiveGraphics3D`",
	"MathWorld`Platonic`",
	"MathWorld`Prisms`",
	"MathWorld`PolyhedronOperations`",
	"MathWorld`SolidGeometry`",
	"MathWorld`UniformPolyhedra`"
	}
]

Boat::usage =
"Polyhedron[Boat] gives the boat octahedron."

Csaszar::usage =
"Polyhedron[Csaszar] gives a set of triangulated polygons for the Csaszar polyhedron."

Deltahedron::usage =
"Polyhedron[Deltahedron[n]] gives the nth (1<=n<=8) convex deltahedron."

Dodecadodecahedron::usage =
"Polyhedron[Dodecadodecahedron] gives a list of Polygons with approximate vertices \
representing this uniform solid."

DodecahedronSmallTriambicIcosahedronCompound::usage =
"Polyhedron[DodecahedronSmallTriambicIcosahedronCompound] gives a list of \
Polygons with analytic vertices representing the compound of the dodecahedron \
and small triambic icosahedron."

DuerersSolid::usage =
"Polyhedron[DuerersSolid] gives Duerer's solid."

Echidnahedron::usage =
"Polyhedron[Echidnahedron] gives a list of Polygons with approximate vertices \
representing this icosahedron stellation."

EschersSolid::usage =
"Polyhedron[EschersSolid] gives a list of polygons representing the solid on the \
right pillar of Escher's \"Waterfall\" woodcut."

GoldenRectangles::usage =
"GoldenRectangles."

GreatRhombicTriacontahedron::usage =
"Polyhedron[GreatRhombicTriacontahedron] gives a list of Polygons with \
analytic vertices representing this stellation of the rhombic triacontahedron."

Hexahedron::usage =
"Polyhedron[Hexahedron[{f1,f2,f3,f4,f5,f6}]] gives the hexahedron having faces with the \
specified number of edges (given in sorted order).  Hexahedron[n] gives the \
nth hexahedron for 1<=n<=7."

Jack::usage =
"Polyhedron[Jack]."

JessensOrthogonalPolyhedron::usage =
"Polyhedron[JessensOrthogonalPolyhedron]."

Octahemioctahedron::usage =
"Polyhedron[Octahemioctahedron] gives a list of Polygons with approximate vertices \
representing this uniform polyhedron."

RhombicDodecahedronStellation::usage =
"Polyhedron[RhombicDodecahedronStellation[n]] gives the nth stellation for n=1, 2."

RhombicEnneacontahedron::usage =
"Polyhedron[RhombicEnneacontahedron[n]] gives the rhombic enneacontahedron equilateral zonohedron."

RhombicIcosahedron::usage =
"Polyhedron[RhombicIcosahedron] gives an analytic rhombic icosahedro equilateral zonohedron."

SmallDitrigonalIcosidodecahedron::usage =
"Polyhedron[SmallDitrigonalIcosidodecahedron] gives a list of Polygons with \
analytic vertices representing the small ditrigonal icosidodecahedron."

SmallTriambicIcosahedron::usage =
"Polyhedron[SmallTriambicIcosahedron] gives a list of Polygons with \
analytic vertices representing the small triambic icosahedron."

Spiky::usage =
"Polyhedron[Spiky] gives the icosahedron cumulation by equal tetrahedra that is the logo \
of Mathematica."

SquashedDodecahedron::usage =
"Polyhedron[SquashedDodecahedron[s]] gives the squashed dodecahedron \
(rotated rhombic dodecahedron) of side length s."

Szilassi::usage =
"Polyhedron[Szilassi] gives a set of triangulated polygons for the Szilassi polyhedron."

Tetrahemihexahedron::usage =
"Polyhedron[Tetrahemihexahedron] gives a list of polygons with approximate vertices \
corresponding to this uniform solid."

Trapezohedron::usage =
"Polyhedron[Trapezohedron[n,pitch]] generates a trapezohedron with n sides
of pitch."

UniformPolyhedron::usage =
"UniformPolyhedron[n,<prec>] gives the list of Polygons corresponding to \
the nth uniform polyhedron using prec digits of precision."

UniformPolyhedronName::usage =
"UniformPolyhedronName[n] gives the name of the nth uniform polyhedron."

UniformPolyhedronDualName::usage =
"UniformPolyhedronDualName[n] gives the name of the dual of the nth uniform \
polyhedron."

UniformPolyhedronHullName::usage =
"UniformPolyhedronHullName[n] gives the name of the convex hull of the \
nth uniform polyhedron."

UnitAntiprism::usage =
"UnitAntiprism."

UnitCuboid::usage =
"UnitCuboid[{a,b,c}] gives a cuboid with center {a,b,c}."

WythoffSymbol::usage =
"WythoffSymbol[n] gives the Wythoff symbol of the nth numbered uniform \
polyhedron."

Zome::usage =
"Zome gives the solid corresponding to the Zome construction unit.  \
To turn into an actual Zome, use Bevel on it."

Zonohedron::usage =
"Zonohedron[vector-list] gives the zonohedron corresponding to the \
specified list of vectors."


(* Options *)

Options[HiddenWireframe]={
	ViewPoint->{1.3,-2.4,2},
	PlotStyles->{Automatic,Dashing[{.02,.02}]}
}


Begin["`Private`"]


addn[n_,k_,m_]:=Mod[n+k-1,m]+1

PolyhedronName[Boat]:="Boat"
PolyhedronPolyhedra[Boat]:={Range[8]}
PolyhedronVertices[Boat]:=
 {{0, -1/2, 1/Sqrt[2]}, {1/2, 0, 0}, {5/6, 0, (2*Sqrt[2])/3}, 
 {0, 1/2, 1/Sqrt[2]}, {-5/6, 0, (2*Sqrt[2])/3}, {-1/2, 0, 0}}
PolyhedronFaces[Boat]:=
	{{1,2,3},{3,2,4},{1,3,4},{4,5,1},{1,5,6},{1,6,2},{4,2,6},{4,6,5}}

CsaszarPolygons:=Module[{
	coords={{0,0,15},{-3,3,0},{3,-3,0},{-3,-3,1},{3,3,1},{1,2,2},{-1,-2,2}},
	faces={{1,2,3},{2,1,4},{3,5,1},{4,6,2},{6,1,5},{5,2,6},{3,4,5},{4,3,6},
		{0,1,6},{0,2,5},{1,0,4},{2,0,3},{3,0,6},{4,0,5}}
	},
	Polygon[coords[[#+1]]]&/@faces
]

Csaszar:=Module[{},
	(*TriangulateSimplePolygons3D[Graphics3D[*)
	CsaszarPolygons
	(*,TriangleEdges->Automatic][[1]]*)
]
      
Deltahedron[1]:=Polyhedron[Tetrahedron]
Deltahedron[2]:=Dipyramid[3]
Deltahedron[3]:=Polyhedron[Octahedron]
Deltahedron[4]:=Dipyramid[5]
Deltahedron[5]:=JohnsonExact[84]
Deltahedron[6]:=AugmentedPrism[3,{1,2,3}] (* JohnsonExact[51] *)
Deltahedron[7]:=JohnsonExact[17]
Deltahedron[8]:=Polyhedron[Icosahedron]

Diagonals[{a_List,b_List,c_List}]:={prec[a,b],prec[a,c],prec[b,c]};

dodeccube:=Module[
	{top,bot,topV,botV,
			i,r=1/Sqrt[10-2Sqrt[5]]},
	top=Table[r{2Cos[2/5 Pi i],2Sin[2/5 Pi i],GoldenRatio+1},{i,5}];
	bot=-top;
	topV=Table[r{
		2GoldenRatio Cos[2/5 Pi i],
		2GoldenRatio Sin[2/5 Pi i],
		GoldenRatio-1},{i,5}];
	botV=-topV;
	{
			Polygon[{top[[2]],top[[4]],botV[[1]],botV[[5]]}],
			Polygon[{top[[4]],topV[[5]],bot[[2]],botV[[1]]}],
			Polygon[{topV[[5]],topV[[1]],bot[[4]],bot[[2]]}],
			Polygon[{top[[2]],topV[[1]],bot[[4]],botV[[5]]}],
			Polygon[{top[[2]],top[[4]],topV[[5]],topV[[1]]}],
			Polygon[{bot[[2]],bot[[4]],botV[[5]],botV[[1]]}]
	}		
]

(* Dodecadodecahedron *)

(* netlib 141 *)
PolyhedronName[Dodecadodecahedron]:="dodecadodecahedron"
NetFaces[Dodecadodecahedron]:={{65,52,42,43,29,44,45,53,67,60},{64,51,52,65},{27,28,43,42},{21,34,44,29},{55,62,53,45},{88,77,60,67},{58,64,59,76,87,85,103,84,83,75},{41,51,64,58},{60,77,76,59},{105,104,85,87},{112,93,84,103},{74,63,75,83},{15,27,39,38,50,37,36,26,14,23},{16,28,27,15},{52,51,38,39},{61,49,37,50},{25,18,26,36},{6,9,23,14},{22,21,33,20,19,11,3,8,4,12},{35,34,21,22},{43,28,20,33},{10,7,11,19},{1,2,8,3},{5,13,12,4},{69,55,54,46,30,40,31,47,57,56},{79,62,55,69},{44,34,46,54},{17,24,40,30},{32,48,47,31},{72,71,56,57},{95,88,66,78,68,89,98,97,115,96},{86,77,88,95},{53,62,78,66},{70,90,89,68},{117,116,97,98},{124,106,96,115},{132,145,155,154,168,153,152,144,130,137},{133,146,145,132},{170,169,154,155},{176,163,153,168},{142,135,144,152},{109,120,137,130},{139,133,138,121,110,112,94,113,114,122},{156,146,133,139},{137,120,121,138},{92,93,112,110},{85,104,113,94},{123,134,122,114},{182,170,158,159,147,160,161,171,183,174},{181,169,170,182},{145,146,159,158},{136,148,160,147},{172,179,171,161},{191,188,174,183},{175,176,164,177,178,186,194,189,193,185},{162,163,176,175},{154,169,177,164},{187,190,186,178},{196,195,189,194},{192,184,185,193},{128,142,143,151,167,157,166,150,140,141},{118,135,142,128},{153,163,151,143},{180,173,157,167},{165,149,150,166},{125,126,141,140},{102,109,131,119,129,108,99,100,82,101},{111,120,109,102},{144,135,119,131},{127,107,108,129},{80,81,100,99},{73,91,101,82}}
NetVertices[Dodecadodecahedron]:={{-5.56475613201,-0.828230060909},{-4.97418185902,-1.63521334951},{-4.61477193005,-0.515932100581},{-4.60918688543,-2.13395645022},{-4.60573513803,-3.13395049292},{-4.39822942266,1.79384610387},{-4.25322870386,-0.0146811585885},{-4.02419765706,-1.32291538918},{-3.80765514967,0.986862815265},{-3.66823947549,0.796359902449},{-3.66265443087,-0.821664447191},{-3.65920268347,-1.82165848989},{-3.65575093607,-2.8216525326},{-3.4482452207,2.1061440642},{-3.44266017609,0.488119714556},{-3.43920842868,-0.511874328146},{-3.43149008685,-2.74792898472},{-3.08670199452,2.60739500619},{-3.0776652025,-0.0106233861535},{-3.0742134551,-1.01061742886},{-3.07208015788,-1.62864773579},{-3.06862841048,-2.6286417785},{-2.85767094771,1.29916077559},{-2.84091581386,-3.55491227333},{-2.50171276614,3.41843606723},{-2.49612772153,1.80041171759},{-2.49267597413,0.800417674884},{-2.48922422672,-0.199576367818},{-2.48709092951,-0.817606674756},{-2.48150588489,-2.4356310244},{-2.47592084027,-4.05365537403},{-2.47246909287,-5.05364941674},{-2.12209595592,-1.31634977547},{-2.11996265871,-1.9343800824},{-2.1165109113,-2.9343741251},{-1.91113849316,2.61145277862},{-1.90768674575,1.61145873592},{-1.90555344854,0.993428428984},{-1.90210170114,-0.00656561371846},{-1.8909316119,-3.242614313},{-1.54401022235,1.49467937098},{-1.54055847495,0.494685328274},{-1.53710672755,-0.505308714428},{-1.53497343033,-1.12333902137},{-1.53152168293,-2.12333306407},{-1.52938838572,-2.741363371},{-1.52593663831,-3.74135741371},{-1.52248489091,-4.74135145641},{-1.32269751738,2.42249979696},{-0.955569246578,1.30572638931},{-0.953435949363,0.687696082374},{-0.949984201961,-0.312297960328},{-0.946532454559,-1.31229200303},{-0.944399157344,-1.93032230997},{-0.940947409942,-2.93031635267},{-0.938814112727,-3.54834665961},{-0.9353623653250001,-4.54834070231},{-0.594026020391,1.8069773313},{-0.588440975774,0.188952981665},{-0.584989228372,-0.811041061037},{-0.370580018205,2.11676745035},{-0.35595818157,-2.11927529163},{-0.232482794204,2.3082282733},{-0.00345174740195,0.999994042702},{2.43975923148^-16,2.77555756156^-17},{0.00345174740195,-0.999994042702},{0.00558504461694,-1.61802434964},{0.00903679201889,-2.61801839234},{0.0111700892339,-3.23604869928},{0.0124885394208,-3.61801243504},{0.0133033864489,-3.85407900622},{0.0167551338508,-4.85407304892},{0.351187983981,3.5012330701},{0.352506434168,3.11926933433},{0.358091478785,1.5012449847},{0.361543226187,0.501250941993},{0.364994973589,-0.498743100709},{0.594026020391,-1.8069773313},{0.5961593176059999,-2.42500763824},{0.930592167736,5.93029848078},{0.934043915138,4.93030443807},{0.936177212353,4.31227413114},{0.943080707157,2.31228604573},{0.946532454559,1.31229200303},{0.948665751774,0.694261696093},{0.949984201961,0.312297960328},{0.952117499176,-0.305732346609},{0.955569246578,-1.30572638931},{0.95902099398,-2.30572043201},{0.962472741382,-3.30571447472},{1.30330548316,3.19550072349},{1.52806993553,3.12332710677},{1.53152168293,2.12333306407},{1.53365498015,1.50530275713},{1.54055847495,-0.494685328274},{1.54401022235,-1.49467937098},{1.54614351957,-2.11270967791},{1.54959526697,-3.11270372062},{1.88270966691,5.62456613417},{1.88616141431,4.62457209146},{1.88829471153,4.00654178453},{1.89174645893,3.00654774183},{1.89864995373,1.00655965642},{1.90078325095,0.388529349483},{1.90423499835,-0.611464693219},{2.12899945072,-0.683638309939},{2.4698321925,5.81757688827},{2.4732839399,4.81758284556},{2.4767356873,3.81758880286},{2.4801874347,2.81759476016},{2.48232073192,2.19956445322},{2.48363918211,1.81760071746},{2.48577247932,1.19957041052},{2.48922422672,0.199576367818},{2.49612772153,-1.80041171759},{2.49826101874,-2.41844202452},{2.50171276614,-3.41843606723},{2.83614561627,4.93687005179},{2.83827891349,4.31883974486},{3.06730996029,3.01060551426},{3.07076170769,2.01061147156},{3.0742134551,1.01061742886},{3.07979849971,-0.6074069207839999},{3.0811169499,-0.989370656549},{3.41554980003,7.36593546247},{3.41900154743,6.36594141977},{3.41981639446,6.12987484859},{3.42113484465,5.74791111283},{3.42326814186,5.12988080589},{3.42671988926,4.12988676319},{3.42885318648,3.51185645625},{3.43230493388,2.51186241355},{3.43575668128,1.51186837085},{3.66478772809,0.203634140253},{3.78826311545,4.63113770518},{3.80288495209,0.395094963202},{4.01729416225,3.32290347459},{4.02074590965,2.32290943189},{4.02633095427,0.704885082246},{4.36766729921,7.06020311586},{4.37111904661,6.06020907316},{4.37325234382,5.44217876622},{4.37670409122,4.44218472352},{4.37883738844,3.82415441658},{4.38228913584,2.82416037388},{4.38574088324,1.82416633118},{4.38787418046,1.20613602424},{4.75500245126,0.08936261659229999},{4.95478982479,7.25321386996},{4.9582415721899995,6.25321982726},{4.9616933196,5.25322578456},{4.96382661681,4.63519547762},{4.96727836421,3.63520143492},{4.96941166143,3.01717112798},{4.97286340883,2.01717708528},{4.97631515623,1.01718304257},{5.32323654578,5.75447672655},{5.33440663502,2.51842802727},{5.33785838242,1.51843398457},{5.33999167963,0.90040367763},{5.34344342704,-0.0995903650726},{5.54881584518,5.44623653866},{5.55226759259,4.44624249595},{5.5544008898,3.82821218902},{5.90477402675,7.56551183029},{5.90822577416,6.56551778759},{5.91381081877,4.94749343795},{5.91939586339,3.32946908831},{5.9215291606,2.71143878137},{5.92498090801,1.71144473867},{5.92843265541,0.711450695965},{5.93401770003,-0.906573653675},{6.27322074774,6.06677468688},{6.28997588159,1.21270163796},{6.50093334436,5.14050419205},{6.50438509176,4.14051014934},{6.50651838898,3.52247984241},{6.50997013638,2.5224857997},{6.5190069284,-0.0955325926376},{6.86379502073,5.25979139827},{6.87151336257,3.0237367417},{6.87496510997,2.02374269899},{6.88055015458,0.405718349355},{7.08805586995,5.33351494615},{7.09150761735,4.33352090344},{7.09495936475,3.33352686074},{7.10054440937,1.7155025111},{7.23996008356,1.52499959829},{7.45650259094,3.83477780273},{7.68553363774,2.52654357214},{7.83053435654,0.718016309683},{8.03804007191,5.64581290647},{8.04149181931,4.64581886377},{8.04707686393,3.02779451413},{8.4064867929,4.14707576306},{8.99706106589,3.34009247446}}
PolyhedronFaces[Dodecadodecahedron]:=
{{15,39,41,54,77,62,73,52,36,37},{15,39,30,9},{48,61,54,41},{93,76,62,77},{80,
    56,52,73},{36,37,28,26},{5,2,3,1,4,6,8,15,9,11},{11,9,30,33},{37,28,8,
    15},{22,19,4,6},{1,3,16,10},{2,5,24,17},{53,60,44,23,29,11,33,41,48,
    71},{71,48,61,86},{41,33,30,39},{11,29,24,5},{27,46,44,23},{79,69,53,
    60},{104,93,77,86,71,91,97,99,110,101},{104,93,76,90},{77,86,61,54},{53,
    69,91,71},{89,92,99,97},{108,95,101,110},{64,80,73,90,104,98,109,94,88,
    84},{64,80,56,43},{62,76,90,73},{101,95,98,104},{106,87,94,109},{67,65,84,
    88},{6,26,36,43,64,45,51,32,14,22},{6,26,28,8},{52,56,43,36},{84,65,45,
    64},{58,42,32,51},{14,22,19,12},{34,57,70,72,96,74,75,59,38,49},{25,50,57,
    34},{78,81,72,70},{103,83,74,96},{75,59,55,68},{21,35,49,38},{40,25,34,18,
    7,10,1,12,14,20},{63,50,25,40},{34,18,35,49},{7,10,16,13},{4,19,12,1},{32,
    42,20,14},{100,78,70,63,40,58,51,67,88,82},{102,81,78,100},{57,50,63,
    70},{20,42,58,40},{45,65,67,51},{94,87,82,88},{105,103,96,102,100,106,109,
    108,110,107},{85,83,103,105},{72,81,102,96},{82,87,106,100},{98,95,108,
    109},{99,92,107,110},{47,68,75,85,105,89,97,79,60,66},{31,55,68,47},{74,
    83,85,75},{107,92,89,105},{97,79,69,91},{60,66,46,44},{13,2,17,23,27,47,
    31,38,21,7},{18,35,21,7},{59,55,31,38},{47,27,46,66},{23,17,24,29},{3,16,
    13,2}}
PolyhedronVertices[Dodecadodecahedron]:={{.45520409720613175,2.9852283637161371,2.6144275734495506},
{.58179973673199928,2.2572778195070664,1.1749500508768096},{.70384217804085175,2.1302163802643068,2.1593082342515283},
{.77926855484144585,2.0516880921526706,2.7676750486808425},{.8575089628990247,1.6017899137072284,1.8780290136906538},
{.90131099615631922,1.9246266529093337,3.7520332320537766},{.97189527864441998,3.8200764392512153,1.3283121421808006},
{.97955140420890048,1.4747284744594158,2.8623871970630421},{1.0279066356797787,1.1966761087008048,2.3125557094819529},
{1.0939377199533824,3.6930150000148285,2.3126703255567359},{1.1061470437340354,.74677793025596729,1.4229096744962048},
{1.1693640967494892,3.6144867118997792,2.9210371399850335},{1.2205333594812287,2.9650644558017301,.87319280298369513},
{1.2914065380596199,3.4874252726583223,3.9053953233586283},{1.3036158618467372,.54118820289558111,3.0156346722940855},
{1.3425758007893254,2.8380030165593695,1.8575509863569636},{1.3742001443322159,2.4366379892411308,.59191358242875417},
{1.4886968085650687,4.1304511980221102,2.1261740900099247},{1.4934285543883417,2.6809464403356961,3.0742846152166112},
{1.6107392498754231,4.0033897587706312,3.1105322733773448},{1.6152924480962415,3.4025006538134397,.68669656743984357},
{1.6154709957049333,2.5538850010939468,4.0586427985878945},{1.6228382251729344,1.5816260057930731,.13679424323185869},
{1.6499093704963192,1.7811500834405759,1.2949925452399425},{1.8080977194941701,4.3222733481925575,2.6192798920968782},
{1.815753845068957,1.9769253834013426,4.1533549469682934},{1.8639305289320914,2.5474886703691335,.23157722824608955},
{1.8939942531204276,1.5270272049502334,3.2637089119796471},{1.8985474513297968,.92613809998854291,.83987320604279201},
{1.9907047160631734,.9709224734326156,2.1640459368196959},{2.012933767086773,3.1444246255294558,.29015633453361755},
{2.0568461488202782,2.9427880479749864,4.2481379319812301},{2.0689451241129468,.52102429498614984,1.2743999018270516},
{2.1274304313135176,4.8382378343155974,1.824416842118707},{2.1320939780177756,3.7128754125827315,1.4845585152653016},
{2.1398183027068947,1.0433851118376966,4.3066024222019305},{2.2180587107578335,.59348693338622656,3.4169563872113253},
{2.254026070846694,4.110287290107932,.38493931954816921},{2.2664139422299668,.31543456762587712,2.8671248996305763},
{2.324899249413065,4.6326481069468664,3.4171418399219891},{2.3446543502809833,-.13446361082333168,1.9774788646371091},
{2.3761788606350306,3.4587525340878999,3.4532748820039454},{2.3809106064643181,2.0092477764108843,4.4013854072100722},
{2.38827783592909,1.0369887811145474,.47953685185230608},{2.5299138446109556,2.6061837315690217,4.4599645134990032},
{2.629370139695431,2.0028514456807202,.57431983687334038},{2.6563309365415271,2.7268488400917616,-.35145924020781244},
{2.6639870620948955,.38150087530588697,1.1826158146568895},{2.7708276007658609,4.4206620488763623,1.1828012673737016},
{2.7708957998775103,4.0965197129156974,2.4707701194348434},{2.7710061483667732,3.5720463961406041,4.5547474985156036},
{2.783215472155238,.62580932639782655,3.6649868474529528},{2.8613455317089272,.7003844647205117,.69136343337414602},
{2.8614558802083911,.17591114794891793,2.7753408124633814},{2.9273766159948184,3.1967233560210148,.69147804945948222},
{3.0243077759137573,1.591671990971467,3.759769832462269},{3.0902285116983836,4.6124841990418598,1.6759070694612663},
{3.0903388601741156,4.0880108822659339,3.7598844485437655},{3.1684689197568727,4.1625860205938754,.78626103447398181},
{3.1806782435302152,1.2163489508438729,-.10349961659975425},{3.1807885920168031,.69187563407209235,1.9804777624867657},
{3.1808567911409713,.36773329812162553,3.2684466145518805},{3.2876973297962622,4.4068944716808591,3.2686320672669036},
{3.2953534553776631,2.0615465069025299,4.8027071221287244},{3.3223142522137699,2.785543901307857,3.8769280450469763},
{3.421770547298598,2.1822116154068134,-.0087166315777274517},{3.563406555963508,3.7514065658733105,3.9717110300712499},
{3.5707737854469616,2.7791475705808998,.049862474713326277},{3.5755055312566543,1.3296428128961506,.99797299991339115},
{3.6070300416174714,4.9228589578097036,2.4737690172914596},{3.6267851424735208,.15574724003574678,1.0341060419927111},
{3.6852704496781916,4.4729607793613874,1.5841229822981473},{3.6976583210672397,.67810805688952575,4.0663085623726483},
{3.7336256811478202,4.1949084136010725,1.0342914947163669},{3.8118660892073978,3.7450102351538262,.14464545972670177},
{3.8195904138877132,1.0755199344103559,2.9666893666501863},{3.8242539605902212,-.049842487320355914,2.6268310398010572},
{3.8827392677800083,4.2673710520001217,3.1768479801036826},{3.8948382430741539,1.8456072990194657,.20310994994185199},
{3.9387506248264248,1.6439707214624181,4.1610915473833509},{3.9609796758362109,3.8174728735588932,2.2872019451091533},
{4.0531369405562652,3.8622572470000651,3.6113746758966273},{4.0576901387884129,3.2613681420409887,1.1875389699547241},
{4.0877538629758141,2.2409066766364805,4.2196706536731217},{4.1359305468435269,2.8114699635859948,.29789293495985688},
{4.1435866724015691,.46612199880503193,1.8319679898199568},{4.3017750214019875,3.0072452635534903,3.1562553366890112},
{4.3288461667252122,3.2067693411986797,4.3144536386967114},{4.3362133962005825,2.2345103459002076,.39260508333602426},
{4.336391943810731,1.3858946931784969,3.7645513144745954},{4.3409451420167143,.78500558821424986,1.3407156085395418},
{4.4582558375087477,2.1074489066538958,1.3769632667096769},{4.4629875833305886,.65794414897266496,2.3250737919033033},
{4.5774842475670285,2.3517573577548114,3.8593342994954853},{4.6091085911116855,1.9503923304393757,2.5936968955620964},
{4.6480685300601302,4.2472071440918674,1.4356132096377544},{4.6602778538366618,1.3009700743364316,.54585255856501941},
{4.7311510324221123,1.8233308911925904,3.5780550789361882},{4.7823202951479375,1.1739086350975202,1.5302107419304483},
{4.8455373481633345,4.0416174167376849,3.0283382074527415},{4.8577466719473249,1.0953803469865592,2.1385775563531789},
{4.9237777562256251,3.5917192382865544,2.1386921724527742},{4.9721329876946906,3.313666872533425,1.5888606848737742},
{4.9797891132616726,.96831890774651186,3.1229357397307436},{5.050373395750848,2.8637686940760191,.69921464988160198},
{5.094175428999096,3.1866054332898235,2.5732188682363366},{5.1724158370592106,2.7367072548361664,1.6835728332585984},
{5.2478422138590212,2.6581789667273683,2.2919396476781985},{5.369884655165212,2.531117527482082,3.2762978310481135},
{5.4964802946937007,1.8031669832782948,1.8368203084732746}}

(* Dodecahedron-Small Triambic Icosahedron Compound *)

NetPieces[DodecahedronSmallTriambicIcosahedronCompound]:={{48,{1}},{48,{2}},{30,{3,4}}}
NetFaces[DodecahedronSmallTriambicIcosahedronCompound]:={{2,9,8},{1,10,7},{5,4,6},{6,3,5}}
NetVertices[DodecahedronSmallTriambicIcosahedronCompound]:=
{{0,-1/10},{0,1/10},{1,2-Sqrt[5]},{1,-2+Sqrt[5]},
{(15-Sqrt[5])/20,0},{(25+Sqrt[5])/20,0},
{Root[2697881-35185320*#1^2+57524400*#1^4-5968000*#1^6+160000*#1^8&,5,0],-1/10},
{Root[2697881-35185320*#1^2+57524400*#1^4-5968000*#1^6+160000*#1^8&,5,0],1/10},
{Root[1042441-582195720*#1^2+33911604400*#1^4-234151248000*#1^6+431660960000*#1^8&,7,0],Root[642840555601+2681053399320*#1-9558056088200*#1^2-36275909836000*#1^3+63668944950000*#1^4+156081724400000*#1^5-209775762000000*#1^6-215830480000000*#1^7+269788100000000*#1^8&,6,0]},
{Root[1042441-582195720*#1^2+33911604400*#1^4-234151248000*#1^6+431660960000*#1^8&,7,0],Root[642840555601-2681053399320*#1-9558056088200*#1^2+36275909836000*#1^3+63668944950000*#1^4-156081724400000*#1^5-209775762000000*#1^6+215830480000000*#1^7+269788100000000*#1^8&,3,0]}}
PolyhedronFaces[DodecahedronSmallTriambicIcosahedronCompound]:=
{{27,17,64},{64,1,27},{1,64,17},{17,18,37},{37,1,17},{1,37,18},{18,28,68},{68,
    1,18},{1,68,28},{28,14,65},{65,1,28},{1,65,14},{14,27,61},{61,1,14},{1,61,
    27},{19,20,39},{39,2,19},{2,39,20},{20,24,49},{49,2,20},{2,49,24},{24,13,
    47},{47,2,24},{2,47,13},{13,23,41},{41,2,13},{2,41,23},{23,19,43},{43,2,
    23},{2,43,19},{18,17,38},{38,3,18},{3,38,17},{17,30,77},{77,3,17},{3,77,
    30},{30,15,75},{75,3,30},{3,75,15},{15,29,69},{69,3,15},{3,69,29},{29,18,
    71},{71,3,29},{3,71,18},{25,20,56},{56,4,25},{4,56,20},{20,19,40},{40,4,
    20},{4,40,19},{19,26,60},{60,4,19},{4,60,26},{26,16,57},{57,4,26},{4,57,
    16},{16,25,53},{53,4,16},{4,53,25},{21,30,80},{80,5,21},{5,80,30},{30,17,
    78},{78,5,30},{5,78,17},{17,27,63},{63,5,17},{5,63,27},{27,31,90},{90,5,
    27},{5,90,31},{31,21,81},{81,5,31},{5,81,21},{28,18,67},{67,6,28},{6,67,
    18},{18,29,72},{72,6,18},{6,72,29},{29,22,74},{74,6,29},{6,74,22},{22,32,
    82},{82,6,22},{6,82,32},{32,28,92},{92,6,32},{6,92,28},{26,19,59},{59,7,
    26},{7,59,19},{19,23,44},{44,7,19},{7,44,23},{23,21,46},{46,7,23},{7,46,
    21},{21,31,83},{83,7,21},{7,83,31},{31,26,88},{88,7,31},{7,88,26},{25,32,
    86},{86,8,25},{8,86,32},{32,22,84},{84,8,32},{8,84,22},{22,24,52},{52,8,
    22},{8,52,24},{24,20,50},{50,8,24},{8,50,20},{20,25,55},{55,8,20},{8,55,
    25},{23,13,42},{42,9,23},{9,42,13},{13,15,33},{33,9,13},{9,33,15},{15,30,
    76},{76,9,15},{9,76,30},{30,21,79},{79,9,30},{9,79,21},{21,23,45},{45,9,
    21},{9,45,23},{24,22,51},{51,10,24},{10,51,22},{22,29,73},{73,10,22},{10,
    73,29},{29,15,70},{70,10,29},{10,70,15},{15,13,34},{34,10,15},{10,34,
    13},{13,24,48},{48,10,13},{10,48,24},{26,31,87},{87,11,26},{11,87,31},{31,
    27,89},{89,11,31},{11,89,27},{27,14,62},{62,11,27},{11,62,14},{14,16,
    35},{35,11,14},{11,35,16},{16,26,58},{58,11,16},{11,58,26},{14,28,66},{66,
    12,14},{12,66,28},{28,32,91},{91,12,28},{12,91,32},{32,25,85},{85,12,
    32},{12,85,25},{25,16,54},{54,12,25},{12,54,16},{16,14,36},{36,12,16},{12,
    36,14}}
PolyhedronVertices[DodecahedronSmallTriambicIcosahedronCompound]:=
{{0,0,Root[5-20*#1^2+16*#1^4&,1,0]},{0,0,Root[5-20*#1^2+16*#1^4&,4,0]},
{Root[1-5*#1^2+5*#1^4&,1,0],0,Root[1-20*#1^2+80*#1^4&,1,0]},{Root[1-5*#1^2+5*#1^4&,4,0],0,Root[1-20*#1^2+80*#1^4&,4,0]},
{Root[1-20*#1^2+80*#1^4&,2,0],(-1-Sqrt[5])/4,Root[1-20*#1^2+80*#1^4&,1,0]},{Root[1-20*#1^2+80*#1^4&,2,0],(1+Sqrt[5])/4,Root[1-20*#1^2+80*#1^4&,1,0]},
{Root[1-20*#1^2+80*#1^4&,3,0],(-1-Sqrt[5])/4,Root[1-20*#1^2+80*#1^4&,4,0]},{Root[1-20*#1^2+80*#1^4&,3,0],(1+Sqrt[5])/4,Root[1-20*#1^2+80*#1^4&,4,0]},
{Root[1-40*#1^2+80*#1^4&,1,0],-1/2,Root[1-20*#1^2+80*#1^4&,4,0]},{Root[1-40*#1^2+80*#1^4&,1,0],1/2,Root[1-20*#1^2+80*#1^4&,4,0]},
{Root[1-40*#1^2+80*#1^4&,4,0],-1/2,Root[1-20*#1^2+80*#1^4&,1,0]},{Root[1-40*#1^2+80*#1^4&,4,0],1/2,Root[1-20*#1^2+80*#1^4&,1,0]},
{Root[1-50*#1^2+125*#1^4&,1,0],0,Root[1-1300*#1^2+2000*#1^4&,4,0]},
{Root[1-50*#1^2+125*#1^4&,4,0],0,Root[1-1300*#1^2+2000*#1^4&,1,0]},
{Root[1-125*#1^2+125*#1^4&,1,0],0,Root[1-100*#1^2+2000*#1^4&,4,0]},
{Root[1-125*#1^2+125*#1^4&,4,0],0,Root[1-100*#1^2+2000*#1^4&,1,0]},
{Root[1-500*#1^2+2000*#1^4&,1,0],(-5-Sqrt[5])/20,Root[1-1300*#1^2+2000*#1^4&,1,0]},
{Root[1-500*#1^2+2000*#1^4&,1,0],(5+Sqrt[5])/20,Root[1-1300*#1^2+2000*#1^4&,1,0]},
{Root[1-500*#1^2+2000*#1^4&,4,0],(-5-Sqrt[5])/20,Root[1-1300*#1^2+2000*#1^4&,4,0]},
{Root[1-500*#1^2+2000*#1^4&,4,0],(5+Sqrt[5])/20,Root[1-1300*#1^2+2000*#1^4&,4,0]},
{Root[1-200*#1^2+2000*#1^4&,1,0],(-5-2*Sqrt[5])/10,Root[1-100*#1^2+2000*#1^4&,4,0]},
{Root[1-200*#1^2+2000*#1^4&,1,0],(5+2*Sqrt[5])/10,Root[1-100*#1^2+2000*#1^4&,4,0]},
{Root[1-100*#1^2+2000*#1^4&,1,0],(-5-3*Sqrt[5])/20,Root[1-1300*#1^2+2000*#1^4&,4,0]},
{Root[1-100*#1^2+2000*#1^4&,1,0],(5+3*Sqrt[5])/20,Root[1-1300*#1^2+2000*#1^4&,4,0]},
{Root[1-1300*#1^2+2000*#1^4&,4,0],(5+3*Sqrt[5])/20,Root[1-100*#1^2+2000*#1^4&,4,0]},
{Root[1-1300*#1^2+2000*#1^4&,4,0],(-5-3*Sqrt[5])/20,Root[1-100*#1^2+2000*#1^4&,4,0]},
{Root[1-100*#1^2+2000*#1^4&,4,0],(-5-3*Sqrt[5])/20,Root[1-1300*#1^2+2000*#1^4&,1,0]},
{Root[1-100*#1^2+2000*#1^4&,4,0],(5+3*Sqrt[5])/20,Root[1-1300*#1^2+2000*#1^4&,1,0]},
{Root[1-1300*#1^2+2000*#1^4&,1,0],(5+3*Sqrt[5])/20,Root[1-100*#1^2+2000*#1^4&,1,0]},
{Root[1-1300*#1^2+2000*#1^4&,1,0],(-5-3*Sqrt[5])/20,Root[1-100*#1^2+2000*#1^4&,1,0]},
{Root[1-200*#1^2+2000*#1^4&,4,0],(-5-2*Sqrt[5])/10,Root[1-100*#1^2+2000*#1^4&,1,0]},
{Root[1-200*#1^2+2000*#1^4&,4,0],(5+2*Sqrt[5])/10,Root[1-100*#1^2+2000*#1^4&,1,0]},
{Root[-4579+3320*#1+7500*#1^2-8000*#1^3+2000*#1^4&,1,0],Root[1-25*#1^2+5*#1^4&,2,0],Root[-4579+5440*#1+16700*#1^2-12000*#1^3+2000*#1^4&,2,0]},
{Root[-4579+3320*#1+7500*#1^2-8000*#1^3+2000*#1^4&,1,0],Root[1-25*#1^2+5*#1^4&,3,0],Root[-4579+5440*#1+16700*#1^2-12000*#1^3+2000*#1^4&,2,0]},
{Root[-4579-3320*#1+7500*#1^2+8000*#1^3+2000*#1^4&,4,0],Root[1-25*#1^2+5*#1^4&,2,0],Root[-4579-5440*#1+16700*#1^2+12000*#1^3+2000*#1^4&,3,0]},
{Root[-4579-3320*#1+7500*#1^2+8000*#1^3+2000*#1^4&,4,0],Root[1-25*#1^2+5*#1^4&,3,0],Root[-4579-5440*#1+16700*#1^2+12000*#1^3+2000*#1^4&,3,0]},
{Root[-6899-20200*#1+27500*#1^2+16000*#1^3+2000*#1^4&,3,0],0,Root[1-1300*#1^2+2000*#1^4&,1,0]},
{Root[-1699-280*#1+8300*#1^2+8000*#1^3+2000*#1^4&,3,0],0,Root[-17359-2960*#1+33900*#1^2-16000*#1^3+2000*#1^4&,1,0]},
{Root[-6899+20200*#1+27500*#1^2-16000*#1^3+2000*#1^4&,2,0],0,Root[1-1300*#1^2+2000*#1^4&,4,0]},
{Root[-1699+280*#1+8300*#1^2-8000*#1^3+2000*#1^4&,2,0],0,Root[-17359+2960*#1+33900*#1^2+16000*#1^3+2000*#1^4&,4,0]},
{Root[-6899-10400*#1+82800*#1^2-96000*#1^3+32000*#1^4&,1,0],Root[-6899-60840*#1-103760*#1^2+3200*#1^3+6400*#1^4&,3,0],Root[1-1300*#1^2+2000*#1^4&,4,0]},
{Root[-1699+7680*#1-400*#1^2-32000*#1^3+32000*#1^4&,1,0],Root[-1699-12200*#1-20560*#1^2+3200*#1^3+6400*#1^4&,2,0],Root[-17359+2960*#1+33900*#1^2+16000*#1^3+2000*#1^4&,4,0]},
{Root[-6899+50800*#1+407200*#1^2+224000*#1^3+32000*#1^4&,4,0],Root[-6899-37520*#1-38240*#1^2+6400*#1^3+6400*#1^4&,3,0],Root[1-1300*#1^2+2000*#1^4&,4,0]},
{Root[-1699-7120*#1+68000*#1^2+96000*#1^3+32000*#1^4&,4,0],Root[-1699-7440*#1-6240*#1^2+6400*#1^3+6400*#1^4&,2,0],Root[-17359+2960*#1+33900*#1^2+16000*#1^3+2000*#1^4&,4,0]},
{Root[-9539+75120*#1+314800*#1^2+192000*#1^3+32000*#1^4&,3,0],Root[-1699-4760*#1+240*#1^2+9600*#1^3+6400*#1^4&,3,0],Root[-4579+5440*#1+16700*#1^2-12000*#1^3+2000*#1^4&,2,0]},
{Root[-19-880*#1-5200*#1^2+32000*#1^4&,3,0],Root[-31859-83480*#1-50960*#1^2+9600*#1^3+6400*#1^4&,3,0],Root[-4579+5440*#1+16700*#1^2-12000*#1^3+2000*#1^4&,2,0]},
{Root[-6899-10400*#1+82800*#1^2-96000*#1^3+32000*#1^4&,1,0],Root[-6899+60840*#1-103760*#1^2-3200*#1^3+6400*#1^4&,2,0],Root[1-1300*#1^2+2000*#1^4&,4,0]},
{Root[-1699+7680*#1-400*#1^2-32000*#1^3+32000*#1^4&,1,0],Root[-1699+12200*#1-20560*#1^2-3200*#1^3+6400*#1^4&,3,0],Root[-17359+2960*#1+33900*#1^2+16000*#1^3+2000*#1^4&,4,0]},
{Root[-6899+50800*#1+407200*#1^2+224000*#1^3+32000*#1^4&,4,0],Root[-6899+37520*#1-38240*#1^2-6400*#1^3+6400*#1^4&,2,0],Root[1-1300*#1^2+2000*#1^4&,4,0]},
{Root[-1699-7120*#1+68000*#1^2+96000*#1^3+32000*#1^4&,4,0],Root[-1699+7440*#1-6240*#1^2-6400*#1^3+6400*#1^4&,3,0],Root[-17359+2960*#1+33900*#1^2+16000*#1^3+2000*#1^4&,4,0]},
{Root[-9539+75120*#1+314800*#1^2+192000*#1^3+32000*#1^4&,3,0],Root[-1699+4760*#1+240*#1^2-9600*#1^3+6400*#1^4&,2,0],Root[-4579+5440*#1+16700*#1^2-12000*#1^3+2000*#1^4&,2,0]},
{Root[-19-880*#1-5200*#1^2+32000*#1^4&,3,0],Root[-31859+83480*#1-50960*#1^2-9600*#1^3+6400*#1^4&,2,0],Root[-4579+5440*#1+16700*#1^2-12000*#1^3+2000*#1^4&,2,0]},
{Root[-1759+6400*#1-4240*#1^2-6400*#1^3+6400*#1^4&,4,0],Root[-6899+60840*#1-103760*#1^2-3200*#1^3+6400*#1^4&,2,0],(-5+3*Sqrt[5])/10},
{Root[-67279+62560*#1+66160*#1^2-44800*#1^3+6400*#1^4&,2,0],Root[-1699+12200*#1-20560*#1^2-3200*#1^3+6400*#1^4&,3,0],(5-3*Sqrt[5])/10},
{Root[-89699+161520*#1+151200*#1^2-160000*#1^3+32000*#1^4&,2,0],Root[-1699+7440*#1-6240*#1^2-6400*#1^3+6400*#1^4&,3,0],Root[-4579+5440*#1+16700*#1^2-12000*#1^3+2000*#1^4&,2,0]},
{Root[-31859-46480*#1+55200*#1^2+96000*#1^3+32000*#1^4&,4,0],Root[-6899+37520*#1-38240*#1^2-6400*#1^3+6400*#1^4&,2,0],Root[-4579+5440*#1+16700*#1^2-12000*#1^3+2000*#1^4&,2,0]},
{Root[-1759+6400*#1-4240*#1^2-6400*#1^3+6400*#1^4&,4,0],Root[-6899-60840*#1-103760*#1^2+3200*#1^3+6400*#1^4&,3,0],(-5+3*Sqrt[5])/10},
{Root[-67279+62560*#1+66160*#1^2-44800*#1^3+6400*#1^4&,2,0],Root[-1699-12200*#1-20560*#1^2+3200*#1^3+6400*#1^4&,2,0],(5-3*Sqrt[5])/10},
{Root[-89699+161520*#1+151200*#1^2-160000*#1^3+32000*#1^4&,2,0],Root[-1699-7440*#1-6240*#1^2+6400*#1^3+6400*#1^4&,2,0],Root[-4579+5440*#1+16700*#1^2-12000*#1^3+2000*#1^4&,2,0]},
{Root[-31859-46480*#1+55200*#1^2+96000*#1^3+32000*#1^4&,4,0],Root[-6899-37520*#1-38240*#1^2+6400*#1^3+6400*#1^4&,3,0],Root[-4579+5440*#1+16700*#1^2-12000*#1^3+2000*#1^4&,2,0]},
{Root[-6899+10400*#1+82800*#1^2+96000*#1^3+32000*#1^4&,4,0],Root[-6899-60840*#1-103760*#1^2+3200*#1^3+6400*#1^4&,3,0],Root[1-1300*#1^2+2000*#1^4&,1,0]},
{Root[-1699-7680*#1-400*#1^2+32000*#1^3+32000*#1^4&,4,0],Root[-1699-12200*#1-20560*#1^2+3200*#1^3+6400*#1^4&,2,0],Root[-17359-2960*#1+33900*#1^2-16000*#1^3+2000*#1^4&,1,0]},
{Root[-1699+7120*#1+68000*#1^2-96000*#1^3+32000*#1^4&,1,0],Root[-1699-7440*#1-6240*#1^2+6400*#1^3+6400*#1^4&,2,0],Root[-17359-2960*#1+33900*#1^2-16000*#1^3+2000*#1^4&,1,0]},
{Root[-6899-50800*#1+407200*#1^2-224000*#1^3+32000*#1^4&,1,0],Root[-6899-37520*#1-38240*#1^2+6400*#1^3+6400*#1^4&,3,0],Root[1-1300*#1^2+2000*#1^4&,1,0]},
{Root[-6899+10400*#1+82800*#1^2+96000*#1^3+32000*#1^4&,4,0],Root[-6899+60840*#1-103760*#1^2-3200*#1^3+6400*#1^4&,2,0],Root[1-1300*#1^2+2000*#1^4&,1,0]},
{Root[-1699-7680*#1-400*#1^2+32000*#1^3+32000*#1^4&,4,0],Root[-1699+12200*#1-20560*#1^2-3200*#1^3+6400*#1^4&,3,0],Root[-17359-2960*#1+33900*#1^2-16000*#1^3+2000*#1^4&,1,0]},
{Root[-1699+7120*#1+68000*#1^2-96000*#1^3+32000*#1^4&,1,0],Root[-1699+7440*#1-6240*#1^2-6400*#1^3+6400*#1^4&,3,0],Root[-17359-2960*#1+33900*#1^2-16000*#1^3+2000*#1^4&,1,0]},
{Root[-6899-50800*#1+407200*#1^2-224000*#1^3+32000*#1^4&,1,0],Root[-6899+37520*#1-38240*#1^2-6400*#1^3+6400*#1^4&,2,0],Root[1-1300*#1^2+2000*#1^4&,1,0]},
{Root[-1759-6400*#1-4240*#1^2+6400*#1^3+6400*#1^4&,1,0],Root[-6899+60840*#1-103760*#1^2-3200*#1^3+6400*#1^4&,2,0],(5-3*Sqrt[5])/10},
{Root[-67279-62560*#1+66160*#1^2+44800*#1^3+6400*#1^4&,3,0],Root[-1699+12200*#1-20560*#1^2-3200*#1^3+6400*#1^4&,3,0],(-5+3*Sqrt[5])/10},
{Root[-31859+46480*#1+55200*#1^2-96000*#1^3+32000*#1^4&,1,0],Root[-6899+37520*#1-38240*#1^2-6400*#1^3+6400*#1^4&,2,0],Root[-4579-5440*#1+16700*#1^2+12000*#1^3+2000*#1^4&,3,0]},
{Root[-89699-161520*#1+151200*#1^2+160000*#1^3+32000*#1^4&,3,0],Root[-1699+7440*#1-6240*#1^2-6400*#1^3+6400*#1^4&,3,0],Root[-4579-5440*#1+16700*#1^2+12000*#1^3+2000*#1^4&,3,0]},
{Root[-31919+18480*#1+99760*#1^2-51200*#1^3+6400*#1^4&,1,0],Root[-1699+4760*#1+240*#1^2-9600*#1^3+6400*#1^4&,2,0],(-5+3*Sqrt[5])/10},
{Root[-14719-6000*#1+61360*#1^2-38400*#1^3+6400*#1^4&,1,0],Root[-31859+83480*#1-50960*#1^2-9600*#1^3+6400*#1^4&,2,0],(5-3*Sqrt[5])/10},
{Root[-1759-6400*#1-4240*#1^2+6400*#1^3+6400*#1^4&,1,0],Root[-6899-60840*#1-103760*#1^2+3200*#1^3+6400*#1^4&,3,0],(5-3*Sqrt[5])/10},
{Root[-67279-62560*#1+66160*#1^2+44800*#1^3+6400*#1^4&,3,0],Root[-1699-12200*#1-20560*#1^2+3200*#1^3+6400*#1^4&,2,0],(-5+3*Sqrt[5])/10},
{Root[-31859+46480*#1+55200*#1^2-96000*#1^3+32000*#1^4&,1,0],Root[-6899-37520*#1-38240*#1^2+6400*#1^3+6400*#1^4&,3,0],Root[-4579-5440*#1+16700*#1^2+12000*#1^3+2000*#1^4&,3,0]},
{Root[-89699-161520*#1+151200*#1^2+160000*#1^3+32000*#1^4&,3,0],Root[-1699-7440*#1-6240*#1^2+6400*#1^3+6400*#1^4&,2,0],Root[-4579-5440*#1+16700*#1^2+12000*#1^3+2000*#1^4&,3,0]},
{Root[-31919+18480*#1+99760*#1^2-51200*#1^3+6400*#1^4&,1,0],Root[-1699-4760*#1+240*#1^2+9600*#1^3+6400*#1^4&,3,0],(-5+3*Sqrt[5])/10},
{Root[-14719-6000*#1+61360*#1^2-38400*#1^3+6400*#1^4&,1,0],Root[-31859-83480*#1-50960*#1^2+9600*#1^3+6400*#1^4&,3,0],(5-3*Sqrt[5])/10},
{(-5+2*Sqrt[5])/5,Root[-4579-9800*#1-4760*#1^2+800*#1^3+400*#1^4&,3,0],(5-3*Sqrt[5])/10},
{(-5+2*Sqrt[5])/5,Root[-4579+9800*#1-4760*#1^2-800*#1^3+400*#1^4&,2,0],(5-3*Sqrt[5])/10},
{(5-2*Sqrt[5])/5,Root[-4579-9800*#1-4760*#1^2+800*#1^3+400*#1^4&,3,0],(-5+3*Sqrt[5])/10},
{(5-2*Sqrt[5])/5,Root[-4579+9800*#1-4760*#1^2-800*#1^3+400*#1^4&,2,0],(-5+3*Sqrt[5])/10},
{Root[-31919-18480*#1+99760*#1^2+51200*#1^3+6400*#1^4&,4,0],Root[-1699+4760*#1+240*#1^2-9600*#1^3+6400*#1^4&,2,0],(5-3*Sqrt[5])/10},
{Root[-14719+6000*#1+61360*#1^2+38400*#1^3+6400*#1^4&,4,0],Root[-31859+83480*#1-50960*#1^2-9600*#1^3+6400*#1^4&,2,0],(-5+3*Sqrt[5])/10},
{Root[-31919-18480*#1+99760*#1^2+51200*#1^3+6400*#1^4&,4,0],Root[-1699-4760*#1+240*#1^2+9600*#1^3+6400*#1^4&,3,0],(5-3*Sqrt[5])/10},
{Root[-14719+6000*#1+61360*#1^2+38400*#1^3+6400*#1^4&,4,0],Root[-31859-83480*#1-50960*#1^2+9600*#1^3+6400*#1^4&,3,0],(-5+3*Sqrt[5])/10},
{Root[-9539-75120*#1+314800*#1^2-192000*#1^3+32000*#1^4&,2,0],Root[-1699-4760*#1+240*#1^2+9600*#1^3+6400*#1^4&,3,0],Root[-4579-5440*#1+16700*#1^2+12000*#1^3+2000*#1^4&,3,0]},
{Root[-19+880*#1-5200*#1^2+32000*#1^4&,2,0],Root[-31859-83480*#1-50960*#1^2+9600*#1^3+6400*#1^4&,3,0],Root[-4579-5440*#1+16700*#1^2+12000*#1^3+2000*#1^4&,3,0]},
{Root[-9539-75120*#1+314800*#1^2-192000*#1^3+32000*#1^4&,2,0],Root[-1699+4760*#1+240*#1^2-9600*#1^3+6400*#1^4&,2,0],Root[-4579-5440*#1+16700*#1^2+12000*#1^3+2000*#1^4&,3,0]},
{Root[-19+880*#1-5200*#1^2+32000*#1^4&,2,0],Root[-31859+83480*#1-50960*#1^2-9600*#1^3+6400*#1^4&,2,0],Root[-4579-5440*#1+16700*#1^2+12000*#1^3+2000*#1^4&,3,0]}}

dot[x_List,y_List]:=(x-y).(x-y)

PolyhedronName[DueresSolid]:="Dürers solid"
PolyhedronFaces[DuerersSolid]:={{1,7,8,12,3},{9,11,7,1,5},{8,2,6,10,12},{4,9,5,6,2},{
6,5,1,3,10},{11,4,2,8,7},{11,9,4},{12,10,3}}
PolyhedronVertices[DuerersSolid]:=
{{0,-Sqrt[2/3],Sqrt[1/3+1/Sqrt[5]]/2},
{0,Sqrt[2/3],-Sqrt[1/3+1/Sqrt[5]]/2},{0,-Sqrt[1-Sqrt[5]/3],
Sqrt[-1/4+23/(12*Sqrt[5])]},{0,Sqrt[1-Sqrt[5]/3],
-Sqrt[-1+23/(3*Sqrt[5])]/2},{-(1/Sqrt[2]),-(1/Sqrt[6]),
-Sqrt[1/3+1/Sqrt[5]]/2},{-(1/Sqrt[2]),1/Sqrt[6],Sqrt[1/3+1/Sqrt[5]]/2},
{1/Sqrt[2],-(1/Sqrt[6]),-Sqrt[1/3+1/Sqrt[5]]/2},
{1/Sqrt[2],1/Sqrt[6],Sqrt[1/3+1/Sqrt[5]]/2},
{-Sqrt[3-Sqrt[5]]/2,-Sqrt[(3-Sqrt[5])/3]/2,-Sqrt[-1+23/(3*Sqrt[5])]/2},
{-Sqrt[3-Sqrt[5]]/2,Sqrt[1-Sqrt[5]/3]/2,Sqrt[-1/4+23/(12*Sqrt[5])]},
{(-1+Sqrt[5])/(2*Sqrt[2]),-Sqrt[(3-Sqrt[5])/3]/2,
-Sqrt[-1+23/(3*Sqrt[5])]/2},{(-1+Sqrt[5])/(2*Sqrt[2]),
Sqrt[1-Sqrt[5]/3]/2,Sqrt[-1/4+23/(12*Sqrt[5])]}}

(* Echidnahedron *)

(* netlib *)
PolyhedronName[Echidnahedron]:="echidnahedron"
PolyhedronFaces[Echidnahedron]:={{66,65,45},{66,45,49},{66,49,65},{78,65,49},{78,49,60},{78,60,65},{86,65,60},{86,60,64},{86,64,65},{82,65,64},{82,64,53},{82,53,65},{69,65,53},{69,53,45},{69,45,65},{32,31,49},{32,49,45},{32,45,31},{17,31,45},{17,45,34},{17,34,31},{8,31,34},{8,34,30},{8,30,31},{12,31,30},{12,30,41},{12,41,31},{26,31,41},{26,41,49},{26,49,31},{75,57,60},{75,60,49},{75,49,57},{55,57,49},{55,49,41},{55,41,57},{51,57,41},{51,41,47},{51,47,57},{71,57,47},{71,47,58},{71,58,57},{80,57,58},{80,58,60},{80,60,57},{91,77,64},{91,64,60},{91,60,77},{89,77,60},{89,60,58},{89,58,77},{88,77,58},{88,58,59},{88,59,77},{90,77,59},{90,59,63},{90,63,77},{92,77,63},{92,63,64},{92,64,77},{84,68,53},{84,53,64},{84,64,68},{87,68,64},{87,64,63},{87,63,68},{83,68,63},{83,63,52},{83,52,68},{72,68,52},{72,52,46},{72,46,68},{73,68,46},{73,46,53},{73,53,68},{39,37,45},{39,45,53},{39,53,37},{43,37,53},{43,53,46},{43,46,37},{23,37,46},{23,46,35},{23,35,37},{14,37,35},{14,35,34},{14,34,37},{19,37,34},{19,34,45},{19,45,37},{7,28,29},{7,29,33},{7,33,28},{15,28,33},{15,33,44},{15,44,28},{27,28,44},{27,44,48},{27,48,28},{24,28,48},{24,48,40},{24,40,28},{11,28,40},{11,40,29},{11,29,28},{2,16,33},{2,33,29},{2,29,16},{1,16,29},{1,29,30},{1,30,16},{3,16,30},{3,30,34},{3,34,16},{5,16,34},{5,34,35},{5,35,16},{4,16,35},{4,35,33},{4,33,16},{18,36,44},{18,44,33},{18,33,36},{13,36,33},{13,33,35},{13,35,36},{22,36,35},{22,35,46},{22,46,36},{42,36,46},{42,46,52},{42,52,36},{38,36,52},{38,52,44},{38,44,36},{61,62,48},{61,48,44},{61,44,62},{67,62,44},{67,44,52},{67,52,62},{81,62,52},{81,52,63},{81,63,62},{85,62,63},{85,63,59},{85,59,62},{76,62,59},{76,59,48},{76,48,62},{54,56,40},{54,40,48},{54,48,56},{74,56,48},{74,48,59},{74,59,56},{79,56,59},{79,59,58},{79,58,56},{70,56,58},{70,58,47},{70,47,56},{50,56,47},{50,47,40},{50,40,56},{9,25,29},{9,29,40},{9,40,25},{20,25,40},{20,40,47},{20,47,25},{21,25,47},{21,47,41},{21,41,25},{10,25,41},{10,41,30},{10,30,25},{6,25,30},{6,30,29},{6,29,25}}
PolyhedronVertices[Echidnahedron]:={{-.24286409207111663,-5.6786771416950944,8.1837347473096866},{-.088115369316114376,-4.4975443438745647,7.0887406215445589},{-.072551178939206845,-5.0235019839630802,9.6533513684246168},{.17783751420375989,-3.1123889718614542,7.881613655433787},{.18745671285368741,-3.4374486700836984,9.4666302649211391},{.73265673095502754,-8.0574693871615606,7.6899650206192901},{1.1377941468392668,-4.9652235772374612,4.8232331818903529},{1.178541726259877,-6.3421985556541304,11.537471285152537},{1.4903104790713548,-8.3465110492057237,6.2898044211628173},{1.5058746694516904,-8.8724686893079761,8.8544151680445662},{1.7406991722120361,-6.4353980371073688,4.5180667081862504},{1.7814467516108736,-7.8123730155303668,11.232304811440672},{1.8340678353096789,-1.3388397336222347,6.8990017334438593},{1.859251224298732,-2.1898570719807278,11.048629089774796},{2.161400778475034,-3.8691098672843776,4.2159456163355327},{2.1865841675263047,-4.7201272056076969,8.3655729727116674},{2.2117675565644075,-5.5711445439445025,12.515200329096514},{2.591721583421552,-1.6278813956856534,5.498841133974398},{2.6324691627892043,-3.0048563741383726,12.213079237197478},{2.7317841856166116,-9.3401479226696134,6.5889077284006358},{2.7414033842797401,-9.6652076208945392,8.1739243378902371},{2.8672936656157848,-.56778572189402543,7.8767307773712233},{2.882857855926531,-1.0937433620230633,10.441341524220517},{3.1369216015154968,-6.2479021127516421,3.7221758896606974},{3.1621049905475301,-7.0989194510681786,7.8718032460394276},{3.1872883795664756,-7.949936789407854,12.021430602408202},{3.3969294933040476,-4.6618487988708894,3.5354547861761856},{3.4124936836811447,-5.1878064389705486,6.1000655330641108},{3.4221128823438947,-5.5128661371943738,7.6850821425556149},{3.4280578740615139,-5.7137640790724045,8.6646762799552868},{3.4376770727220398,-6.0388237773002668,10.249692889443369},{3.453241263109031,-6.5647814174069077,12.814303636334351},{3.5828065968222095,-4.5326312812512898,7.5696821541869831},{3.5924257954818447,-4.8576909794847608,9.1546987636665091},{3.6880657658602958,-4.1277107652046604,8.4779551764618308},{3.8428144886274833,-2.9465779673718112,7.3829610507034721},{3.85837867897892,-3.4725356074878592,9.9475717975590936},{4.0932031818117353,-1.0354649552679757,5.6112233377276755},{4.1339507611503717,-2.4124399336770487,12.325461440952689},{4.1857116221816367,-6.0028057411205761,7.2645156804877717},{4.1953308208418657,-6.3278654393526949,8.8495322899788174},{4.2635160949223671,-.3802897975460208,7.0808399588529597},{4.2886994838962519,-1.2313071358651384,11.230467315175093},{4.4457195139804156,-4.4167524272467279,7.0777945770049993},{4.4612837043560499,-4.9427100673539263,9.642405323881333},{4.6160324271104244,-3.7615772695375307,8.5474111981092882},{4.6635865888848297,-6.5065030106701062,7.9841854498009011},{4.8183353116428433,-5.3253702128358554,6.8891913240345267},{4.8338995020231994,-5.8513278529465934,9.4538020709156965},{4.990919532099796,-9.0367731443057513,5.3011293326871506},{5.0161029211059979,-9.8877904826468064,9.4507566890932573},{5.0842881951600002,-3.9402148408448384,7.6820643579400012},{5.0939073938121063,-4.2652745390722046,9.2670809674149182},{5.1456682548502634,-7.8556403464830682,4.2061352069250536},{5.1864158342608176,-9.2326153249127696,10.920373310203452},{5.4212403370141005,-6.7955446727086107,6.5840248503336453},{5.4368045273872042,-7.3215023128254215,9.14863559722132},{5.591553250146004,-6.1403695150079954,8.0536414714500555},{5.6871932205226376,-5.4103893007178194,7.3768978842476964},{5.6968124191838512,-5.7354489989458783,8.9619144937288617},{5.826377752903524,-3.7032988627880736,3.7172930115948923},{5.8419419432855818,-4.2292565028874919,6.2819037584849018},{5.8515611419375353,-4.5543162011241645,7.8669203679592954},{5.8575061336513755,-4.7552141429983069,8.8465145053510605},{5.8671253323149433,-5.0802738412128885,10.431531114846621},{5.8826895227010593,-5.6062314813068032,12.996141861731369},{6.0923306364346091,-2.3181434907877006,4.5101660455107433},{6.1175140254483087,-3.1691608291224377,8.6597934018621925},{6.1426974144705168,-4.0201781674161182,12.809420758247508},{6.3967611600559992,-9.1743369181745034,6.090255123655866},{6.4123253504057544,-9.700294558302196,8.654865870561494},{6.5382156317255985,-.60287265929595191,8.3576723100211805},{6.5478348303744034,-.92793235752339249,9.9426889195096606},{6.6471498531892301,-7.2632239060672622,4.3185174106856382},{6.6878974326073163,-8.6401988845077886,11.032755513942223},{7.067851459465004,-4.6969357362656883,4.0163963188475451},{7.0930348484814779,-5.5479530745927637,8.1660236752075566},{7.1182182375278528,-6.3989704128994381,12.315651031573348},{7.4203677916897673,-8.0782232082184795,5.4829675581049782},{7.4455511807004666,-8.9292405465778792,9.63259491450033},{7.498172264393606,-2.4557072646567981,5.2992918364780819},{7.5389198437759436,-3.8326822430597784,12.013529939725552},{7.7737443465489986,-1.3956115908852233,7.6771814798511767},{7.7893085369186862,-1.9215692309770366,10.241792226730851},{8.1010772897591439,-3.9258817245330329,4.9941253627829656},{8.141824869153687,-5.3028567029330248,11.708363466031958},{8.5469622850478912,-2.2106108930381505,8.8416316272682647},{9.0921623031559344,-6.8306316101143144,7.0649663829956729},{9.1017815018074846,-7.1556913083251197,8.6499829924890039},{9.3521701949524845,-5.2445782962367052,6.8782452795244579},{9.367734385319896,-5.770535936311236,9.4428560263843593},{9.522483108077056,-4.5894031385098053,8.3478619006439574}}

EschersSolid:=Cumulate[Polyhedron[RhombicDodecahedron],Sqrt[2/3]]

fs[l_List]:=Plus@@l/5

g[l1_List,l2_List]:=l1+(l2-l1)/GoldenRatio

GoldenRectangles:=Module[
	{top,bot,topV,botV,botRibbon,topRibbon,i,r=1/Sqrt[10-2Sqrt[5]]},
	top=Table[r{2Cos[2/5 Pi i],2Sin[2/5 Pi i],GoldenRatio+1},{i,5}];
	bot=-top;
	topV=Table[r{
		2GoldenRatio Cos[2/5 Pi i],
		2GoldenRatio Sin[2/5 Pi i],
		GoldenRatio-1},{i,5}];
	botV=-topV;
	botRibbon=Table[{
		bot[[i]],
		bot[[p5[i+1]]],
		botV[[p5[i+1]]],
		topV[[p5[i+3]]],
		botV[[p5[i]]]},{i,5}];
	topRibbon=-botRibbon;
(*	Graphics3D[{Polygon/@{top,bot},Polygon/@botRibbon,Polygon/@topRibbon}//
          Flatten]
			*)
		{Polygon[{fs[topRibbon[[1]]],fs[topRibbon[[2]]],
				fs[botRibbon[[1]]],fs[botRibbon[[2]]]}],
			Polygon[{fs[topRibbon[[3]]],fs[botRibbon[[5]]],
					fs[botRibbon[[3]]],fs[topRibbon[[5]]]}],
			Polygon[{fs[top],fs[topRibbon[[4]]],
					fs[bot],fs[botRibbon[[4]]]}]
			}
]
    
(* Great Rhombic Triacontahedron *)

PolyhedronName[GreatRhombicTriacontahedron]:="great rhombic triacontahedron"
PolyhedronFaces[GreatRhombicTriacontahedron]:=
{{138,71,69},{138,85,137},{148,80,78},{148,87,147},{140,139,70,72,31,32},{150,
    149,77,79,37,38},{144,143,86},{144,73,75},{154,153,88},{154,82,84},{156,
    42,41,83,81,155},{146,36,35,76,74,145},{1,22,34},{1,33,25},{166,158,
    141},{166,142,157},{96,51,73,93,86,126},{89,125,85,92,69,47},{170,160,
    151},{170,152,159},{2,26,40},{2,39,29},{97,135,87,100,78,57},{104,61,82,
    101,88,136},{46,107,64},{46,23,19},{1,53,17},{1,34,63},{179,135,67,111,77,
    149},{181,167,117,161,113,173},{1,63,33},{1,18,54},{50,20,24},{50,65,
    108},{180,155,81,112,68,136},{181,174,114,162,121,167},{56,111,67},{56,27,
    15},{2,43,13},{2,40,66},{177,125,64,107,70,139},{182,171,128,163,115,
    175},{2,66,39},{2,14,44},{60,16,28},{60,68,112},{178,145,74,108,65,
    126},{182,176,116,164,132,171},{138,92,85},{138,113,161},{2,44,21},{2,13,
    26},{178,174,141,158,122,145},{97,57,5,19,23,3},{2,29,14},{2,21,43},{144,
    162,114},{144,86,93},{177,139,118,157,142,173},{104,12,24,20,9,61},{148,
    100,87},{148,115,163},{1,54,30},{1,17,22},{180,176,151,160,131,155},{89,
    47,4,15,27,3},{1,25,18},{1,30,53},{154,164,116},{154,88,101},{179,149,127,
    159,152,175},{96,12,28,16,8,51},{56,55,27},{56,4,6},{166,165,142},{166,
    118,120},{168,91,90,119,117,167},{58,32,31,7,5,57},{60,10,8},{60,28,
    59},{166,124,122},{166,141,165},{62,61,9,11,35,36},{168,167,121,123,94,
    95},{170,169,152},{170,127,129},{46,45,23},{46,5,7},{48,38,37,6,4,
    47},{172,99,98,130,128,171},{170,133,131},{170,151,169},{50,11,9},{50,24,
    49},{172,171,132,134,102,103},{52,51,8,10,41,42},{46,7,31},{46,70,
    107},{50,108,74},{50,35,11},{181,173,142,165,141,174},{172,103,39,66,40,
    99},{56,6,37},{56,77,111},{60,112,81},{60,41,10},{182,175,152,169,151,
    176},{168,95,33,63,34,91},{148,130,98},{148,78,100},{166,157,118},{166,
    105,124},{89,3,23,45,64,125},{62,36,14,29,39,103},{154,101,82},{154,102,
    134},{166,120,105},{166,122,158},{58,99,40,26,13,32},{96,126,65,49,24,
    12},{170,159,127},{170,109,133},{138,119,90},{138,69,92},{52,42,18,25,33,
    95},{97,3,27,55,67,135},{170,129,109},{170,131,160},{144,93,73},{144,94,
    123},{104,136,68,59,28,12},{48,91,34,22,17,38},{56,15,4},{56,37,79},{144,
    75,94},{144,121,162},{156,110,30,54,18,42},{177,173,113,137,85,125},{138,
    161,117},{138,90,71},{60,83,41},{60,8,16},{150,38,17,53,30,110},{178,126,
    86,143,114,174},{46,19,5},{46,31,72},{154,84,102},{154,132,164},{146,106,
    21,44,14,36},{179,175,115,147,87,135},{148,163,128},{148,98,80},{50,76,
    35},{50,9,20},{140,32,13,43,21,106},{180,136,88,153,116,176},{144,123,
    121},{144,114,143},{46,72,70},{46,64,45},{146,145,122,124,105,106},{48,47,
    69,71,90,91},{138,137,113},{138,117,119},{50,49,65},{50,74,76},{52,95,94,
    75,73,51},{140,106,105,120,118,139},{56,79,77},{56,67,55},{154,134,
    132},{154,116,153},{58,57,78,80,98,99},{156,155,131,133,109,110},{60,59,
    68},{60,81,83},{148,147,115},{148,128,130},{150,110,109,129,127,149},{62,
    103,102,84,82,61}}
PolyhedronVertices[GreatRhombicTriacontahedron]:=
{{0, 0, -Sqrt[125/32 + (55*Sqrt[5])/32]}, {0, 0, Sqrt[125/32 + (55*Sqrt[5])/32]}, 
 {0, (-5 - 3*Sqrt[5])/4, 0}, {0, (-5 - Sqrt[5])/4, -Sqrt[25/32 + (5*Sqrt[5])/32]}, 
 {0, (-5 - Sqrt[5])/4, Sqrt[25/32 + (5*Sqrt[5])/32]}, 
 {0, (-5 - Sqrt[5])/4, -Sqrt[25/16 + (5*Sqrt[5])/8]}, 
 {0, (-5 - Sqrt[5])/4, Sqrt[25/16 + (5*Sqrt[5])/8]}, 
 {0, (5 + Sqrt[5])/4, -Sqrt[25/32 + (5*Sqrt[5])/32]}, 
 {0, (5 + Sqrt[5])/4, Sqrt[25/32 + (5*Sqrt[5])/32]}, 
 {0, (5 + Sqrt[5])/4, -Sqrt[25/16 + (5*Sqrt[5])/8]}, {0, (5 + Sqrt[5])/4, Sqrt[25/16 + (5*Sqrt[5])/8]}, 
 {0, (5 + 3*Sqrt[5])/4, 0}, {-Sqrt[5 - 2*Sqrt[5]]/4, -Sqrt[5]/4, Sqrt[65/32 + (29*Sqrt[5])/32]}, 
 {-Sqrt[5 - 2*Sqrt[5]]/4, Sqrt[5]/4, Sqrt[65/32 + (29*Sqrt[5])/32]}, 
 {-Sqrt[5 - 2*Sqrt[5]]/4, (-5 - 2*Sqrt[5])/4, -Sqrt[5/16 + Sqrt[5]/8]}, 
 {-Sqrt[5 - 2*Sqrt[5]]/4, (5 + 2*Sqrt[5])/4, -Sqrt[5/16 + Sqrt[5]/8]}, 
 {Sqrt[5 - 2*Sqrt[5]]/4, -Sqrt[5]/4, -Sqrt[65/32 + (29*Sqrt[5])/32]}, 
 {Sqrt[5 - 2*Sqrt[5]]/4, Sqrt[5]/4, -Sqrt[65/32 + (29*Sqrt[5])/32]}, 
 {Sqrt[5 - 2*Sqrt[5]]/4, (-5 - 2*Sqrt[5])/4, Sqrt[5/16 + Sqrt[5]/8]}, 
 {Sqrt[5 - 2*Sqrt[5]]/4, (5 + 2*Sqrt[5])/4, Sqrt[5/16 + Sqrt[5]/8]}, 
 {-Sqrt[(5 - Sqrt[5])/2]/2, 0, Sqrt[65/32 + (29*Sqrt[5])/32]}, 
 {-Sqrt[(5 - Sqrt[5])/2]/4, (-5 - Sqrt[5])/8, -Sqrt[45/16 + (9*Sqrt[5])/8]}, 
 {-Sqrt[(5 - Sqrt[5])/2]/4, (5*(-1 - Sqrt[5]))/8, Sqrt[5/32 + Sqrt[5]/32]}, 
 {-Sqrt[(5 - Sqrt[5])/2]/4, (5*(1 + Sqrt[5]))/8, Sqrt[5/32 + Sqrt[5]/32]}, 
 {-Sqrt[(5 - Sqrt[5])/2]/4, (5 + Sqrt[5])/8, -Sqrt[45/16 + (9*Sqrt[5])/8]}, 
 {Sqrt[(5 - Sqrt[5])/2]/4, (-5 - Sqrt[5])/8, Sqrt[45/16 + (9*Sqrt[5])/8]}, 
 {Sqrt[(5 - Sqrt[5])/2]/4, (5*(-1 - Sqrt[5]))/8, -Sqrt[5/32 + Sqrt[5]/32]}, 
 {Sqrt[(5 - Sqrt[5])/2]/4, (5*(1 + Sqrt[5]))/8, -Sqrt[5/32 + Sqrt[5]/32]}, 
 {Sqrt[(5 - Sqrt[5])/2]/4, (5 + Sqrt[5])/8, Sqrt[45/16 + (9*Sqrt[5])/8]}, 
 {Sqrt[(5 - Sqrt[5])/2]/2, 0, -Sqrt[65/32 + (29*Sqrt[5])/32]}, 
 {-Sqrt[5/32 + Sqrt[5]/32], (-5 - 3*Sqrt[5])/8, Sqrt[45/32 + (9*Sqrt[5])/32]}, 
 {-Sqrt[5/32 + Sqrt[5]/32], (-5 - 3*Sqrt[5])/8, Sqrt[25/8 + (11*Sqrt[5])/8]}, 
 {-Sqrt[5/32 + Sqrt[5]/32], (5 - Sqrt[5])/8, -Sqrt[65/32 + (29*Sqrt[5])/32]}, 
 {-Sqrt[5/32 + Sqrt[5]/32], (-5 + Sqrt[5])/8, -Sqrt[65/32 + (29*Sqrt[5])/32]}, 
 {-Sqrt[5/32 + Sqrt[5]/32], (5 + 3*Sqrt[5])/8, Sqrt[45/32 + (9*Sqrt[5])/32]}, 
 {-Sqrt[5/32 + Sqrt[5]/32], (5 + 3*Sqrt[5])/8, Sqrt[25/8 + (11*Sqrt[5])/8]}, 
 {Sqrt[5/32 + Sqrt[5]/32], (-5 - 3*Sqrt[5])/8, -Sqrt[45/32 + (9*Sqrt[5])/32]}, 
 {Sqrt[5/32 + Sqrt[5]/32], (-5 - 3*Sqrt[5])/8, -Sqrt[25/8 + (11*Sqrt[5])/8]}, 
 {Sqrt[5/32 + Sqrt[5]/32], (5 - Sqrt[5])/8, Sqrt[65/32 + (29*Sqrt[5])/32]}, 
 {Sqrt[5/32 + Sqrt[5]/32], (-5 + Sqrt[5])/8, Sqrt[65/32 + (29*Sqrt[5])/32]}, 
 {Sqrt[5/32 + Sqrt[5]/32], (5 + 3*Sqrt[5])/8, -Sqrt[45/32 + (9*Sqrt[5])/32]}, 
 {Sqrt[5/32 + Sqrt[5]/32], (5 + 3*Sqrt[5])/8, -Sqrt[25/8 + (11*Sqrt[5])/8]}, 
 {-Sqrt[5/16 + Sqrt[5]/8], -Sqrt[5]/4, Sqrt[45/16 + (9*Sqrt[5])/8]}, 
 {-Sqrt[5/16 + Sqrt[5]/8], Sqrt[5]/4, Sqrt[45/16 + (9*Sqrt[5])/8]}, 
 {-Sqrt[5/16 + Sqrt[5]/8], (-5 - 2*Sqrt[5])/4, Sqrt[5 - 2*Sqrt[5]]/4}, 
 {-Sqrt[5/16 + Sqrt[5]/8], (-5 - 2*Sqrt[5])/4, Sqrt[25/32 + (11*Sqrt[5])/32]}, 
 {-Sqrt[5/16 + Sqrt[5]/8], (-5 - 2*Sqrt[5])/4, -Sqrt[5/4 + Sqrt[5]/2]}, 
 {-Sqrt[5/16 + Sqrt[5]/8], (-5 - 2*Sqrt[5])/4, -Sqrt[85/16 + (19*Sqrt[5])/8]}, 
 {-Sqrt[5/16 + Sqrt[5]/8], (5 + 2*Sqrt[5])/4, Sqrt[5 - 2*Sqrt[5]]/4}, 
 {-Sqrt[5/16 + Sqrt[5]/8], (5 + 2*Sqrt[5])/4, Sqrt[25/32 + (11*Sqrt[5])/32]}, 
 {-Sqrt[5/16 + Sqrt[5]/8], (5 + 2*Sqrt[5])/4, -Sqrt[5/4 + Sqrt[5]/2]}, 
 {-Sqrt[5/16 + Sqrt[5]/8], (5 + 2*Sqrt[5])/4, -Sqrt[85/16 + (19*Sqrt[5])/8]}, 
 {Sqrt[5/16 + Sqrt[5]/8], -Sqrt[5]/4, -Sqrt[45/16 + (9*Sqrt[5])/8]}, 
 {Sqrt[5/16 + Sqrt[5]/8], Sqrt[5]/4, -Sqrt[45/16 + (9*Sqrt[5])/8]}, 
 {Sqrt[5/16 + Sqrt[5]/8], (-5 - 2*Sqrt[5])/4, -Sqrt[5 - 2*Sqrt[5]]/4}, 
 {Sqrt[5/16 + Sqrt[5]/8], (-5 - 2*Sqrt[5])/4, -Sqrt[25/32 + (11*Sqrt[5])/32]}, 
 {Sqrt[5/16 + Sqrt[5]/8], (-5 - 2*Sqrt[5])/4, Sqrt[5/4 + Sqrt[5]/2]}, 
 {Sqrt[5/16 + Sqrt[5]/8], (-5 - 2*Sqrt[5])/4, Sqrt[85/16 + (19*Sqrt[5])/8]}, 
 {Sqrt[5/16 + Sqrt[5]/8], (5 + 2*Sqrt[5])/4, -Sqrt[5 - 2*Sqrt[5]]/4}, 
 {Sqrt[5/16 + Sqrt[5]/8], (5 + 2*Sqrt[5])/4, -Sqrt[25/32 + (11*Sqrt[5])/32]}, 
 {Sqrt[5/16 + Sqrt[5]/8], (5 + 2*Sqrt[5])/4, Sqrt[5/4 + Sqrt[5]/2]}, 
 {Sqrt[5/16 + Sqrt[5]/8], (5 + 2*Sqrt[5])/4, Sqrt[85/16 + (19*Sqrt[5])/8]}, 
 {-Sqrt[5/8 + Sqrt[5]/8], 0, -Sqrt[45/16 + (9*Sqrt[5])/8]}, {-Sqrt[5/8 + Sqrt[5]/8], (-5 - Sqrt[5])/4, 
  Sqrt[5/32 + Sqrt[5]/32]}, {-Sqrt[5/8 + Sqrt[5]/8], (5 + Sqrt[5])/4, Sqrt[5/32 + Sqrt[5]/32]}, 
 {Sqrt[5/8 + Sqrt[5]/8], 0, Sqrt[45/16 + (9*Sqrt[5])/8]}, {Sqrt[5/8 + Sqrt[5]/8], (-5 - Sqrt[5])/4, 
  -Sqrt[5/32 + Sqrt[5]/32]}, {Sqrt[5/8 + Sqrt[5]/8], (5 + Sqrt[5])/4, -Sqrt[5/32 + Sqrt[5]/32]}, 
 {-Sqrt[25/32 + (5*Sqrt[5])/32], (-5 - 3*Sqrt[5])/8, -Sqrt[25/32 + (5*Sqrt[5])/32]}, 
 {-Sqrt[25/32 + (5*Sqrt[5])/32], (-5 - 3*Sqrt[5])/8, Sqrt[25/32 + (5*Sqrt[5])/32]}, 
 {-Sqrt[25/32 + (5*Sqrt[5])/32], (-5 - 3*Sqrt[5])/8, -Sqrt[25/16 + (5*Sqrt[5])/8]}, 
 {-Sqrt[25/32 + (5*Sqrt[5])/32], (-5 - 3*Sqrt[5])/8, Sqrt[25/16 + (5*Sqrt[5])/8]}, 
 {-Sqrt[25/32 + (5*Sqrt[5])/32], (5 + 3*Sqrt[5])/8, -Sqrt[25/32 + (5*Sqrt[5])/32]}, 
 {-Sqrt[25/32 + (5*Sqrt[5])/32], (5 + 3*Sqrt[5])/8, Sqrt[25/32 + (5*Sqrt[5])/32]}, 
 {-Sqrt[25/32 + (5*Sqrt[5])/32], (5 + 3*Sqrt[5])/8, -Sqrt[25/16 + (5*Sqrt[5])/8]}, 
 {-Sqrt[25/32 + (5*Sqrt[5])/32], (5 + 3*Sqrt[5])/8, Sqrt[25/16 + (5*Sqrt[5])/8]}, 
 {Sqrt[25/32 + (5*Sqrt[5])/32], (-5 - 3*Sqrt[5])/8, -Sqrt[25/32 + (5*Sqrt[5])/32]}, 
 {Sqrt[25/32 + (5*Sqrt[5])/32], (-5 - 3*Sqrt[5])/8, Sqrt[25/32 + (5*Sqrt[5])/32]}, 
 {Sqrt[25/32 + (5*Sqrt[5])/32], (-5 - 3*Sqrt[5])/8, -Sqrt[25/16 + (5*Sqrt[5])/8]}, 
 {Sqrt[25/32 + (5*Sqrt[5])/32], (-5 - 3*Sqrt[5])/8, Sqrt[25/16 + (5*Sqrt[5])/8]}, 
 {Sqrt[25/32 + (5*Sqrt[5])/32], (5 + 3*Sqrt[5])/8, -Sqrt[25/32 + (5*Sqrt[5])/32]}, 
 {Sqrt[25/32 + (5*Sqrt[5])/32], (5 + 3*Sqrt[5])/8, Sqrt[25/32 + (5*Sqrt[5])/32]}, 
 {Sqrt[25/32 + (5*Sqrt[5])/32], (5 + 3*Sqrt[5])/8, -Sqrt[25/16 + (5*Sqrt[5])/8]}, 
 {Sqrt[25/32 + (5*Sqrt[5])/32], (5 + 3*Sqrt[5])/8, Sqrt[25/16 + (5*Sqrt[5])/8]}, 
 {-Sqrt[45/32 + (9*Sqrt[5])/32], (-5 - 3*Sqrt[5])/8, -Sqrt[5/32 + Sqrt[5]/32]}, 
 {-Sqrt[45/32 + (9*Sqrt[5])/32], (5 + 3*Sqrt[5])/8, -Sqrt[5/32 + Sqrt[5]/32]}, 
 {Sqrt[45/32 + (9*Sqrt[5])/32], (-5 - 3*Sqrt[5])/8, Sqrt[5/32 + Sqrt[5]/32]}, 
 {Sqrt[45/32 + (9*Sqrt[5])/32], (5 + 3*Sqrt[5])/8, Sqrt[5/32 + Sqrt[5]/32]}, 
 {-Sqrt[25/32 + (11*Sqrt[5])/32], (-15 - 7*Sqrt[5])/8, -Sqrt[5/16 + Sqrt[5]/8]}, 
 {-Sqrt[25/32 + (11*Sqrt[5])/32], (-5 - Sqrt[5])/8, -Sqrt[45/32 + (9*Sqrt[5])/32]}, 
 {-Sqrt[25/32 + (11*Sqrt[5])/32], (-5 - Sqrt[5])/8, -Sqrt[25/8 + (11*Sqrt[5])/8]}, 
 {-Sqrt[25/32 + (11*Sqrt[5])/32], (5*(-1 - Sqrt[5]))/8, -Sqrt[5/16 + Sqrt[5]/8]}, 
 {-Sqrt[25/32 + (11*Sqrt[5])/32], (5*(1 + Sqrt[5]))/8, -Sqrt[5/16 + Sqrt[5]/8]}, 
 {-Sqrt[25/32 + (11*Sqrt[5])/32], (5 + Sqrt[5])/8, -Sqrt[45/32 + (9*Sqrt[5])/32]}, 
 {-Sqrt[25/32 + (11*Sqrt[5])/32], (5 + Sqrt[5])/8, -Sqrt[25/8 + (11*Sqrt[5])/8]}, 
 {-Sqrt[25/32 + (11*Sqrt[5])/32], (15 + 7*Sqrt[5])/8, -Sqrt[5/16 + Sqrt[5]/8]}, 
 {Sqrt[25/32 + (11*Sqrt[5])/32], (-15 - 7*Sqrt[5])/8, Sqrt[5/16 + Sqrt[5]/8]}, 
 {Sqrt[25/32 + (11*Sqrt[5])/32], (-5 - Sqrt[5])/8, Sqrt[45/32 + (9*Sqrt[5])/32]}, 
 {Sqrt[25/32 + (11*Sqrt[5])/32], (-5 - Sqrt[5])/8, Sqrt[25/8 + (11*Sqrt[5])/8]}, 
 {Sqrt[25/32 + (11*Sqrt[5])/32], (5*(-1 - Sqrt[5]))/8, Sqrt[5/16 + Sqrt[5]/8]}, 
 {Sqrt[25/32 + (11*Sqrt[5])/32], (5*(1 + Sqrt[5]))/8, Sqrt[5/16 + Sqrt[5]/8]}, 
 {Sqrt[25/32 + (11*Sqrt[5])/32], (5 + Sqrt[5])/8, Sqrt[45/32 + (9*Sqrt[5])/32]}, 
 {Sqrt[25/32 + (11*Sqrt[5])/32], (5 + Sqrt[5])/8, Sqrt[25/8 + (11*Sqrt[5])/8]}, 
 {Sqrt[25/32 + (11*Sqrt[5])/32], (15 + 7*Sqrt[5])/8, Sqrt[5/16 + Sqrt[5]/8]}, 
 {-Sqrt[5/4 + Sqrt[5]/2], 0, Sqrt[45/32 + (9*Sqrt[5])/32]}, 
 {-Sqrt[5/4 + Sqrt[5]/2], 0, Sqrt[25/8 + (11*Sqrt[5])/8]}, {-Sqrt[5/4 + Sqrt[5]/2], (-5 - Sqrt[5])/4, 
  Sqrt[5/16 + Sqrt[5]/8]}, {-Sqrt[5/4 + Sqrt[5]/2], (5 + Sqrt[5])/4, Sqrt[5/16 + Sqrt[5]/8]}, 
 {Sqrt[5/4 + Sqrt[5]/2], 0, -Sqrt[45/32 + (9*Sqrt[5])/32]}, 
 {Sqrt[5/4 + Sqrt[5]/2], 0, -Sqrt[25/8 + (11*Sqrt[5])/8]}, {Sqrt[5/4 + Sqrt[5]/2], (-5 - Sqrt[5])/4, 
  -Sqrt[5/16 + Sqrt[5]/8]}, {Sqrt[5/4 + Sqrt[5]/2], (5 + Sqrt[5])/4, -Sqrt[5/16 + Sqrt[5]/8]}, 
 {-Sqrt[65/32 + (19*Sqrt[5])/32], (-5 - Sqrt[5])/8, -Sqrt[5/32 + Sqrt[5]/32]}, 
 {-Sqrt[65/32 + (19*Sqrt[5])/32], (5 + Sqrt[5])/8, -Sqrt[5/32 + Sqrt[5]/32]}, 
 {Sqrt[65/32 + (19*Sqrt[5])/32], (-5 - Sqrt[5])/8, Sqrt[5/32 + Sqrt[5]/32]}, 
 {Sqrt[65/32 + (19*Sqrt[5])/32], (5 + Sqrt[5])/8, Sqrt[5/32 + Sqrt[5]/32]}, 
 {-Sqrt[25/16 + (5*Sqrt[5])/8], -Sqrt[5]/4, -Sqrt[25/32 + (5*Sqrt[5])/32]}, 
 {-Sqrt[25/16 + (5*Sqrt[5])/8], -Sqrt[5]/4, Sqrt[25/32 + (5*Sqrt[5])/32]}, 
 {-Sqrt[25/16 + (5*Sqrt[5])/8], -Sqrt[5]/4, -Sqrt[25/16 + (5*Sqrt[5])/8]}, 
 {-Sqrt[25/16 + (5*Sqrt[5])/8], -Sqrt[5]/4, Sqrt[25/16 + (5*Sqrt[5])/8]}, 
 {-Sqrt[25/16 + (5*Sqrt[5])/8], Sqrt[5]/4, -Sqrt[25/32 + (5*Sqrt[5])/32]}, 
 {-Sqrt[25/16 + (5*Sqrt[5])/8], Sqrt[5]/4, Sqrt[25/32 + (5*Sqrt[5])/32]}, 
 {-Sqrt[25/16 + (5*Sqrt[5])/8], Sqrt[5]/4, -Sqrt[25/16 + (5*Sqrt[5])/8]}, 
 {-Sqrt[25/16 + (5*Sqrt[5])/8], Sqrt[5]/4, Sqrt[25/16 + (5*Sqrt[5])/8]}, 
 {-Sqrt[25/16 + (5*Sqrt[5])/8], (-5 - 2*Sqrt[5])/4, 0}, {-Sqrt[25/16 + (5*Sqrt[5])/8], 
  (5 + 2*Sqrt[5])/4, 0}, {Sqrt[25/16 + (5*Sqrt[5])/8], -Sqrt[5]/4, -Sqrt[25/32 + (5*Sqrt[5])/32]}, 
 {Sqrt[25/16 + (5*Sqrt[5])/8], -Sqrt[5]/4, Sqrt[25/32 + (5*Sqrt[5])/32]}, 
 {Sqrt[25/16 + (5*Sqrt[5])/8], -Sqrt[5]/4, -Sqrt[25/16 + (5*Sqrt[5])/8]}, 
 {Sqrt[25/16 + (5*Sqrt[5])/8], -Sqrt[5]/4, Sqrt[25/16 + (5*Sqrt[5])/8]}, 
 {Sqrt[25/16 + (5*Sqrt[5])/8], Sqrt[5]/4, -Sqrt[25/32 + (5*Sqrt[5])/32]}, 
 {Sqrt[25/16 + (5*Sqrt[5])/8], Sqrt[5]/4, Sqrt[25/32 + (5*Sqrt[5])/32]}, 
 {Sqrt[25/16 + (5*Sqrt[5])/8], Sqrt[5]/4, -Sqrt[25/16 + (5*Sqrt[5])/8]}, 
 {Sqrt[25/16 + (5*Sqrt[5])/8], Sqrt[5]/4, Sqrt[25/16 + (5*Sqrt[5])/8]}, 
 {Sqrt[25/16 + (5*Sqrt[5])/8], (-5 - 2*Sqrt[5])/4, 0}, {Sqrt[25/16 + (5*Sqrt[5])/8], (5 + 2*Sqrt[5])/4, 
  0}, {-Sqrt[65/32 + (29*Sqrt[5])/32], (-5 - 3*Sqrt[5])/8, -Sqrt[5 - 2*Sqrt[5]]/4}, 
 {-Sqrt[65/32 + (29*Sqrt[5])/32], (-5 - 3*Sqrt[5])/8, -Sqrt[25/32 + (11*Sqrt[5])/32]}, 
 {-Sqrt[65/32 + (29*Sqrt[5])/32], (-5 - 3*Sqrt[5])/8, Sqrt[5/4 + Sqrt[5]/2]}, 
 {-Sqrt[65/32 + (29*Sqrt[5])/32], (-5 - 3*Sqrt[5])/8, Sqrt[85/16 + (19*Sqrt[5])/8]}, 
 {-Sqrt[65/32 + (29*Sqrt[5])/32], (5 - Sqrt[5])/8, Sqrt[5/32 + Sqrt[5]/32]}, 
 {-Sqrt[65/32 + (29*Sqrt[5])/32], (-5 + Sqrt[5])/8, Sqrt[5/32 + Sqrt[5]/32]}, 
 {-Sqrt[65/32 + (29*Sqrt[5])/32], (5 + 3*Sqrt[5])/8, -Sqrt[5 - 2*Sqrt[5]]/4}, 
 {-Sqrt[65/32 + (29*Sqrt[5])/32], (5 + 3*Sqrt[5])/8, -Sqrt[25/32 + (11*Sqrt[5])/32]}, 
 {-Sqrt[65/32 + (29*Sqrt[5])/32], (5 + 3*Sqrt[5])/8, Sqrt[5/4 + Sqrt[5]/2]}, 
 {-Sqrt[65/32 + (29*Sqrt[5])/32], (5 + 3*Sqrt[5])/8, Sqrt[85/16 + (19*Sqrt[5])/8]}, 
 {Sqrt[65/32 + (29*Sqrt[5])/32], (-5 - 3*Sqrt[5])/8, Sqrt[5 - 2*Sqrt[5]]/4}, 
 {Sqrt[65/32 + (29*Sqrt[5])/32], (-5 - 3*Sqrt[5])/8, Sqrt[25/32 + (11*Sqrt[5])/32]}, 
 {Sqrt[65/32 + (29*Sqrt[5])/32], (-5 - 3*Sqrt[5])/8, -Sqrt[5/4 + Sqrt[5]/2]}, 
 {Sqrt[65/32 + (29*Sqrt[5])/32], (-5 - 3*Sqrt[5])/8, -Sqrt[85/16 + (19*Sqrt[5])/8]}, 
 {Sqrt[65/32 + (29*Sqrt[5])/32], (5 - Sqrt[5])/8, -Sqrt[5/32 + Sqrt[5]/32]}, 
 {Sqrt[65/32 + (29*Sqrt[5])/32], (-5 + Sqrt[5])/8, -Sqrt[5/32 + Sqrt[5]/32]}, 
 {Sqrt[65/32 + (29*Sqrt[5])/32], (5 + 3*Sqrt[5])/8, Sqrt[5 - 2*Sqrt[5]]/4}, 
 {Sqrt[65/32 + (29*Sqrt[5])/32], (5 + 3*Sqrt[5])/8, Sqrt[25/32 + (11*Sqrt[5])/32]}, 
 {Sqrt[65/32 + (29*Sqrt[5])/32], (5 + 3*Sqrt[5])/8, -Sqrt[5/4 + Sqrt[5]/2]}, 
 {Sqrt[65/32 + (29*Sqrt[5])/32], (5 + 3*Sqrt[5])/8, -Sqrt[85/16 + (19*Sqrt[5])/8]}, 
 {-Sqrt[85/32 + (31*Sqrt[5])/32], (-5 - Sqrt[5])/8, Sqrt[5/16 + Sqrt[5]/8]}, 
 {-Sqrt[85/32 + (31*Sqrt[5])/32], (5 + Sqrt[5])/8, Sqrt[5/16 + Sqrt[5]/8]}, 
 {Sqrt[85/32 + (31*Sqrt[5])/32], (-5 - Sqrt[5])/8, -Sqrt[5/16 + Sqrt[5]/8]}, 
 {Sqrt[85/32 + (31*Sqrt[5])/32], (5 + Sqrt[5])/8, -Sqrt[5/16 + Sqrt[5]/8]}, 
 {-Sqrt[45/16 + (9*Sqrt[5])/8], -Sqrt[5]/4, -Sqrt[5/16 + Sqrt[5]/8]}, 
 {-Sqrt[45/16 + (9*Sqrt[5])/8], Sqrt[5]/4, -Sqrt[5/16 + Sqrt[5]/8]}, 
 {Sqrt[45/16 + (9*Sqrt[5])/8], -Sqrt[5]/4, Sqrt[5/16 + Sqrt[5]/8]}, 
 {Sqrt[45/16 + (9*Sqrt[5])/8], Sqrt[5]/4, Sqrt[5/16 + Sqrt[5]/8]}, 
 {-Sqrt[25/8 + (11*Sqrt[5])/8], 0, Sqrt[5 - 2*Sqrt[5]]/4}, {-Sqrt[25/8 + (11*Sqrt[5])/8], 0, 
  Sqrt[25/32 + (11*Sqrt[5])/32]}, {-Sqrt[25/8 + (11*Sqrt[5])/8], 0, -Sqrt[5/4 + Sqrt[5]/2]}, 
 {-Sqrt[25/8 + (11*Sqrt[5])/8], 0, -Sqrt[85/16 + (19*Sqrt[5])/8]}, 
 {Sqrt[25/8 + (11*Sqrt[5])/8], 0, -Sqrt[5 - 2*Sqrt[5]]/4}, {Sqrt[25/8 + (11*Sqrt[5])/8], 0, 
  -Sqrt[25/32 + (11*Sqrt[5])/32]}, {Sqrt[25/8 + (11*Sqrt[5])/8], 0, Sqrt[5/4 + Sqrt[5]/2]}, 
 {Sqrt[25/8 + (11*Sqrt[5])/8], 0, Sqrt[85/16 + (19*Sqrt[5])/8]}, 
 {-Sqrt[125/32 + (55*Sqrt[5])/32], (-5 - Sqrt[5])/8, 0}, {-Sqrt[125/32 + (55*Sqrt[5])/32], 
  (5 + Sqrt[5])/8, 0}, {Sqrt[125/32 + (55*Sqrt[5])/32], (-5 - Sqrt[5])/8, 0}, 
 {Sqrt[125/32 + (55*Sqrt[5])/32], (5 + Sqrt[5])/8, 0}, {-Sqrt[85/16 + (19*Sqrt[5])/8], 
  (-5 - 2*Sqrt[5])/4, Sqrt[5/16 + Sqrt[5]/8]}, {-Sqrt[85/16 + (19*Sqrt[5])/8], (5 + 2*Sqrt[5])/4, 
  Sqrt[5/16 + Sqrt[5]/8]}, {Sqrt[85/16 + (19*Sqrt[5])/8], (-5 - 2*Sqrt[5])/4, -Sqrt[5/16 + Sqrt[5]/8]}, 
 {Sqrt[85/16 + (19*Sqrt[5])/8], (5 + 2*Sqrt[5])/4, -Sqrt[5/16 + Sqrt[5]/8]}, 
 {-Sqrt[65/8 + (29*Sqrt[5])/8], 0, -Sqrt[5/16 + Sqrt[5]/8]}, 
 {Sqrt[65/8 + (29*Sqrt[5])/8], 0, Sqrt[5/16 + Sqrt[5]/8]}}

Hexahedron[{3,3,3,3,3,3}] :=With[{
      a ={0,0,0},b={-1,0,1},c={.5,.7,1},d={.5,-.7,1},e ={0,0,2}
      },
     Polygon/@{{a,b,c},{a,b,d},{a,c,d},{e,b,c},{e,b,d},{e,c,d}}
    ]

Hexahedron[{3,3,3,3,3,5}] :=With[{
      a ={0,0,1},b={0.,1.,0.},c={0.95,0.31,0.},d={0.59,-0.81,0.},
      e ={-0.59,-0.81,0.},f={-0.95,0.31,0.}
      }, 
    Polygon/@{{a,b,c},{a,c,d},{a,d,e},{a,e,f},{a,f,b},{b,c,d,e,f}}
    ]

Hexahedron[{3,3,3,3,4,4}]:=With[{
      a ={0,0,0},b1={-1,.3,1},b2 ={-1,-.04,.4},c={.5,.7,1},d={.5,-.7,1},
      e ={0,.2,2}
      },
    Polygon/@{{a,b2,b1,c},{a,b2,d},{a,c,d},{e,b1,c},{e,b1,b2,d},{e,c,d}}
    ]

Hexahedron[{3,3,3,4,4,5}]:= With[{
      a ={0,0,0},b1={-1,.3,1},b2 ={-1,-.04,.4},c1={.5,.7,1},c2={1,.2667,0},
      d={.5,-.7,1},e ={0,.2,2}
      },
    Polygon/@{{a,b2,b1,c1,c2},{a,b2,d},{a,c2,d},{e,b1,c1},{e,b1,b2,d},{e,c1,
          c2,d}}
    ]

Hexahedron[{3,3,4,4,4,4}]:= With[{
      a ={0,0,0},b1={-1,.3,1},b2 ={-.571429,-.228571,.571429},c1={.5,.7,1},
      c2 = {0,.933333,.0},d={.5,-.7,1},e ={0,.2,2}
      },
    Polygon/@{{a,b2,b1,c2},{a,b2,d},{a,c2,c1,d},{e,b1,c2,c1},{e,b1,b2,d},{e,
          c1,d}}
    ]

Hexahedron[{3,3,4,4,5,5}]:= With[{
      a ={0,0,0},b1={-1,.3,1},b2 ={-1,-.04,.4},c1={.65,.57,.7},c2={.5,.7,1},
      d1={.5,-.7,1},d2={.65,-.55,.7},e ={0,.2,2}
      },
    Polygon/@{{a,b2,b1,c2,c1},{a,b2,d1,d2},{a,c1,d2},{e,b1,c2},{e,b1,b2,
          d1},{e,c2,c1,d2,d1}}
    ]

Hexahedron[{4,4,4,4,4,4}]:=With[{
      a={0,0,0},b={0,0,1},c={0,1,0},d={1,0,0},e={0,1,1},f={1,0,1},g={1,1,0},
      h={1,1,1}
      },
    Polygon/@{{a,b,e,c},{a,b,f,d},{a,c,g,d},{h,e,c,g},{h,e,b,f},{h,f,d,g}}
    ]

Hexahedron[n_Integer]:=
  Orient[Hexahedron[{{3,3,3,3,3,3},{3,3,3,3,3,5},{3,3,3,3,4,4},{3,3,3,4,4,5},{3,3,4,
            4,4,4},{3,3,4,4,5,5},{4,4,4,4,4,4}}[[n]]]]/; 1<=n<=7

h[l1_List,l2_List]:=g[l2,l1]

Jack:=Cuboid/@{{-1.5,-.5,-.5},{-.5,-.5,-.5},{.5,-.5,-.5},{-.5,-1.5,-.5},
	{-.5,.5,-.5},{-.5,-.5,-1.5},{-.5,-.5,.5}}

(*
JessensOrthogonalPolyhedron:=oi[2]
*)

PolyhedronName[JessensOrthogonalPolyhedron]:="Jessen's orthogonal polyhedron"
PolyhedronFaces[JessensOrthogonalPolyhedron]:=
{{6,1,3},{6,1,12},{6,12,5},{6,3,11},{6,5,11},{12,1,10},{12,10,2},{5,12,2},{3,
    11,9},{1,3,8},{1,10,8},{10,2,7},{10,8,7},{3,9,8},{9,8,7},{5,2,4},{2,7,
    4},{7,9,4},{11,9,4},{11,4,5}}
PolyhedronVertices[JessensOrthogonalPolyhedron]:=
{{0, -GoldenRatio^(-1), 1 - GoldenRatio^(-1)}, {0, -GoldenRatio^(-1), 
  -1 + GoldenRatio^(-1)}, {0, GoldenRatio^(-1), 1 - GoldenRatio^(-1)}, 
 {0, GoldenRatio^(-1), -1 + GoldenRatio^(-1)}, {1 - GoldenRatio^(-1), 0, -GoldenRatio^(-1)}, 
 {1 - GoldenRatio^(-1), 0, GoldenRatio^(-1)}, {-1 + GoldenRatio^(-1), 0, -GoldenRatio^(-1)}, 
 {-1 + GoldenRatio^(-1), 0, GoldenRatio^(-1)}, {-GoldenRatio^(-1), 1 - GoldenRatio^(-1), 0}, 
 {-GoldenRatio^(-1), -1 + GoldenRatio^(-1), 0}, {GoldenRatio^(-1), 1 - GoldenRatio^(-1), 0}, 
 {GoldenRatio^(-1), -1 + GoldenRatio^(-1), 0}}

line[l_List]:=Line[Join[l,{l[[1]]}]]

lline[l_List]:=Module[{i,j},
	Table[cline[Table[rotz[j Pi/2].l[[i]],{i,Length[l]}]],{j,0,3}]
]

(* Norm *)

norm[a_List,b_List]:=norm[a-b]

norm[a_List]:=Sqrt[normsq[a]]

normsq[a_List,b_List]:=normsq[a-b]

normsq[a_List]:=a.a

OctahedronIcosahedron[1]:=oi[1]

OctahedronIcosahedron[2]:=Module[{v=OctahedronIcosahedronPoints[2]},
		Polygon/@{v[[{1,3,6}]],v[[{1,2,4}]],v[[{1,4,6}]],
				v[[{2,1,9}]],v[[{4,5,6}]],v[[{2,4,8}]],v[[{1,3,9}]],
				v[[{4,5,8}]],v[[{9,2,7}]],v[[{8,2,7}]],v[[{9,11,7}]],
				v[[{7,8,10}]],v[[{7,10,11}]],v[[{3,9,11}]],v[[{3,11,12}]],
				v[[{10,11,12}]],v[[{12,3,6}]],v[[{12,6,5}]],
				v[[{5,10,12}]],v[[{5,10,8}]]}
]

OctahedronIcosahedronPoints[n_]:=Module[
	{
		xp={1,0,0},xm={-1,0,0},
		yp={0,1,0},ym={0,-1,0},
		zp={0,0,1},zm={0,0,-1},
		v
	},
	v[1]={
		g[zp,xp],g[ym,zp],g[yp,zp],
		g[xp,ym],g[zm,xp],g[xp,yp],
		g[xm,ym],g[ym,zm],g[zp,xm],
		g[zm,xm],g[xm,yp],g[yp,zm]
	};
	v[2]={
		h[zp,xp],h[ym,zp],h[yp,zp],
		h[xp,ym],h[zm,xp],h[xp,yp],
		h[xm,ym],h[ym,zm],h[zp,xm],
		h[zm,xm],h[xm,yp],h[yp,zm]
	};
	v[n]
]

OctahedronVertices:={PointSize[.03],
	Point/@{{0,0,1},{0,0,-1},{1,0,0},{-1,0,0},{0,1,0},{0,-1,0}}
}

(* Octahemioctahedron *)

PolyhedronName[Octahemioctahedron]:="octahemioctahedron"
NetFaces[Octahemioctahedron]:={{7,9,12},{1,2,4},{2,6,4},{4,6,9},{6,11,9},{4,9,7},{13,8,10},{8,5,10},{10,5,7},{5,3,7},{10,7,12},{16,19,14},{19,17,14},{14,17,12},{17,15,12},{14,12,9},{22,27,25},{21,26,24},{26,29,24},{24,29,27},{29,31,27},{24,27,22},{18,15,20},{15,17,20},{20,17,22},{17,19,22},{20,22,25},{33,32,30},{32,28,30},{30,28,25},{28,23,25},{30,25,27}}
NetVertices[Octahemioctahedron]:={{-2,-(1/Sqrt[3])},{-3/2,-5/(2Sqrt[3])},{-3/2,1/(2Sqrt[3])},{-1,-(1/Sqrt[3])},{-1,2/Sqrt[3]},{-1/2,-5/(2Sqrt[3])},{-1/2,1/(2Sqrt[3])},{-1/2,7/(2Sqrt[3])},{0,-(1/Sqrt[3])},{0,2/Sqrt[3]},{1/2,-5/(2Sqrt[3])},{1/2,1/(2Sqrt[3])},{1/2,7/(2Sqrt[3])},{1,-(1/Sqrt[3])},{1,2/Sqrt[3]},{3/2,-5/(2Sqrt[3])},{3/2,1/(2Sqrt[3])},{3/2,7/(2Sqrt[3])},{2,-(1/Sqrt[3])},{2,2/Sqrt[3]},{5/2,-5/(2Sqrt[3])},{5/2,1/(2Sqrt[3])},{5/2,7/(2Sqrt[3])},{3,-(1/Sqrt[3])},{3,2/Sqrt[3]},{7/2,-5/(2Sqrt[3])},{7/2,1/(2Sqrt[3])},{7/2,7/(2Sqrt[3])},{4,-(1/Sqrt[3])},{4,2/Sqrt[3]},{9/2,1/(2Sqrt[3])},{9/2,7/(2Sqrt[3])},{5,2/Sqrt[3]}}
PolyhedronFaces[Octahemioctahedron]:={{6,2,9},{6,3,7},{3,1,7},{7,1,2},{1,4,2},{7,2,6},{9,13,7},{13,10,7},{7,10,6},{10,3,6},{7,6,9},{2,4,7},{4,11,7},{7,11,9},{11,13,9},{7,9,2},{8,5,12},{8,4,7},{4,1,7},{7,1,5},{1,3,5},{7,5,8},{12,13,7},{13,11,7},{7,11,8},{11,4,8},{7,8,12},{5,3,7},{3,10,7},{7,10,12},{10,13,12},{7,12,5}}
PolyhedronVertices[Octahemioctahedron]:={{-1.8124523700696942,3.7178418226141683,-.68831620013553378},{-1.5331504343927934,2.7602992293855104,-.75974876186145617},{-1.3879997967813838,3.7742334241201059,.21537622558519083},{-1.2759995935655044,3.4107899190515568,-1.474406237659885},{-1.1308489559560146,4.424724113788677,-.49928125021377046},{-1.1086978611039091,2.8166908308896541,.1439436638619199},{-.85154702027722935,3.4671815205576127,-.57071381193861869},{-.59439617944975366,4.1176722102251516,-1.2853712877383561},{-.57224508459965393,2.50963892732693,-.64214637366434788},{-.42709444698891891,3.5235731220635502,.33297861378210607},{-.31509424377152633,3.1601296169942332,-1.3568038494646975},{-.16994360616160805,4.1740638117297629,-.38167886201616724},{.10935832951571622,3.2165212185008803,-.45311142374191835}}

oi[n_]:=Module[{v=OctahedronIcosahedronPoints[n]},
		Polygon/@{v[[{1,2,3}]],v[[{1,2,4}]],v[[{1,4,5}]],v[[{1,3,6}]],
				v[[{1,5,6}]],v[[{4,2,7}]],v[[{4,7,8}]],v[[{5,4,8}]],
				v[[{3,6,11}]],v[[{2,3,9}]],v[[{2,7,9}]],v[[{7,8,10}]],
				v[[{7,9,10}]],v[[{3,11,9}]],v[[{11,9,10}]],v[[{5,8,12}]],
				v[[{8,10,12}]],v[[{10,11,12}]],v[[{6,11,12}]],v[[{6,12,5}]]}
]

p5[n_]:=Mod[n-1,5]+1

polygon[{a_,b_,c_,d_}]:=Polygon[{a,c,b,d}];

feq[x_,y_,eps_:10^-10]:=Plus@@Abs[x-y]<eps

rotmm::Singular="Singular value ArcTan[0,0] encounted: (`1`,`2`,`3`)"

rotmm[{x_, y_, z_}] := Module[{a, b,eps=.001},
	If[Sqrt[N[y^2+z^2]]<eps && Abs[N[x]]<eps,
		Message[rotmm::Singular,x,y,z],
		{a, b} = {If[Abs[N[y]]<eps && Abs[N[z]]<eps,0,ArcTan[z, y]], ArcTan[Sqrt[y^2 + z^2], x]};
		    {{Cos[b],0,-Sin[b]},{0,1,0},{Sin[b],0,Cos[b]}}.
    		{{1,0,0},{0,Cos[a],-Sin[a]},{0,Sin[a],Cos[a]}}
	]
]

prec[a_,b_]:=Sort[{a,b}];

(*
ppolygon[l_List]:=Module[{i,j},
	Table[Polygon[Table[rotz[j Pi/2].l[[i]],{i,Length[l]}]],{j,0,3}]
]

ppolygon2[l_List]:=Module[{i,j},
	Table[Polygon[Table[rotz[j Pi/2].rotx[k Pi/2].l[[i]],{i,Length[l]}]],{k,0,2,2},
    {j,0,3}]
]

RhombicDodecahedronConstruction:=Graphics3D[{
	RhombicDodecahedron[2][[1]],
	{PointSize[.04],Point/@{{1,0,0},{-1,0,0},{0,1,0},{0,-1,0},{0,0,1},{0,0,-1}}},
	frame[Jack]
}]
*)

polygon[l_List]:=Module[{i},
	{Polygon[l],Polygon[Table[{l[[i,1]],l[[i,2]],-l[[i,3]]},{i,3}]]}
]

RhombicDodecahedronStellation[1]:=Module[
	{
	v=Join[NetlistSolidVertices[10],
	{{0,0,Sqrt[2]},{-1,-1,0},{1,-1,0},{1,1,0},
	{0,-1,1/Sqrt[2]},{1,0,1/Sqrt[2]},
	{-1,1,0},{0,1,1/Sqrt[2]},{-1,0,1/Sqrt[2]}}//N]
	},
	Graphics3D[polygon/@{
		v[[{10,14,17}]],v[[{17,14,8}]],v[[{13,17,7}]],
		v[[{17,7,15}]],v[[{8,13,17}]],v[[{10,15,17}]],		
		v[[{7,13,18}]],v[[{6,16,18}]],v[[{9,18,16}]],
		v[[{9,18,15}]],v[[{6,13,18}]],v[[{7,15,18}]],
		v[[{20,13,6}]],v[[{20,5,13}]],v[[{20,19,12}]],
		v[[{20,12,16}]],v[[{20,6,16}]],v[[{20,5,19}]],	
		v[[{21,5,13}]],v[[{21,8,13}]],v[[{21,11,14}]],
		v[[{11,21,19}]],v[[{21,5,19}]],v[[{21,8,14}]]
	}]
]

RhombicDodecahedronStellation[2]:=Module[
	{
	v=NetlistSolidVertices[12],
	c={{0,-1,Sqrt[3]/2},{0,-1,-Sqrt[3]/2}},
	d={{0,0,Sqrt[3]},{0,0,-Sqrt[3]},{-Sqrt[3/2],-Sqrt[3/2],0}},
	l
	},
	l={LineLineIntersection[v[[{2,6}]],v[[{21,9}]]],
		LineLineIntersection[v[[{1,5}]],v[[{17,12}]]],
		LineLineIntersection[v[[{1,5}]],v[[{15,19}]]],
		LineLineIntersection[v[[{2,6}]],v[[{11,23}]]],
		LineLineIntersection[v[[{10,24}]],v[[{14,20}]]]};
	Graphics3D[{
		lline[{c[[1]],l[[1]],v[[2]],v[[2]]+c[[1]]-l[[1]],v[[1]],l[[2]]}],
		lline[{c[[1]],l[[2]],v[[17]],v[[17]]+c[[1]]-l[[2]],v[[20]],l[[5]]}],
		lline[{c[[1]],l[[5]],v[[24]],v[[24]]+c[[1]]-l[[5]],v[[21]],l[[1]]}],
	
		lline[{c[[2]],l[[5]],v[[24]],v[[23]]+c[[2]]-l[[4]],v[[23]],l[[4]]}],
		lline[{c[[2]],l[[5]],v[[20]],v[[20]]+c[[2]]-l[[5]],v[[19]],l[[3]]}],
		lline[{c[[2]],l[[3]],v[[5]],v[[6]]+c[[2]]-l[[4]],v[[6]],l[[4]]}],

		ppolygon[{v[[1]],{0,-1,2.598076},d[[1]],rotz[-Pi/2].{0,-1,2.598076}}],
		ppolygon[{v[[5]],{0,-1,-2.598076},d[[2]],rotz[-Pi/2].{0,-1,-2.598076}}],

		ppolygon2[{v[[24]],v[[23]]+c[[2]]-l[[4]],d[[3]],v[[24]]+c[[1]]-l[[5]]}],
		ppolygon2[{v[[21]],v[[24]]+c[[1]]-l[[5]],d[[3]],rotz[Pi/2].(v[[17]]+c[[1]]-l[[2]])}],

		{EdgeForm[],
		ppolygon[{v[[2]],v[[2]]+c[[1]]-l[[1]],c[[1]],l[[1]]}],
		ppolygon[{v[[2]]+c[[1]]-l[[1]],v[[1]],l[[2]],c[[1]]}],
		ppolygon[{v[[24]],v[[24]]+c[[1]]-l[[5]],c[[1]],l[[5]]}],
		ppolygon[{v[[21]],l[[1]],c[[1]],v[[24]]+c[[1]]-l[[5]]}],
		ppolygon[{v[[17]],v[[17]]+c[[1]]-l[[2]],c[[1]],l[[2]]}],
		ppolygon[{v[[20]],l[[5]],c[[1]],v[[17]]+c[[1]]-l[[2]]}],

		ppolygon[{v[[23]],v[[23]]+c[[2]]-l[[4]],c[[2]],l[[4]]}],
		ppolygon[{v[[24]],v[[23]]+c[[2]]-l[[4]],c[[2]],l[[5]]}],
		ppolygon[{v[[20]],v[[20]]+c[[2]]-l[[5]],c[[2]],l[[5]]}],
		ppolygon[{v[[19]],v[[20]]+c[[2]]-l[[5]],c[[2]],l[[3]]}],
		ppolygon[{v[[6]],v[[6]]+c[[2]]-l[[4]],c[[2]],l[[4]]}],
		ppolygon[{v[[5]],v[[6]]+c[[2]]-l[[4]],c[[2]],l[[3]]}]
		}
	}]
]

(* Rhombic Enneacontahedron *)

PolyhedronName[RhombicEnneacontahedron]:="rhombic enneacontahedron"
PolyhedronFaces[RhombicEnneacontahedron]:=
{{81,82,45,46},{48,47,88,87},{85,86,47,48},{46,45,84,83},{47,88,44,86},{39,83,
    46,81},{61,62,82,81},{87,88,68,67},{62,77,45,82},{48,87,67,80},{60,73,65,
    79},{64,78,51,72},{85,86,66,65},{63,64,84,83},{70,51,77,62},{80,67,75,
    60},{79,65,85,48},{84,45,78,64},{51,78,45,77},{48,80,60,79},{32,31,62,
    61},{67,68,37,38},{58,28,2,9},{1,8,55,25},{49,69,54,24},{59,29,50,76},{88,
    68,37,92},{32,89,81,61},{17,5,30,43},{23,40,12,4},{65,66,35,36},{34,33,64,
    63},{74,50,28,58},{25,55,71,49},{24,54,7,1},{10,2,29,59},{6,14,41,26},{42,
    27,3,15},{63,83,90,34},{91,35,66,86},{50,29,2,28},{1,25,49,24},{13,14,33,
    34},{36,35,16,15},{57,30,8,1},{9,2,52,23},{72,51,53,26},{56,27,73,60},{65,
    73,27,42},{26,41,64,72},{4,12,21,58},{20,55,5,17},{30,5,55,8},{58,9,23,
    4},{33,64,41,14},{42,15,36,65},{38,37,18,17},{11,12,31,32},{60,75,30,
    57},{23,52,51,70},{26,53,2,10},{1,7,27,56},{15,3,54,19},{59,22,14,6},{70,
    62,40,23},{43,30,75,67},{67,38,17,43},{12,40,62,31},{6,26,10,59},{7,54,3,
    27},{56,60,57,1},{52,2,53,51},{17,18,14,13},{15,16,12,11},{49,71,83,
    39},{86,44,50,74},{76,50,44,88},{39,81,69,49},{19,54,89,32},{92,37,22,
    59},{58,21,35,91},{34,90,55,20},{54,69,81,89},{88,92,59,76},{74,58,91,
    86},{90,83,71,55},{20,17,13,34},{16,35,21,12},{14,22,37,18},{32,11,15,19}}
PolyhedronVertices[RhombicEnneacontahedron]:=
{{0, 0, Sqrt[45/2 + (15*Sqrt[5])/2]}, {0, 0, -Sqrt[81/2 + (27*Sqrt[5])/2]}, 
 {0, -3 - Sqrt[5], 2*Sqrt[3]}, {0, -3 - Sqrt[5], -Sqrt[6*(3 + Sqrt[5])]}, 
 {0, 3 + Sqrt[5], 2*Sqrt[3]}, {0, 3 + Sqrt[5], -Sqrt[6*(3 + Sqrt[5])]}, 
 {-Sqrt[(3 - Sqrt[5])/6], (-3 - Sqrt[5])/2, Sqrt[(2*(27 + 7*Sqrt[5]))/3]}, 
 {-Sqrt[(3 - Sqrt[5])/6], (3 + Sqrt[5])/2, Sqrt[(2*(27 + 7*Sqrt[5]))/3]}, 
 {Sqrt[(3 - Sqrt[5])/6], (-3 - Sqrt[5])/2, -Sqrt[32 + (32*Sqrt[5])/3]}, 
 {Sqrt[(3 - Sqrt[5])/6], (3 + Sqrt[5])/2, -Sqrt[32 + (32*Sqrt[5])/3]}, 
 {-Sqrt[1/2 + Sqrt[5]/6], (-7 - 3*Sqrt[5])/2, -Sqrt[(2*(3 - Sqrt[5]))/3]}, 
 {-Sqrt[1/2 + Sqrt[5]/6], (-7 - 3*Sqrt[5])/2, -Sqrt[21/2 + (5*Sqrt[5])/6]}, 
 {-Sqrt[1/2 + Sqrt[5]/6], (7 + 3*Sqrt[5])/2, -Sqrt[(2*(3 - Sqrt[5]))/3]}, 
 {-Sqrt[1/2 + Sqrt[5]/6], (7 + 3*Sqrt[5])/2, -Sqrt[21/2 + (5*Sqrt[5])/6]}, 
 {Sqrt[1/2 + Sqrt[5]/6], (-7 - 3*Sqrt[5])/2, Sqrt[(27 - 7*Sqrt[5])/6]}, 
 {Sqrt[1/2 + Sqrt[5]/6], (-7 - 3*Sqrt[5])/2, -2*Sqrt[(2*(3 - Sqrt[5]))/3]}, 
 {Sqrt[1/2 + Sqrt[5]/6], (7 + 3*Sqrt[5])/2, Sqrt[(27 - 7*Sqrt[5])/6]}, 
 {Sqrt[1/2 + Sqrt[5]/6], (7 + 3*Sqrt[5])/2, -2*Sqrt[(2*(3 - Sqrt[5]))/3]}, 
 {-Sqrt[7/6 + Sqrt[5]/2], (-5 - 3*Sqrt[5])/2, 4/Sqrt[3]}, 
 {-Sqrt[7/6 + Sqrt[5]/2], (5 + 3*Sqrt[5])/2, 4/Sqrt[3]}, 
 {Sqrt[7/6 + Sqrt[5]/2], (-5 - 3*Sqrt[5])/2, -Sqrt[(2*(23 + 3*Sqrt[5]))/3]}, 
 {Sqrt[7/6 + Sqrt[5]/2], (5 + 3*Sqrt[5])/2, -Sqrt[(2*(23 + 3*Sqrt[5]))/3]}, 
 {-Sqrt[5/2 + (5*Sqrt[5])/6], (-5 - Sqrt[5])/2, -Sqrt[49/2 + (49*Sqrt[5])/6]}, 
 {-Sqrt[5/2 + (5*Sqrt[5])/6], (-1 - Sqrt[5])/2, Sqrt[(2*(27 + 7*Sqrt[5]))/3]}, 
 {-Sqrt[5/2 + (5*Sqrt[5])/6], (1 + Sqrt[5])/2, Sqrt[(2*(27 + 7*Sqrt[5]))/3]}, 
 {-Sqrt[5/2 + (5*Sqrt[5])/6], (5 + Sqrt[5])/2, -Sqrt[49/2 + (49*Sqrt[5])/6]}, 
 {Sqrt[5/2 + (5*Sqrt[5])/6], (-5 - Sqrt[5])/2, Sqrt[29/2 + (13*Sqrt[5])/6]}, 
 {Sqrt[5/2 + (5*Sqrt[5])/6], (-1 - Sqrt[5])/2, -Sqrt[32 + (32*Sqrt[5])/3]}, 
 {Sqrt[5/2 + (5*Sqrt[5])/6], (1 + Sqrt[5])/2, -Sqrt[32 + (32*Sqrt[5])/3]}, 
 {Sqrt[5/2 + (5*Sqrt[5])/6], (5 + Sqrt[5])/2, Sqrt[29/2 + (13*Sqrt[5])/6]}, 
 {-Sqrt[35/6 + (5*Sqrt[5])/2], (-5 - 3*Sqrt[5])/2, -2*Sqrt[5/3]}, 
 {-Sqrt[35/6 + (5*Sqrt[5])/2], (-5 - 3*Sqrt[5])/2, Sqrt[(7 - 3*Sqrt[5])/6]}, 
 {-Sqrt[35/6 + (5*Sqrt[5])/2], (5 + 3*Sqrt[5])/2, -2*Sqrt[5/3]}, 
 {-Sqrt[35/6 + (5*Sqrt[5])/2], (5 + 3*Sqrt[5])/2, Sqrt[(7 - 3*Sqrt[5])/6]}, 
 {Sqrt[35/6 + (5*Sqrt[5])/2], (-5 - 3*Sqrt[5])/2, -Sqrt[(67 - 15*Sqrt[5])/6]}, 
 {Sqrt[35/6 + (5*Sqrt[5])/2], (-5 - 3*Sqrt[5])/2, Sqrt[(2*(7 - 3*Sqrt[5]))/3]}, 
 {Sqrt[35/6 + (5*Sqrt[5])/2], (5 + 3*Sqrt[5])/2, -Sqrt[(67 - 15*Sqrt[5])/6]}, 
 {Sqrt[35/6 + (5*Sqrt[5])/2], (5 + 3*Sqrt[5])/2, Sqrt[(2*(7 - 3*Sqrt[5]))/3]}, 
 {-2*Sqrt[(2*(7 + 3*Sqrt[5]))/3], 0, 4/Sqrt[3]}, {-Sqrt[(2*(7 + 3*Sqrt[5]))/3], 
  -3 - Sqrt[5], -Sqrt[(2*(23 + 3*Sqrt[5]))/3]}, {-Sqrt[(2*(7 + 3*Sqrt[5]))/3], 
  3 + Sqrt[5], -Sqrt[(2*(23 + 3*Sqrt[5]))/3]}, {Sqrt[(2*(7 + 3*Sqrt[5]))/3], 
  -3 - Sqrt[5], 4/Sqrt[3]}, {Sqrt[(2*(7 + 3*Sqrt[5]))/3], 3 + Sqrt[5], 4/Sqrt[3]}, 
 {2*Sqrt[(2*(7 + 3*Sqrt[5]))/3], 0, -Sqrt[(2*(23 + 3*Sqrt[5]))/3]}, 
 {-Sqrt[(10*(7 + 3*Sqrt[5]))/3], 0, -Sqrt[(67 - 15*Sqrt[5])/6]}, 
 {-Sqrt[(10*(7 + 3*Sqrt[5]))/3], 0, Sqrt[(2*(7 - 3*Sqrt[5]))/3]}, 
 {Sqrt[(10*(7 + 3*Sqrt[5]))/3], 0, -2*Sqrt[5/3]}, {Sqrt[(10*(7 + 3*Sqrt[5]))/3], 0, 
  Sqrt[(7 - 3*Sqrt[5])/6]}, {-Sqrt[10 + (10*Sqrt[5])/3], 0, 
  Sqrt[29/2 + (13*Sqrt[5])/6]}, {Sqrt[10 + (10*Sqrt[5])/3], 0, 
  -Sqrt[49/2 + (49*Sqrt[5])/6]}, {-2*Sqrt[(9 + 4*Sqrt[5])/3], 0, 
  -Sqrt[49/2 + (49*Sqrt[5])/6]}, {-Sqrt[(9 + 4*Sqrt[5])/3], -1, 
  -Sqrt[32 + (32*Sqrt[5])/3]}, {-Sqrt[(9 + 4*Sqrt[5])/3], 1, 
  -Sqrt[32 + (32*Sqrt[5])/3]}, {-Sqrt[(9 + 4*Sqrt[5])/3], -2 - Sqrt[5], 
  Sqrt[29/2 + (13*Sqrt[5])/6]}, {-Sqrt[(9 + 4*Sqrt[5])/3], 2 + Sqrt[5], 
  Sqrt[29/2 + (13*Sqrt[5])/6]}, {Sqrt[(9 + 4*Sqrt[5])/3], -1, 
  Sqrt[(2*(27 + 7*Sqrt[5]))/3]}, {Sqrt[(9 + 4*Sqrt[5])/3], 1, 
  Sqrt[(2*(27 + 7*Sqrt[5]))/3]}, {Sqrt[(9 + 4*Sqrt[5])/3], -2 - Sqrt[5], 
  -Sqrt[49/2 + (49*Sqrt[5])/6]}, {Sqrt[(9 + 4*Sqrt[5])/3], 2 + Sqrt[5], 
  -Sqrt[49/2 + (49*Sqrt[5])/6]}, {2*Sqrt[(9 + 4*Sqrt[5])/3], 0, 
  Sqrt[29/2 + (13*Sqrt[5])/6]}, {-Sqrt[(5*(9 + 4*Sqrt[5]))/3], -2 - Sqrt[5], 
  -Sqrt[(2*(3 - Sqrt[5]))/3]}, {-Sqrt[(5*(9 + 4*Sqrt[5]))/3], -2 - Sqrt[5], 
  -Sqrt[21/2 + (5*Sqrt[5])/6]}, {-Sqrt[(5*(9 + 4*Sqrt[5]))/3], 2 + Sqrt[5], 
  -Sqrt[(2*(3 - Sqrt[5]))/3]}, {-Sqrt[(5*(9 + 4*Sqrt[5]))/3], 2 + Sqrt[5], 
  -Sqrt[21/2 + (5*Sqrt[5])/6]}, {Sqrt[(5*(9 + 4*Sqrt[5]))/3], -2 - Sqrt[5], 
  Sqrt[(27 - 7*Sqrt[5])/6]}, {Sqrt[(5*(9 + 4*Sqrt[5]))/3], -2 - Sqrt[5], 
  -2*Sqrt[(2*(3 - Sqrt[5]))/3]}, {Sqrt[(5*(9 + 4*Sqrt[5]))/3], 2 + Sqrt[5], 
  Sqrt[(27 - 7*Sqrt[5])/6]}, {Sqrt[(5*(9 + 4*Sqrt[5]))/3], 2 + Sqrt[5], 
  -2*Sqrt[(2*(3 - Sqrt[5]))/3]}, {-Sqrt[21/2 + (9*Sqrt[5])/2], (-3 - Sqrt[5])/2, 
  2*Sqrt[3]}, {-Sqrt[21/2 + (9*Sqrt[5])/2], (-3 - Sqrt[5])/2, -Sqrt[6*(3 + Sqrt[5])]}, 
 {-Sqrt[21/2 + (9*Sqrt[5])/2], (3 + Sqrt[5])/2, 2*Sqrt[3]}, 
 {-Sqrt[21/2 + (9*Sqrt[5])/2], (3 + Sqrt[5])/2, -Sqrt[6*(3 + Sqrt[5])]}, 
 {Sqrt[21/2 + (9*Sqrt[5])/2], (-3 - Sqrt[5])/2, 2*Sqrt[3]}, 
 {Sqrt[21/2 + (9*Sqrt[5])/2], (-3 - Sqrt[5])/2, -Sqrt[6*(3 + Sqrt[5])]}, 
 {Sqrt[21/2 + (9*Sqrt[5])/2], (3 + Sqrt[5])/2, 2*Sqrt[3]}, 
 {Sqrt[21/2 + (9*Sqrt[5])/2], (3 + Sqrt[5])/2, -Sqrt[6*(3 + Sqrt[5])]}, 
 {-Sqrt[103/6 + (15*Sqrt[5])/2], (-1 - Sqrt[5])/2, -Sqrt[(2*(23 + 3*Sqrt[5]))/3]}, 
 {-Sqrt[103/6 + (15*Sqrt[5])/2], (1 + Sqrt[5])/2, -Sqrt[(2*(23 + 3*Sqrt[5]))/3]}, 
 {Sqrt[103/6 + (15*Sqrt[5])/2], (-1 - Sqrt[5])/2, 4/Sqrt[3]}, 
 {Sqrt[103/6 + (15*Sqrt[5])/2], (1 + Sqrt[5])/2, 4/Sqrt[3]}, 
 {-Sqrt[41/2 + (55*Sqrt[5])/6], (-3 - Sqrt[5])/2, Sqrt[(27 - 7*Sqrt[5])/6]}, 
 {-Sqrt[41/2 + (55*Sqrt[5])/6], (-3 - Sqrt[5])/2, -2*Sqrt[(2*(3 - Sqrt[5]))/3]}, 
 {-Sqrt[41/2 + (55*Sqrt[5])/6], (3 + Sqrt[5])/2, Sqrt[(27 - 7*Sqrt[5])/6]}, 
 {-Sqrt[41/2 + (55*Sqrt[5])/6], (3 + Sqrt[5])/2, -2*Sqrt[(2*(3 - Sqrt[5]))/3]}, 
 {Sqrt[41/2 + (55*Sqrt[5])/6], (-3 - Sqrt[5])/2, -Sqrt[(2*(3 - Sqrt[5]))/3]}, 
 {Sqrt[41/2 + (55*Sqrt[5])/6], (-3 - Sqrt[5])/2, -Sqrt[21/2 + (5*Sqrt[5])/6]}, 
 {Sqrt[41/2 + (55*Sqrt[5])/6], (3 + Sqrt[5])/2, -Sqrt[(2*(3 - Sqrt[5]))/3]}, 
 {Sqrt[41/2 + (55*Sqrt[5])/6], (3 + Sqrt[5])/2, -Sqrt[21/2 + (5*Sqrt[5])/6]}, 
 {-Sqrt[(29 + 12*Sqrt[5])/3], -2 - Sqrt[5], 4/Sqrt[3]}, 
 {-Sqrt[(29 + 12*Sqrt[5])/3], 2 + Sqrt[5], 4/Sqrt[3]}, 
 {Sqrt[(29 + 12*Sqrt[5])/3], -2 - Sqrt[5], -Sqrt[(2*(23 + 3*Sqrt[5]))/3]}, 
 {Sqrt[(29 + 12*Sqrt[5])/3], 2 + Sqrt[5], -Sqrt[(2*(23 + 3*Sqrt[5]))/3]}}/Sqrt[9/2+(3Sqrt[5])/2]

(* Rhombic Icosahedron *)

PolyhedronName[RhombicIcosahedron]:="rhombic icosahedron"
PolyhedronFaces[RhombicIcosahedron]:=
{{19,5,7,9},{20,10,8,6},{7,11,21,23},{8,24,21,12},{20,16,19,22},{2,12,21,
    11},{5,2,11,7},{6,8,12,2},{19,16,2,5},{20,6,2,16},{7,17,3,9},{14,10,20,
    22},{21,18,15,17},{4,18,8,10},{19,9,13,22},{15,18,4,1},{17,15,1,3},{1,4,
    10,14},{3,1,13,9},{13,1,14,22}}
PolyhedronVertices[RhombicIcosahedron]:=
{{0, 0, -Sqrt[125/128 + (25*Sqrt[5])/128]}, {0, 0, Sqrt[125/128 + (25*Sqrt[5])/128]}, 
 {-Sqrt[(5 - Sqrt[5])/2]/4, (-5 - Sqrt[5])/8, -Sqrt[45/128 + (9*Sqrt[5])/128]}, 
 {-Sqrt[(5 - Sqrt[5])/2]/4, (5 + Sqrt[5])/8, -Sqrt[45/128 + (9*Sqrt[5])/128]}, 
 {Sqrt[(5 - Sqrt[5])/2]/4, (-5 - Sqrt[5])/8, Sqrt[45/128 + (9*Sqrt[5])/128]}, 
 {Sqrt[(5 - Sqrt[5])/2]/4, (5 + Sqrt[5])/8, Sqrt[45/128 + (9*Sqrt[5])/128]}, 
 {-Sqrt[5/32 + Sqrt[5]/32], (-5 - 3*Sqrt[5])/8, Sqrt[5/128 + Sqrt[5]/128]}, 
 {-Sqrt[5/32 + Sqrt[5]/32], (5 + 3*Sqrt[5])/8, Sqrt[5/128 + Sqrt[5]/128]}, 
 {Sqrt[5/32 + Sqrt[5]/32], (-5 - 3*Sqrt[5])/8, -Sqrt[5/128 + Sqrt[5]/128]}, 
 {Sqrt[5/32 + Sqrt[5]/32], (5 + 3*Sqrt[5])/8, -Sqrt[5/128 + Sqrt[5]/128]}, 
 {-Sqrt[5/16 + Sqrt[5]/8], -Sqrt[5]/4, Sqrt[45/128 + (9*Sqrt[5])/128]}, 
 {-Sqrt[5/16 + Sqrt[5]/8], Sqrt[5]/4, Sqrt[45/128 + (9*Sqrt[5])/128]}, 
 {Sqrt[5/16 + Sqrt[5]/8], -Sqrt[5]/4, -Sqrt[45/128 + (9*Sqrt[5])/128]}, 
 {Sqrt[5/16 + Sqrt[5]/8], Sqrt[5]/4, -Sqrt[45/128 + (9*Sqrt[5])/128]}, 
 {-Sqrt[5/8 + Sqrt[5]/8], 0, -Sqrt[45/128 + (9*Sqrt[5])/128]}, 
 {Sqrt[5/8 + Sqrt[5]/8], 0, Sqrt[45/128 + (9*Sqrt[5])/128]}, 
 {-Sqrt[25/32 + (11*Sqrt[5])/32], (-5 - Sqrt[5])/8, -Sqrt[5/128 + Sqrt[5]/128]}, 
 {-Sqrt[25/32 + (11*Sqrt[5])/32], (5 + Sqrt[5])/8, -Sqrt[5/128 + Sqrt[5]/128]}, 
 {Sqrt[25/32 + (11*Sqrt[5])/32], (-5 - Sqrt[5])/8, Sqrt[5/128 + Sqrt[5]/128]}, 
 {Sqrt[25/32 + (11*Sqrt[5])/32], (5 + Sqrt[5])/8, Sqrt[5/128 + Sqrt[5]/128]}, 
 {-Sqrt[5/4 + Sqrt[5]/2], 0, Sqrt[5/128 + Sqrt[5]/128]}, 
 {Sqrt[5/4 + Sqrt[5]/2], 0, -Sqrt[5/128 + Sqrt[5]/128]}, 
 {Root[5 - 400*#1^2 + 256*#1^4 & , 1, 0], (-5 - Sqrt[5])/8, -Sqrt[5/128 + Sqrt[5]/128]}, 
 {Root[5 - 400*#1^2 + 256*#1^4 & , 1, 0], (5 + Sqrt[5])/8, -Sqrt[5/128 + Sqrt[5]/128]}}

ribbon[n_,rot_:0,z_:0]:=Module[{c,i,top,bot,R=Csc[Pi/n]/2},
    c=Table[R{Cos[#],Sin[#]}&[2Pi(i+rot/2)/n],{i,n}];
    top=Append[#,z+1/2]&/@c;
    bot=Append[#,z-1/2]&/@c;
    Polygon/@Reverse/@
      Join@@@Transpose[{Partition[top,2,1,1],Reverse/@Partition[bot,2,1,1]}]
]

(* Rim stuff *)

rim1[z_,R_,n_]:=Module[{i},
	Table[{R Cos[2 i Pi/n],R Sin[2 i Pi/n],z},{i,0,n}]
]

rim2[z_,R_,n_]:=Module[{i},
	Table[{R Cos[(2i+1) Pi/n],R Sin[(2i+1) Pi/n],z},{i,0,n}]
]

rim[z_,R_,n_]:=Module[{i},
	Table[{R Cos[2 i Pi/n],R Sin[2 i Pi/n],z},{i,0,n}]
]

Rot[x_List?VectorQ]:=Module[
	{i,rot={{Cos[Pi/3],Sin[Pi/3],0},{-Sin[Pi/3],Cos[Pi/3],0},{0,0,1}}//N},
	Table[rot.x[[i]],{i,1,Length[x]}]
]

Rot[m_,x_List?VectorQ]:=Module[{i,j},
	Table[Table[m.x[[j]][[i]],{i,Length[x[[j]]]}],{j,Length[x]}]
]

rotx[x_]:={{1,0,0},{0,Cos[x],Sin[x]},{0,-Sin[x],Cos[x]}}
roty[x_]:={{Cos[x],0,-Sin[x]},{0,1,0},{Sin[x],0,Cos[x]}}
rotz[x_]:={{Cos[x],Sin[x],0},{-Sin[x],Cos[x],0},{0,0,1}}

RotationFormula[r_List?VectorQ,n0_List?VectorQ,ang_]:=Module[
	{n=Unit[n0]},
	r Cos[ang]+n(n.r)(1-Cos[ang])+Cross[r,n]Sin[ang]
]

RotateAboutAxis[Polygon[l_List],n_List,ang_]:=
	Polygon[RotationFormula[#,n,ang]&/@l]
		
RotateAboutAxis[l_List?VectorQ,n_List,ang_]:=RotateAboutAxis[#,n,ang]&/@l

RotateE[g_,n_,ang_]:=Module[{i},
	Polygon/@Table[RotateAboutAxis[g[[i]],n,ang],{i,Length[g]}]
]

(* Small Ditrigonal Icosidodecahedron *)

PolyhedronName[SmallDitrigonalIcosidodecahedron]:="small ditrigonal icosidodecahedron"
NetFaces[SmallDitrigonalIcosidodecahedron]:={{99,111,103,116,129,126,139,124,122,114},{111,99,96},{116,103,115},{126,129,140},{124,139,134},{114,122,107},{108,96,102,89,81,83,77,86,87,91},{96,108,111},{89,102,90},{83,81,76},{86,77,79},{91,87,100},{131,115,106,109,94,110,113,120,135,123},{115,131,116},{109,106,93},{110,94,101},{120,113,128},{123,135,138},{137,140,127,142,145,149,157,152,155,147},{140,137,126},{142,127,132},{149,145,154},{152,157,158},{147,155,148},{119,134,136,144,153,146,151,141,130,133},{134,119,124},{144,136,150},{146,153,156},{141,151,143},{133,130,118},{98,107,121,112,117,104,92,95,85,97},{107,98,114},{112,121,125},{104,117,105},{95,92,84},{97,85,88},{60,48,56,43,30,33,20,35,37,45},{48,60,63},{43,56,44},{33,30,19},{35,20,25},{45,37,52},{51,63,57,70,78,76,82,73,72,68},{63,51,48},{70,57,69},{76,78,83},{73,82,80},{68,72,59},{28,44,53,50,65,49,46,39,24,36},{44,28,43},{50,53,66},{49,65,58},{39,46,31},{36,24,21},{22,19,32,17,14,10,2,7,4,12},{19,22,33},{17,32,27},{10,14,5},{7,2,1},{12,4,11},{40,25,23,15,6,13,8,18,29,26},{25,40,35},{15,23,9},{13,6,3},{18,8,16},{26,29,41},{61,52,38,47,42,55,67,64,74,62},{52,61,45},{47,38,34},{55,42,54},{64,67,75},{62,74,71}}
NetVertices[SmallDitrigonalIcosidodecahedron]:={{-8.00858491306,-0.141090157243},{-7.74976586795,0.824835669046},{-7.51096124479,-2.4974539978},{-7.49664958651,-0.7732776363280001},{-7.30671656237,1.50707177195},{-7.25214219969,-1.53152817152},{-7.04265908677,0.11772888786},{-6.99902591825,-3.12964147689},{-6.80909289411,-0.8492920686100001},{-6.76207752736,0.668401204006},{-6.65797901856,-1.31791667134},{-6.60564306232,-0.319287136589},{-6.54503541851,-2.2386349527},{-6.30808702762,1.55940772819},{-6.2644538591,-1.68796263656},{-6.1603553503,-3.67428051191},{-6.15165256258,0.571719387599},{-6.10801939406,-2.67565097715},{-6.05497074617,-0.0387055771803},{-6.01239684386,-0.851064026465},{-5.92132389802,1.06033454678},{-5.89853628113,-1.02639391778},{-5.81046335936,-0.796956112367},{-5.66250485292,2.02626037307},{-5.65402889432,-1.78464445296},{-5.55734707791,-2.39506941774},{-5.52233217153,1.34886534906},{-5.40938857147,0.428147067698},{-5.40091261287,-3.38275775834},{-5.27782478471,0.59061481387},{-5.21945554734,2.70849647598},{-5.16396422198,0.415284922559},{-5.12139031967,-0.397073526726},{-5.11615188917,-2.75234223711},{-5.02470850327,-1.00749849151},{-4.95539807173,1.31915359189},{-4.86827403823,-1.9951868321},{-4.85733284407,-1.78641641082},{-4.67481651233,1.86982590803},{-4.66634055372,-1.941078918},{-4.62376665141,-2.75343736729},{-4.60421656262,-3.38452971619},{-4.57071800353,-0.116491967317},{-4.51838204728,0.882137567438},{-4.41428353849,-1.10418030791},{-4.22082601259,2.76083243222},{-4.15022606288,-2.493523192},{-4.13370197908,-0.553507991766},{-4.06439154755,1.77314409163},{-3.96770973114,1.16271912685},{-3.87488293398,0.412417834523},{-3.86964450347,-1.94285087586},{-3.8112752661,0.175030786251},{-3.76554599468,-3.92916875121},{-3.71321003843,-2.93053921645},{-3.67971147934,0.337498532423},{-3.62176665253,-1.18569547085},{-3.4350711565,2.55029005308},{-3.4318336284,1.09465393743},{-3.42659519789,-1.26061477295},{-3.41565400373,-1.05184435167},{-3.25921953869,-2.03953269226},{-3.16777615279,-0.294688946663},{-3.16253772229,-2.64995765704},{-3.07670320695,1.61670962659},{-3.03412930464,0.804351177301},{-3.00610325725,-3.63764599764},{-2.88719459338,0.255983369483},{-2.78309608458,-1.73033450587},{-2.73076012834,-0.731704971112},{-2.62989914764,-1.26238673081},{-2.43320409364,1.14698989367},{-2.2767696286,0.159301553076},{-2.2715311981,-2.1959671573},{-2.22895729579,-3.00832560659},{-2.18008781219,-0.451123411704},{-2.13751390989,-1.26348186099},{-2.02365334715,-1.4388117523},{-1.77914596034,-2.19706228749},{-1.64744923755,0.936447514533},{-1.40294185074,0.178196979346},{-1.28908128801,0.00286708803601},{-1.2465073857,-0.809491361249},{-1.1976379021,1.74771083364},{-1.15506399979,0.935352384352},{-1.14982556929,-1.41991632603},{-0.993391104251,-2.40760466662},{-0.79669605025,0.00177195785499},{-0.695835069551,-0.52890980184},{-0.643499113308,0.469719732914},{-0.539400604511,-1.51659814244},{-0.420491940647,2.37703122469},{-0.392465893251,-2.06496595025},{-0.349891990942,-2.87732439954},{-0.264057475606,1.38934288409},{-0.258819045103,-0.965925826289},{-0.1673756592,0.778917919312},{-0.0109411941594,-0.208770421283},{0,0},{0.00523843050392,-2.35526871038},{0.00847595860304,-3.81090482604},{0.195171454637,-0.0749193021007},{0.253116281447,-1.59811330537},{0.28661484054,1.6699244435},{0.338950796783,2.66855397825},{0.384680068206,-1.4356455592},{0.44304930558,0.682236102905},{0.448287736084,-1.67303260748},{0.541114533246,-2.4233338998},{0.637796349653,-3.03375886458},{1/Sqrt[2],-(1/Sqrt[2])},{0.723630864989,1.23290841905},{0.794230814693,-4.02144720517},{0.987688340595,-0.15643446504},{1.09178684939,-2.14275234039},{1.14412280564,-1.14412280564},{1.17762136473,2.12391494324},{1.19717145352,1.49282259434},{1.23974535583,0.68046414505},{1.24822131443,-3.13044068099},{1.43073764618,0.525801637865},{1.44167884033,0.734572059148},{1.52880287384,-2.57976836484},{1.59811330537,-0.253116281447},{1.68955669128,1.49172746415},{1.69479512178,-0.863541246227},{1.73736902409,-1.67589969551},{1.79286034945,-3.96911124893},{1.85122958682,-1.85122958682},{1.97431741498,2.12214298538},{1.98279337358,-1.68876184065},{2.09573697364,-2.60948012201},{2.13075188002,1.13445464479},{2.22743369642,0.52402968001},{2.23590965503,-3.28687514603},{2.38386816146,-0.463658660585},{2.47194108324,-0.234220855177},{2.49472870013,-2.32094931974},{2.58580164597,-0.409550746487},{2.62837554828,-1.22190919577},{2.68142419616,1.4150362042},{2.72505736469,-1.83233416055},{2.73376015241,2.41366573895},{2.8378586612,0.427347863603},{2.88149182973,-2.82002250115},{3.11844022061,0.97802017975},{3.17904786443,-0.941327636363},{3.23138382067,0.0573018983911},{3.33548232947,-1.92901597696},{3.38249769622,-0.411322704342},{3.57243072035,1.86902670394},{3.61606388887,-1.37834366081},{3.8255470018,0.270913398563},{3.88012136448,-2.7676865449},{4.07005438861,-0.487337136624},{4.0843660469,1.23683922485},{4.32317067006,-2.085450442},{4.58198971516,-1.11952461571}}
PolyhedronFaces[SmallDitrigonalIcosidodecahedron]:={{76,74,80,72,69,64,50,60,54,66},{74,76,77},{72,80,73},{64,69,51},{60,50,41},{66,54,56},{80,77,76,70,62,68,63,71,79,78},{77,80,74},{70,76,58},{68,62,49},{71,63,59},{78,79,75},{69,73,80,75,79,67,55,61,53,65},{73,69,72},{75,80,78},{67,79,57},{61,55,42},{65,53,52},{50,51,69,52,53,38,19,32,18,37},{51,50,64},{52,69,65},{38,53,34},{32,19,13},{37,18,33},{54,41,50,33,18,22,2,24,26,35},{41,54,60},{33,50,37},{22,18,10},{24,2,14},{35,26,45},{76,56,54,45,26,39,28,47,62,58},{56,76,66},{45,54,35},{39,26,20},{47,28,43},{58,62,70},{31,17,12,9,1,7,5,15,27,21},{17,31,30},{9,12,8},{7,1,4},{15,5,25},{21,27,40},{12,30,31,44,63,49,62,43,28,29},{30,12,17},{44,31,48},{49,63,68},{43,62,47},{29,28,16},{1,8,12,16,28,20,26,14,2,6},{8,1,9},{16,12,29},{20,28,39},{14,26,24},{6,2,3},{5,4,1,3,2,10,18,13,19,11},{4,5,7},{3,1,6},{10,2,22},{13,18,32},{11,19,23},{27,25,5,23,19,34,53,42,55,36},{25,27,15},{23,5,11},{34,19,38},{42,53,61},{36,55,46},{31,40,27,46,55,57,79,59,63,48},{40,31,21},{46,27,36},{57,55,67},{59,79,71},{48,63,44}}
PolyhedronVertices[SmallDitrigonalIcosidodecahedron]:={{3.8142072830715827,-.81702976668938636,-.37447380102938546},{4.2522073993440187,.64051051459563811,.17485871697124851},{4.2674145762346055,.038818925002720999,-.62372501237940248},{4.2768131284263428,-.33304692811189364,-1.1172768999792269},{4.2920203053169793,-.93473851770639707,-1.9158606293310159},{4.3891805389119151,-.34306224643574281,.292429898780014},{4.4137862679965821,-1.316619689147173,-.99970571817338747},{4.4738345947109401,-.95094364316660147,.36509288523175667},{4.4890417716018315,-1.5526352327556344,-.43349084412317084},{4.5475120873634149,.56776250569589131,-.77777073272980612},{4.5627192642503249,-.033929083899913282,-1.5763544620807946},{4.6108077342797958,-1.9345164041946795,.48266406703176767},{4.7300204215947125,.52280176357925805,-1.3665281113279497},{4.744533553637733,-.050134209356329799,.70459905247852517},{4.7843464596195053,-1.6253832416601309,-1.3861202938274526},{4.8815066932106591,-1.0337069703868084,.82217023428412115},{4.9061124222961267,-2.0072644130978952,-.46996538266793143},{5.0007193805305611,1.4236111973806155,-1.0270219440770895},{5.0253251096090864,.45005375467233446,-2.3191575610293277},{5.0488078505503352,-.47697612290651231,1.0319965850392081},{5.0886207565249615,-2.0522251552119372,-1.0587227612662091},{5.1224853432076011,1.0417300259423349,-.1108670329218358},{5.1622982491748195,-.53351900635735234,-2.2015863792252335},{5.1977408468116275,.8057144823306104,.45534784113629043},{5.2469523049687222,-1.1414004030845322,-2.1289233927725321},{5.3195068094869729,.42383331089551936,1.3715027522872832},{5.3839254445408598,-2.1249731641177964,-2.0113522109686168},{5.5411340048537392,-1.167620846864733,1.561736920547566},{5.55634118174639,-1.7693124364602178,.7631531911941431},{5.5657397339366832,-2.1411782895752646,.26960130359281835},{5.5809469108289035,-2.7428698791690416,-.52898242575551644},{5.6002983654556576,.92402127492296677,-1.6522538612199221},{5.6603466921759223,1.2896973209025664,-.28745525781643294},{5.7001595981352678,-.28555171139573099,-2.3781746041201458},{5.782112654851231,.90781614946685603,.62869965333724576},{5.8371327377023645,-1.2691244724300328,-2.2606034223168127},{5.9556513801857517,1.216949311996242,-1.2400847075187449},{5.970858557069894,.61525772240082881,-2.0386684368710664},{5.9943412980166725,-.31177215517620437,1.3124857092006881},{6.0435527561873251,-2.2588870405922869,-1.2717855247068784},{6.0680187906787902,1.2069339936725966,.16962209123998295},{6.1172302488285827,-.74018089173562633,-2.4146491426688927},{6.1407129897781325,-1.667210769324966,.93650500340248841},{6.1559201666708567,-2.2689023589207871,.13792127404968941},{6.2744388091501878,.21717142550979676,1.1584399888486142},{6.3294588920053033,-1.9597691963854324,-1.7308630868059856},{6.4114119487158653,-.76640133552081486,1.2760111706525124},{6.4512248546852634,-2.3416503678227789,-.81470817564958762},{6.511273181405472,-1.9759743218429815,.550090427754781},{6.5306246360326429,1.6909168322466329,-.57318100771147952},{6.5458318129187841,1.0892252426527436,-1.3717647370603347},{6.5552303651089385,.71735938953950208,-1.8653166246636205},{6.5704375419959242,.11566779994730272,-2.663900354017589},{6.7276461023166965,1.0730201171967393,.90918877749807379},{6.7920647373607977,-1.4757863578090223,-2.4736661857574236},{6.8646192418821502,.08944735616468876,1.0267599593044298},{6.9138307000351786,-1.8576675292477805,-1.5575112746004959},{6.949273297678081,-.51843404056282814,1.0994229457571464},{6.9890862036437517,-2.0936830728605916,-.99129640054888553},{7.0229507903335158,1.0002721082882285,-.043440672200877901},{7.0627636962970651,-.57497692400993581,-2.1341600185077798},{7.0862464372406073,-1.5020068015951067,1.2169941275610437},{7.110852166323505,-2.4755642443032284,-.075141489390550209},{7.2054591245621166,.95531136617324854,-.63219805080141255},{7.2300648536401395,-.018246076533788597,-1.9243336677534936},{7.3272250872376555,.57343019473605032,.28395686035384921},{7.3670379932011695,-1.0018188375603693,-1.8067624859516601},{7.3815511252572915,-1.5747548105036022,.26436467786167347},{7.5007638125774819,.88256335727005557,-1.584827500501068},{7.5488522825991796,-1.0180239630240873,.47419102861264976},{7.5640594594887381,-1.6197155526132525,-.32439270073964462},{7.6225297752516591,.50068218583340824,-.66867258935298807},{7.6377369521422468,-.10100940376338166,-1.4672563186958243},{7.6977852788593567,.26466664221858676,-.10245771529335814},{7.7223910079369182,-.70889080048998855,-1.3945933322437907},{7.8195512415356802,-.11721452922034223,.81369719586252742},{7.8347584184248569,-.71890611881523542,.01511346651326767},{7.8441569706142394,-1.0907719719292259,-.47843842109030055},{7.8593641475002861,-1.6924635615180231,-1.2770221504396647},{8.2973642637807672,-.23492328024235003,-.72768963243513997}}

(* Small Triambic Icosahedron *)

PolyhedronFaces[SmallTriambicIcosahedron]:=
{{24,2,8},{24,8,10},{24,10,2},{13,2,10},{13,10,9},{13,9,2},{23,2,9},{23,9,
    7},{23,7,2},{19,2,7},{19,7,4},{19,4,2},{20,2,4},{20,4,8},{20,8,2},{27,11,
    5},{27,5,1},{27,1,11},{14,12,11},{14,11,1},{14,1,12},{28,6,12},{28,12,
    1},{28,1,6},{18,3,6},{18,6,1},{18,1,3},{17,5,3},{17,3,1},{17,1,5},{22,8,
    6},{22,6,10},{22,10,8},{15,10,3},{15,3,9},{15,9,10},{21,9,5},{21,5,7},{21,
    7,9},{26,7,11},{26,11,4},{26,4,7},{25,4,12},{25,12,8},{25,8,4},{31,11,
    7},{31,7,5},{31,5,11},{16,12,4},{16,4,11},{16,11,12},{32,6,8},{32,8,
    12},{32,12,6},{29,3,10},{29,10,6},{29,6,3},{30,5,9},{30,9,3},{30,3,5}}
PolyhedronVertices[SmallTriambicIcosahedron]:=
{{0, 0, -5/Sqrt[50 - 10*Sqrt[5]]}, {0, 0, 5/Sqrt[50 - 10*Sqrt[5]]}, {Root[1 - 5*#1^2 + 5*#1^4 & , 1, 0], 0, 
  -(1/Sqrt[10 - 2*Sqrt[5]])}, {Sqrt[(5 + Sqrt[5])/10], 0, 1/Sqrt[10 - 2*Sqrt[5]]}, 
 {Root[1 - 20*#1^2 + 80*#1^4 & , 2, 0], (-1 - Sqrt[5])/4, -(1/Sqrt[10 - 2*Sqrt[5]])}, 
 {Root[1 - 20*#1^2 + 80*#1^4 & , 2, 0], (1 + Sqrt[5])/4, -(1/Sqrt[10 - 2*Sqrt[5]])}, 
 {Sqrt[1/8 - 1/(8*Sqrt[5])], (-1 - Sqrt[5])/4, 1/Sqrt[10 - 2*Sqrt[5]]}, 
 {Sqrt[1/8 - 1/(8*Sqrt[5])], (1 + Sqrt[5])/4, 1/Sqrt[10 - 2*Sqrt[5]]}, 
 {-Sqrt[1 + 2/Sqrt[5]]/2, -1/2, 1/Sqrt[10 - 2*Sqrt[5]]}, {-Sqrt[1 + 2/Sqrt[5]]/2, 1/2, 1/Sqrt[10 - 2*Sqrt[5]]}, 
 {Sqrt[1/4 + 1/(2*Sqrt[5])], -1/2, -(1/Sqrt[10 - 2*Sqrt[5]])}, {Sqrt[1/4 + 1/(2*Sqrt[5])], 1/2, 
  -(1/Sqrt[10 - 2*Sqrt[5]])}, {-Sqrt[5 + 2*Sqrt[5]]/5, 0, Root[5 - 65*#1^2 + #1^4 & , 4, 0]/10}, 
 {Sqrt[5 + 2*Sqrt[5]]/5, 0, -Root[5 - 65*#1^2 + #1^4 & , 4, 0]/10}, {Root[1 - 125*#1^2 + 125*#1^4 & , 1, 0], 0, 
  Sqrt[(5 + Sqrt[5])/2]/10}, {Sqrt[1/2 + 11/(10*Sqrt[5])], 0, -(1/Sqrt[50 - 10*Sqrt[5]])}, 
 {-(1/Sqrt[250 - 110*Sqrt[5]]), (-5 + Sqrt[5])^(-1), Root[1 - 1300*#1^2 + 2000*#1^4 & , 1, 0]}, 
 {-(1/Sqrt[250 - 110*Sqrt[5]]), (5 + Sqrt[5])/20, Root[1 - 1300*#1^2 + 2000*#1^4 & , 1, 0]}, 
 {1/Sqrt[250 - 110*Sqrt[5]], (-5 + Sqrt[5])^(-1), Sqrt[13/40 + 29/(40*Sqrt[5])]}, 
 {1/Sqrt[250 - 110*Sqrt[5]], (5 + Sqrt[5])/20, Sqrt[13/40 + 29/(40*Sqrt[5])]}, 
 {-Sqrt[5 + 2*Sqrt[5]]/10, -1/2 - 1/Sqrt[5], Sqrt[(5 + Sqrt[5])/2]/10}, 
 {-Sqrt[5 + 2*Sqrt[5]]/10, 1/2 + 1/Sqrt[5], Sqrt[(5 + Sqrt[5])/2]/10}, 
 {-Sqrt[(5 + Sqrt[5])/2]/10, (-5 - 3*Sqrt[5])/20, Root[5 - 65*#1^2 + #1^4 & , 4, 0]/10}, 
 {-Sqrt[(5 + Sqrt[5])/2]/10, (5 + 3*Sqrt[5])/20, Root[5 - 65*#1^2 + #1^4 & , 4, 0]/10}, 
 {Sqrt[130 + 58*Sqrt[5]]/20, (5 + 3*Sqrt[5])/20, Sqrt[(5 + Sqrt[5])/2]/10}, 
 {Sqrt[130 + 58*Sqrt[5]]/20, (-5 - 3*Sqrt[5])/20, Sqrt[(5 + Sqrt[5])/2]/10}, 
 {1/Sqrt[50 - 10*Sqrt[5]], (-5 - 3*Sqrt[5])/20, -Root[5 - 65*#1^2 + #1^4 & , 4, 0]/10}, 
 {1/Sqrt[50 - 10*Sqrt[5]], (5 + 3*Sqrt[5])/20, -Root[5 - 65*#1^2 + #1^4 & , 4, 0]/10}, 
 {Root[1 - 1300*#1^2 + 2000*#1^4 & , 1, 0], (5 + 3*Sqrt[5])/20, -Sqrt[(5 + Sqrt[5])/2]/10}, 
 {Root[1 - 1300*#1^2 + 2000*#1^4 & , 1, 0], (-5 - 3*Sqrt[5])/20, -Sqrt[(5 + Sqrt[5])/2]/10}, 
 {Sqrt[5 + 2*Sqrt[5]]/10, -1/2 - 1/Sqrt[5], -(1/Sqrt[50 - 10*Sqrt[5]])}, 
 {Sqrt[5 + 2*Sqrt[5]]/10, 1/2 + 1/Sqrt[5], -(1/Sqrt[50 - 10*Sqrt[5]])}}

(* Spiky *)

PolyhedronName[Spiky]:="spiky"
PolyhedronFaces[Spiky]:=
{{21,2,12},{21,12,6},{21,6,2},{15,2,6},{15,6,5},{15,5,2},{22,2,5},{22,5,11},{22,11,2},
{28,2,11},{28,11,4},{28,4,2},{27,2,4},{27,4,12},{27,12,2},{24,7,9},{24,9,1},{24,1,7},
{16,8,7},{16,7,1},{16,1,8},{23,10,8},{23,8,1},{23,1,10},{25,3,10},{25,10,1},{25,1,3},
{26,9,3},{26,3,1},{26,1,9},{29,12,10},{29,10,6},{29,6,12},{13,6,3},{13,3,5},{13,5,6},
{30,5,9},{30,9,11},{30,11,5},{20,11,7},{20,7,4},{20,4,11},{19,4,8},{19,8,12},{19,12,4},
{32,7,11},{32,11,9},{32,9,7},{14,8,4},{14,4,7},{14,7,8},{31,10,12},{31,12,8},{31,8,10},
{17,3,6},{17,6,10},{17,10,3},{18,9,5},{18,5,3},{18,3,9}}
PolyhedronVertices[Spiky]:=
{{0,0,Root[5-20*#1^2+16*#1^4&,1]},{0,0,Root[5-20*#1^2+16*#1^4&,4]},
{Root[1-5*#1^2+5*#1^4&,1],0,Root[1-20*#1^2+80*#1^4&,1]},
{Root[1-5*#1^2+5*#1^4&,4],0,Root[1-20*#1^2+80*#1^4&,4]},
{Root[1-40*#1^2+80*#1^4&,1],-1/2,Root[1-20*#1^2+80*#1^4&,4]},
{Root[1-40*#1^2+80*#1^4&,1],1/2,Root[1-20*#1^2+80*#1^4&,4]},
{Root[1-40*#1^2+80*#1^4&,4],-1/2,Root[1-20*#1^2+80*#1^4&,1]},
{Root[1-40*#1^2+80*#1^4&,4],1/2,Root[1-20*#1^2+80*#1^4&,1]},
{Root[1-20*#1^2+80*#1^4&,2],(-1-Sqrt[5])/4,Root[1-20*#1^2+80*#1^4&,1]},
{Root[1-20*#1^2+80*#1^4&,2],(1+Sqrt[5])/4,Root[1-20*#1^2+80*#1^4&,1]},
{Root[1-20*#1^2+80*#1^4&,3],(-1-Sqrt[5])/4,Root[1-20*#1^2+80*#1^4&,4]},
{Root[1-20*#1^2+80*#1^4&,3],(1+Sqrt[5])/4,Root[1-20*#1^2+80*#1^4&,4]},
{Root[1-290*#1^2+2555*#1^4-5850*#1^6+2025*#1^8&,1],0,Root[1-7720*#1^2+127280*#1^4-489600*#1^6+518400*#1^8&,6]},
{Root[1-290*#1^2+2555*#1^4-5850*#1^6+2025*#1^8&,8],0,Root[1-7720*#1^2+127280*#1^4-489600*#1^6+518400*#1^8&,3]},
{Root[1-740*#1^2+3230*#1^4-4500*#1^6+2025*#1^8&,1],0,Root[1-520*#1^2+40880*#1^4-835200*#1^6+518400*#1^8&,8]},
{Root[1-740*#1^2+3230*#1^4-4500*#1^6+2025*#1^8&,8],0,Root[1-520*#1^2+40880*#1^4-835200*#1^6+518400*#1^8&,1]},
{Root[1-520*#1^2+40880*#1^4-835200*#1^6+518400*#1^8&,1],Root[1+28*#1-20*#1^2-144*#1^3+144*#1^4&,4],Root[1-7720*#1^2+127280*#1^4-489600*#1^6+518400*#1^8&,3]},
{Root[1-520*#1^2+40880*#1^4-835200*#1^6+518400*#1^8&,1],Root[1-28*#1-20*#1^2+144*#1^3+144*#1^4&,1],Root[1-7720*#1^2+127280*#1^4-489600*#1^6+518400*#1^8&,3]},
{Root[1-520*#1^2+40880*#1^4-835200*#1^6+518400*#1^8&,8],Root[1+28*#1-20*#1^2-144*#1^3+144*#1^4&,4],Root[1-7720*#1^2+127280*#1^4-489600*#1^6+518400*#1^8&,6]},
{Root[1-520*#1^2+40880*#1^4-835200*#1^6+518400*#1^8&,8],Root[1-28*#1-20*#1^2+144*#1^3+144*#1^4&,1],Root[1-7720*#1^2+127280*#1^4-489600*#1^6+518400*#1^8&,6]},
{Root[1-7720*#1^2+127280*#1^4-489600*#1^6+518400*#1^8&,3],Root[1+28*#1-20*#1^2-144*#1^3+144*#1^4&,4],Root[1-520*#1^2+40880*#1^4-835200*#1^6+518400*#1^8&,8]},
{Root[1-7720*#1^2+127280*#1^4-489600*#1^6+518400*#1^8&,3],Root[1-28*#1-20*#1^2+144*#1^3+144*#1^4&,1],Root[1-520*#1^2+40880*#1^4-835200*#1^6+518400*#1^8&,8]},
{Root[1-7720*#1^2+127280*#1^4-489600*#1^6+518400*#1^8&,6],Root[1+28*#1-20*#1^2-144*#1^3+144*#1^4&,4],Root[1-520*#1^2+40880*#1^4-835200*#1^6+518400*#1^8&,1]},
{Root[1-7720*#1^2+127280*#1^4-489600*#1^6+518400*#1^8&,6],Root[1-28*#1-20*#1^2+144*#1^3+144*#1^4&,1],Root[1-520*#1^2+40880*#1^4-835200*#1^6+518400*#1^8&,1]},
{Root[1-1160*#1^2+40880*#1^4-374400*#1^6+518400*#1^8&,1],Root[1+44*#1-100*#1^2-48*#1^3+144*#1^4&,3],Root[1-520*#1^2+40880*#1^4-835200*#1^6+518400*#1^8&,1]},
{Root[1-1160*#1^2+40880*#1^4-374400*#1^6+518400*#1^8&,1],Root[1-44*#1-100*#1^2+48*#1^3+144*#1^4&,2],Root[1-520*#1^2+40880*#1^4-835200*#1^6+518400*#1^8&,1]},
{Root[1-1160*#1^2+40880*#1^4-374400*#1^6+518400*#1^8&,8],Root[1+44*#1-100*#1^2-48*#1^3+144*#1^4&,3],Root[1-520*#1^2+40880*#1^4-835200*#1^6+518400*#1^8&,8]},
{Root[1-1160*#1^2+40880*#1^4-374400*#1^6+518400*#1^8&,8],Root[1-44*#1-100*#1^2+48*#1^3+144*#1^4&,2],Root[1-520*#1^2+40880*#1^4-835200*#1^6+518400*#1^8&,8]},
{Root[1-2960*#1^2+51680*#1^4-288000*#1^6+518400*#1^8&,1],Root[1+16*#1-40*#1^2-192*#1^3+144*#1^4&,4],Root[1-7720*#1^2+127280*#1^4-489600*#1^6+518400*#1^8&,6]},
{Root[1-2960*#1^2+51680*#1^4-288000*#1^6+518400*#1^8&,1],Root[1-16*#1-40*#1^2+192*#1^3+144*#1^4&,1],Root[1-7720*#1^2+127280*#1^4-489600*#1^6+518400*#1^8&,6]},
{Root[1-2960*#1^2+51680*#1^4-288000*#1^6+518400*#1^8&,8],Root[1+16*#1-40*#1^2-192*#1^3+144*#1^4&,4],Root[1-7720*#1^2+127280*#1^4-489600*#1^6+518400*#1^8&,3]},
{Root[1-2960*#1^2+51680*#1^4-288000*#1^6+518400*#1^8&,8],Root[1-16*#1-40*#1^2+192*#1^3+144*#1^4&,1],Root[1-7720*#1^2+127280*#1^4-489600*#1^6+518400*#1^8&,3]}}

(* SquashedDodecahedron *)

SquashedDodecahedron[s_:1]:=Module[
	{
		i,
		v,u,w,
		h=2/3Sqrt[3],a,b,c,
		short={2,5,12},long={1,9,7}
	},
	(* Rhombic dodecahedron vertices *)
	v=Union[Flatten[
		{Table[{
		{{1,1,i},{0,0,2i},{1,-1,i},{2,0,0}}/s,
		{{-1,1,i},{0,0,2i},{1,1,i},{0,2,0}}/s,
		{{-1,-1,i},{0,0,2i},{-1,1,i},{-2,0,0}}/s,
		{{1,-1,i},{0,0,2i},{-1,-1,i},{0,-2,0}}/s
		},{i,-1,1,2}],
		{{{{1,-1,-1},{2,0,0},{1,-1,1},{0,-2,0}}/s,
		{{1,1,-1},{0,2,0},{1,1,1},{2,0,0}}/s,
		{{-1,1,-1},{-2,0,0},{-1,1,1},{0,2,0}}/s,
		{{-1,-1,-1},{0,-2,0},{-1,-1,1},{-2,0,0}}/s}}
	},3]];
	a=Centroid[v[[{1,7,9}]]];
	b=Centroid[v[[{2,12,5}]]];
	c=v[[4]];
	u=(a-c)/Sqrt[(c-a).(c-a)];
	w=Table[{0,0,0},{Length[v]}];
	w[[4]]=v[[4]]+2(b-c)+h u;
	Do[
		w[[short[[i]]]]=v[[short[[i]]]]+h u,
		{i,Length[short]}
	];
	Do[
		w[[long[[i]]]]=v[[long[[i]]]]+2(b-a)+h u,
		{i,Length[long]}
	];
	{
	Polygon/@{v[[{4,9,5,1}]],v[[{4,9,12,7}]],v[[{4,7,2,1}]]},
	Polygon/@{w[[{4,9,5,1}]],w[[{4,9,12,7}]],w[[{4,7,2,1}]]},
	Polygon/@{
		{v[[5]],v[[9]],w[[9]],w[[5]]},{v[[9]],v[[12]],w[[12]],w[[9]]},
		{v[[12]],v[[7]],w[[7]],w[[12]]},{v[[7]],v[[2]],w[[2]],w[[7]]},
		{v[[2]],v[[1]],w[[1]],w[[2]]},{v[[1]],v[[5]],w[[5]],w[[1]]}}
	}
]

StellaOctangulaCubeEdges:=Module[
	{diags=Flatten[StellaOctangulaDiagonals,1,Line],l={},i=2},
    While[diags!={},
		While[(diags[[1,2]]+diags[[1,1]])!=(diags[[i,2]]+diags[[i,1]]),i++];
		l=Append[l,{diags[[1,1]],diags[[i,1]],diags[[1,2]],diags[[i,2]],diags[[1,1]]}];
		diags=Drop[Drop[diags,{i}],1];
		i=2;
	];
	Line/@l
]

StellaOctangulaDiagonals:=
	Line/@Union[Flatten[Diagonals/@Flatten[StellaOctangula[[1]],1,Polygon],1]];
  
StellaOctangulaPoints:=Union[Table[Point[Together[(
	StellaOctangulaDiagonals[[i,1,1]]+StellaOctangulaDiagonals[[i,1,2]])/2]],
	{i,12}]
]

StellaOctangulaOctahedron:=Module[{v=Flatten[StellaOctangulaPoints,1,Point],i},
	Graphics3D[Table[{
		Polygon[{v[[i]],v[[3]],v[[4]]}],
		Polygon[{v[[i]],v[[5]],v[[6]]}],
		Polygon[{v[[i]],v[[3]],v[[5]]}],
		Polygon[{v[[i]],v[[4]],v[[6]]}]},
	{i,2}]]
]

SzilassiPolygons:=Module[{
	coords={{-1.200,0.000,1.200},{1.200,0.000,1.200},{0.000,-1.260,-1.200},
		{0.000,1.260,-1.200},{0.200,-0.500,-0.800},{-0.200,0.500,-0.800},
		{-0.375,-0.375,-0.300},{0.375,0.375,-0.300},{0.450,-0.250,0.200},
		{-0.450,0.250,0.200},{-0.700,0.000,0.200},{0.700,0.000,0.200},
		{-0.700,-0.250,0.200},{0.700,0.250,0.200}
		},
	faces={{9,7,5,0,1,13},{10,12,0,5,3,2},{6,7,9,10,2,4},{7,6,8,11,3,5},
		{11,13,1,4,2,3},{8,6,4,1,0,12},{13,11,8,12,10,9}
		}
	},
	Polygon[coords[[#+1]]]&/@faces
]

Szilassi:=TriangulateSimplePolygons3D[
	Graphics3D[SzilassiPolygons],TriangleEdges->Automatic][[1]]

(* Tetrahemihexahedron *)

PolyhedronName[Tetrahemihexahedron]:="tetrahemihexahedron"
NetFaces[Tetrahemihexahedron]:={{4,6,7},{1,3,4},{4,3,6},{6,3,2},{10,6,5},{12,8,10},{10,8,6},{6,8,7},{17,14,13},{18,16,17},{17,16,14},{14,16,15},{10,14,12},{9,11,10},{10,11,14},{14,11,13}}
NetVertices[Tetrahemihexahedron]:={{(-1-Sqrt[3])/2,(-3+Sqrt[3])/6},{-Sqrt[3]/2,(-3-2Sqrt[3])/6},{(-1-Sqrt[3])/4,(-3-Sqrt[3])/12},{-1/2,1/(2Sqrt[3])},{0,(-3-Sqrt[3])/3},{0,-(1/Sqrt[3])},{1/2,1/(2Sqrt[3])},{(1+Sqrt[3])/4,(-3-Sqrt[3])/12},{Sqrt[3]/2,(-9-2Sqrt[3])/6},{Sqrt[3]/2,(-3-2Sqrt[3])/6},{(1+Sqrt[3])/2,(-3-Sqrt[3])/3},{(1+Sqrt[3])/2,(-3+Sqrt[3])/6},{(2+Sqrt[3])/2,(-9-2Sqrt[3])/6},{(2+Sqrt[3])/2,(-3-2Sqrt[3])/6},{(3+Sqrt[3])/2,(-3+Sqrt[3])/6},{(5+3Sqrt[3])/4,(-9-Sqrt[3])/12},{1+Sqrt[3],(-3-Sqrt[3])/3},{(3+2Sqrt[3])/2,(-6+Sqrt[3])/6}}
PolyhedronFaces[Tetrahemihexahedron]:={{6,5,7},{1,4,6},{6,4,5},{5,4,1},{2,5,1},{7,4,2},{2,4,5},{5,4,7},{6,3,1},{7,4,6},{6,4,3},{3,4,7},{2,3,7},{1,4,2},{2,4,3},{3,4,1}}
PolyhedronVertices[Tetrahemihexahedron]:={{1.9940169358628587,-.35566243269371522,1.84557863754038},{2.1316414545285817,-.93427707211306912,1.041672802936344},{2.4880338717151991,0,1.0521988413179606},{2.5557511543173783,-.56572292787741363,1.4709731701758223},{2.6234684369145995,-1.1314458557558181,1.88974749902929},{2.9798608541069637,-.19716878363837495,1.9002735374132989},{3.1174853727715124,-.77578342305886739,1.0963677028091161}}

(* This code due to Russell Towle *)

Trapezohedron[n_Integer,pitch_]:=Block[
	{c=1/(2*(Sin[Pi/(2*n)])),d=Tan[Degree pitch]/2,
		faceslope=(ArcTan[Tan[Degree pitch]/Cos[(Pi-(Pi/n))/2]]),
		apex=(Tan[faceslope]*c*Cos[Pi/n])+d},
		Graphics3D[
			{N[Table[
				{Polygon[{{c Cos[2Pi i/n], c Sin[2Pi i/n], -d},
				{0,0,-apex},
			 	{c Cos[2Pi (i+1)/n], c Sin[2Pi (i+1)/n], -d},
			 	{c Cos[(Pi/n)+(2Pi i/n)], c Sin[(Pi/n)+(2Pi i/n)], d}}],
				Polygon[{{c Cos[(Pi/n)+(2Pi i/n)], c Sin[(Pi/n)+(2Pi i/n)], d},
			 	{c Cos[(2Pi (i+1)/n)], c Sin[(2Pi (i+1)/n)], -d},
			 	{c Cos[(Pi/n)+(2Pi (i+1)/n)], c Sin[(Pi/n)+(2Pi (i+1)/n)], d},
			 	{0,0,apex}}]},
				{i, n}]
			]}
		]
	]/; n > 2
	
Trapezohedron[n_:7, pitch_:30] := Trapezohedron[n, pitch]

(* standard list of {name, Wythoff symbol} *)

UniformPolyhedronName[n_]:=standard[[n,1]]

UniformPolyhedronHullName[n_]:=standard[[n,3]]

UniformPolyhedronDualName[n_]:=dualnames[[n]]

UniformPolyhedron[n_Integer,prec_:25] /; 1 <= n <= Length[standard]:=
	Graphics3D[MakeUniform[standard[[n, 2]],prec]][[1]]

standard = {
{"tetrahedron", w1[3, 2, 3], "tetrahedron"},
{"truncated tetrahedron", w2[2, 3, 3], "truncated tetrahedron"},
{"octahemioctahedron", w2[3/2, 3, 3], "cuboctahedron"},
{"tetrahemihexahedron", w2[3/2, 3, 2], "octahedron"},
{"octahedron", w1[4, 2, 3], "octahedron"},
{"cube", w1[3, 2, 4], "cube", "cube"},
{"cuboctahedron", w1[2, 3, 4], "cuboctahedron"},
{"truncated octahedron", w2[2, 4, 3], "truncated octahedron"},
{"truncated cube", w2[2, 3, 4], "truncated cube"},
{"small rhombicuboctahedron", w2[3, 4, 2], "small rhombicuboctahedron"},
{"great rhombicuboctahedron (Archimedean)", w3[2, 3, 4], "great rhombicuboctahedron (Archimedean)"},
{"snub cube", w0[2, 3, 4], "snub cube"},
{"small cubicuboctahedron", w2[3/2, 4, 4], "small rhombicuboctahedron"},
{"great cubicuboctahedron", w2[3, 4, 4/3], "truncated cube"},
{"cubohemioctahedron", w2[4/3, 4, 3], "cuboctahedron"},
{"cubitruncated cuboctahedron", w3[4/3, 3, 4], "H1"},
{"great rhombicuboctahedron", w2[3/2, 4, 2], "truncated cube"},
{"small rhombihexahedron", w3[3/2, 2, 4], "small rhombicuboctahedron"},
{"stellated truncated hexahedron", w2[2, 3, 4/3], "small rhombicuboctahedron"},
{"great truncated cuboctahedron", w3[4/3, 2, 3], "H2"},
{"great rhombihexahedron", w3[4/3, 3/2, 2], "truncated cube"},
{"icosahedron", w1[5, 2, 3], "icosahedron"},
{"dodecahedron", w1[3, 2, 5], "dodecahedron"},
{"icosidodecahedron", w1[2, 3, 5], "icosidodecahedron"},
{"truncated icosahedron", w2[2, 5, 3], "truncated icosahedron"},
{"truncated dodecahedron", w2[2, 3, 5], "truncated dodecahedron"},
{"small rhombicosidodecahedron", w2[3, 5, 2], "small rhombicosidodecahedron"},
{"great rhomicosidodecahedron (Archimedean)", w3[2, 3, 5], "great rhomicosidodecahedron (Archimedean)"},
{"snub dodecahedron", w0[2, 3, 5], "snub dodecahedron"},
{"small ditrigonal icosidodecahedron", w1[3, 5/2, 3], "dodecahedron"},
{"small icosicosidodecahedron", w2[5/2, 3, 3], "H3"},
{"small snub icosicosidodecahedron", w0[5/2, 3, 3], "truncated icosahedron"},
{"small dodecicosidodecahedron", w2[3/2, 5, 5], "small rhombicosidodecahedron"},
{"small stellated dodecahedron", w1[5, 2, 5/2], "icosahedron"},
{"great dodecahedron", w1[5/2, 2, 5], "icosahedron"},
{"dodecadodecahedron", w1[2, 5/2, 5], "icosidodecahedron"},
{"truncated great dodecahedron", w2[2, 5/2, 5], "H4"},
{"rhombidodecadodecahedron", w2[5/2, 5, 2], "H5"},
{"small rhombidodecahedron", w3[2, 5/2, 5], "small rhombicosidodecahedron"},
{"snub dodecadodecahedron", w0[2, 5/2, 5], "snub dodecahedron"},
{"ditrigonal dodecadodecahedron", w1[3, 5/3, 5], "dodecahedron"},
{"great ditrigonal dodecicosidodecahedron", w2[3, 5, 5/3], "truncated dodecahedron"},
{"small ditrigonal dodecicosidodecahedron", w2[5/3, 3, 5], "H3"},
{"icosidodecadodecahedron", w2[5/3, 5, 3], "H5"},
{"icositruncated dodecadodecahedron", w3[5/3, 3, 5], "H6"},
{"snub icosidodecadodecahedron", w0[5/3, 3, 5], "H6"},
{"great ditrigonal icosidodecahedron", w1[3/2, 3, 5], "dodecahedron"},
{"great icosicosidodecahedron", w2[3/2, 5, 3], "truncated dodecahedron"},
{"small icosihemidodecahedron", w2[3/2, 3, 5], "icosidodecahedron"},
{"small dodecicosahedron", w3[3/2, 3, 5], "H3"},
{"small dodecahemidodecahedron", w2[5/4, 5, 5], "icosidodecahedron"},
{"great stellated dodecahedron", w1[3, 2, 5/2], "dodecahedron"},
{"great icosahedron", w1[5/2, 2, 3], "icosahedron"},
{"great icosidodecahedron", w1[2, 5/2, 3], "icosidodecahedron"},
{"great truncated icosahedron", w2[2, 5/2, 3], "H7"},
{"rhombicosahedron", w3[2, 5/2, 3], "H5"},
{"great snub icosidodecahedron", w0[2, 5/2, 3], "H7"},
{"small stellated truncated dodecahedron", w2[2, 5, 5/3], "small rhombicosidodecahedron"},
{"truncated dodecadodecahedron", w3[5/3, 2, 5], "H8"},
{"inverted snub dodecadodecahedron", w0[5/3, 2, 5], "H7'"},
{"great dodecicosidodecahedron", w2[5/2, 3, 5/3], "H4"},
{"small dodecahemicosahedron", w2[5/3, 5/2, 3], "icosidodecahedron"},
{"great dodecicosahedron", w3[5/3, 5/2, 3], "truncated dodecahedron"},
{"great snub dodecicosidodecahedron", w0[5/3, 5/2, 3], "small rhombicosidodecahedron"},
{"great dodecahemicosahedron", w2[5/4, 5, 3], "icosidodecahedron"},
{"great stellated truncated dodecahedron", w2[2, 3, 5/3], "H3"},
{"great rhombicosidodecahedron", w2[5/3, 3, 2], "H4"},
{"great truncated icosidodecahedron", w3[5/3, 2, 3], "H9"},
{"great inverted snub icosidodecahedron", w0[5/3, 2, 3], "H7"},
{"great dodecahemidodecahedron", w2[5/3, 5/2, 5/3], "icosidodecahedron"},
{"great icosihemidodecahedron", w2[3/2, 3, 5/3], "icosidodecahedron"},
{"small retrosnub icosicosidodecahedron", w0[3/2, 3/2, 5/2], "H10"},
{"great rhombidodecahedron", w3[3/2, 5/3, 2], "H4"},
{"great retrosnub icosidodecahedron", w0[3/2, 5/3, 2], "H11"},
{"great dirhombicosidodecahedron", w0[3/2, 5/3, 3, 5/2], "small rhombicosidodecahedron"},
{"pentagonal prism", w2[2, 5, 2], "pentagonal prism"},
{"pentagonal antiprism", w0[2, 2, 5], "pentagonal antiprism"},
{"pentagrammic prism", w2[2, 5/2, 2], "pentagonal prism"},
{"pentagrammic antiprism", w0[2, 2, 5/2], "pentagonal prism"},
{"pentagrammic crossed antiprism", w0[2, 2, 5/3], "pentagonal antiprism"}
};

dualnames={
"tetrahedron",
"triakis tetrahedron",
"octahemioctacron",
"tetrahemihexacron",
"cube",
"octahedron",
"rhombic dodecahedron",
"tetrakis hexahedron",
"triakis octahedron",
"deltoidal icositetrahedron",
"disdyakis dodecahedron",
"pentagonal icositetrahedron",
"small hexacronic icositetrahedron",
"great hexacronic icositetrahedron",
"hexahemioctahedron",
"tetradyakis hexahedron",
"great deltoidal icositetrahedron",
"small rhombihexacron",
"great triakis octahedron",
"great disdyakis dodecahedron",
"great rhombihexacron",
"dodecahedron",
"icosahedron",
"rhombic triacontahedron",
"pentakis dodecahedron",
"triakis icosahedron",
"deltoidal hexecontahedron",
"disdyakis triacontahedron",
"pentagonal hexecontahedron",
"small triambic icosahedron",
"small icosacronic hexecontahedron",
"small hexagonal hexecontahedron",
"small dodecacronic hexecontahedron",
"great dodecahedron",
"small stellated dodecahedron",
"medial rhombic triacontahedron",
"small stellapentakis dodecahedron",
"medial deltoidal hexecontahedron",
"small rhombidodecacron",
"medial pentagonal hexecontahedron",
"medial triambic icosahedron",
"great ditrigonal dodecacronic hexecontahedron",
"small ditrigonal dodecacronic hexecontahedron",
"medial icosacronic hexecontahedron",
"tridyakis icosahedron",
"medial hexagonal hexecontahedron",
"great triambic icosahedron",
"great icosacronic hexecontahedron",
"small icosihemidodecacron",
"small dodecicosacron",
"small dodecahemidodecacron",
"great icosahedron",
"great stellated dodecahedron",
"great rhombic triacontahedron",
"great stellapentakis dodecahedron",
"rhombicosacron",
"great pentagonal hexecontahedron",
"great pentakis dodecahedron",
"medial disdyakis triacontahedron",
"medial inverted pentagonal hexecontahedron",
"great dodecacronic hexecontahedron",
"small dodecahemicosacron",
"great dodecicosacron",
"great hexagonal hexecontahedron",
"great dodecahemicosacron",
"great triakis icosahedron",
"great deltoidal hexecontahedron",
"great disdyakis triacontahedron",
"great inverted pentagonal hexecontahedron",
"great dodecahemidodecacron",
"great icosihemidodecacron",
"small hexagrammic hexecontahedron",
"great rhombidodecacron",
"great pentagrammic hexecontahedron",
"great dirhombicosidodecacron",
"pentagonal dipyramid",
"pentagonal deltahedron",
"pentagrammic dipyramid",
"pentagrammic deltahedron",
"pentagrammic concave deltahedron"
};

uniforms:=Module[{n},Table[UniformPolyhedronName[n],{n,Length[standard]}]]

ustring[str_String]:=Module[{p=Position[uniforms,ToLowerCase[str]]},
	If[p=={},"unknown U","U"<>ToString[p[[1,1]]]]
]

udualstring[str_String]:=Module[{p=Position[dualnames,ToLowerCase[str]]},
	If[p=={},"unknown U","U"<>ToString[p[[1,1]]]]
]

unit[v_List]:=v/Sqrt[v.v]
Unit[v_List]:=#/Sqrt[#.#]&[v]

UnitCuboid[{x_,y_,z_}]:={
		Polygon[{{x,y,z},{x+1,y,z},{x+1,y+1,z},{x,y+1,z}}],
		Polygon[{{x,y,z+1},{x+1,y,z+1},{x+1,y+1,z+1},{x,y+1,z+1}}],
		Polygon[{{x,y,z},{x,y+1,z},{x,y+1,z+1},{x,y,z+1}}],
		Polygon[{{x+1,y,z},{x+1,y+1,z},{x+1,y+1,z+1},{x+1,y,z+1}}],
		Polygon[{{x,y,z},{x+1,y,z},{x+1,y,z+1},{x,y,z+1}}],
		Polygon[{{x,y+1,z},{x+1,y+1,z},{x+1,y+1,z+1},{x,y+1,z+1}}]
	}

WenningerDuals:={
{1,"Tetrahedron",1,"Tetrahedron",8},
{2,"Octahedron",2,"Cube",8},
{3,"Cube",3,"Octahedron",9},
{4,"Icosahedron",4,"Dodecahedron",9},
{5,"Dodecahedron",5,"lcosahedron",9},
{6,"Truncated tetrahedron",6,"Triakistetrahedron",14},
{7,"Truncated octahedron",7,"Tetrakishexahedron",14},
{8,"Truncated hexahedron",8,"Triakisoctahedron",17},
{9,"Truncated icosahedron",9,"Pentakisdodecahedron",18},
{10,"Truncated dodecahedron",10,"Triakisicosahedron",19},
{11,"Cuboctahedron",11,"Rhombic dodecahedron",19},
{12,"Icosidodecahedron",12,"Rhombic triacontahedron",22},
{13,"Rhombicuboctahedron",13,"Deltoidal icositetrahedron",22},
{14,"Rhombicosidodecahedron",14,"Deltoidal hexecontahedron",24},
{15,"Rhombitruncated cuboctahedron",15,"Disdyakisdodecahedron",25},
{16,"Rhombitruncated icosidodecahedron",16,"Disdyakistriacontahedron",25},
{17,"Snub cuboctahedron",17,"Pentagonal icositetrahedron",28},
{18,"Snub icosidodecahedron",18,"Pentagonal hexecontahedron",29},
{"","",19,"First stellation of the triakistetrahedron",37},
{"","",20,"Final stellation of the triakistetrahedron",37},
{"","",21,"A stellation of the triakisoctahedron",38},
{"","",22,"Another stellation of the triakisoctahedron",38},
{20,"Small stellated dodecahedron",23,"Great dodecahedron",39},
{21,"Great dodecahedron",24,"Small stellated dodecahedron",39},
{22,"Great stellated dodecahedron",25,"Great icosahedron",40},
{41,"Great icosabedron",26,"Great stellated dodecahedron",40},
{73,"Dodecadodecahedron",27,"Medial rhombic triacontahedron",46},
{80,"Ditrigonal dodecadodecahedron",28,"Medial triambic icosahedron",46},
{87,"Great ditrigonal icosidodecahedron",28,"Great triambic icosahedron",46},
{70,"Small ditrigonal icosidodecahedron",29,"Small triambic icosahedron",46},
{"","",30,"Compound of dodecahedron and icosahedron stellations",51},
{"","",31,"Compound of great stellated dodecahedron and great icosahedron",52},
{94,"Great icosidodecahedron",32,"Great rhombic triacontahedron",54},
{69,"Small cubicuboctahedron",33,"Small hexacronic icositetrahedron",57},
{86,"Small rhombihexahedron",33,"Small rhombihexacron",57},
{92,"Stellated truncated hexahedron",34,"Great triakisoctahedron",57},
{77,"Great cubicuboctahedron",35,"Great hexacronic icositetrahedron",58},
{85,"Great rhombicuboctahedron",36,"Great deltoidal icositetrahedron",59},
{103,"Great rhombihexahedron",37,"Great rhombihexacron",60},
{81,"Great ditrigonal dodecicosidodecahedron",38,"Great ditrigonal dodecacronic hexecontahedron",62},
{88,"Great icosicosidodecahedron",39,"Great icosacronic hexecontahedron",65},
{101,"Great dodecicosahedron",40,"Great dodecicosacron",67},
{72,"Small dodecicosidodecahedron",41,"Small dodecacronic hexecontahedron",70},
{74,"Small rhombidodecahedron",41,"Small rhombidodecacron",70},
{97," Small stellated truncated dodecahedron",42,"Great pentakisdodecahedron",70},
{71,"Small icosicosidodecahedron",43,"Hidden vertices of the great pentakisdodecahedron",74},
{8,"Small ditrigonal dodecicosidodecahedron",44,"Small icosacronic hexecontabedron",74},
{90,"Small dodecicosahedron",45,"Small ditrigonal dodecacronic hexecontahedron",74},
{95,"Great truncated icosahedron",45,"Small dodecicosacron",74},
{104,"Great stellated truncated dodecahedron",46,"Great stellapentakisdodecahedron",75},
{75,"Great truncated dodecahedron",47,"Great triakisicosahedron",77},
{76,"Rhombidodecadodecahedron",48,"Small stellapentakisdodecahedron",84},
{83,"lcosidodecadodecahedron",49,"Medial deltoidal hexecontahedron",84},
{96,"Rhombicosahedron",50,"Medial icosacronic hexecontahedron",85},
{99,"Great dodecicosidodecahedron",51,"Rhombicosacron",85},
{105,"Great rhombicosidodecahedron",52,"Great dodecacronic hexecontahedron",88},
{109,"Great rhombidodecahedron",53,"Great deltoidal hexecontahedron",88},
{79,"Cubitruncated cuboctahedron",53,"Great rhombidodecacron",88},
{93,"Great truncated cuboctahedron",54,"Tetradyakishexabedron",92},
{84,"Icositruncated dodecadodecahedron",55,"Great disdyakisdodecahedron",92},
{98,"Truncated dodecadodecahedron",56,"Tridyakisicosahedron",96},
{108,"Great truncated icosidodecahedron",57,"Medial disdyakistriacontahedron",96},
{67,"Tetrahemihexahedron",58,"Great disdyakistriacontahedron",96},
{68,"Octahemioctahedron",59,"Tetrahemihexacron",101},
{78,"Cubohemioctahedron",60,"Octahemioctacron",104},
{89,"Small icosihemidodecahedron",60,"Hexahemioctacron",104},
{91,"Small dodecahemidodecahedron",61,"Small icosihemidodecacron",104},
{100,"Small dodecahemicosahedron",61,"Small dodecahemidodecacron",104},
{102,"Great dodecahemicosahedron",62,"Small dodecahemicosacron",107},
{106,"Great icosihemidodecahedron",62,"Great dodecahemicosacron",107},
{107,"Great dodecahemidodecahedron",63,"Great icosihemidodecacron",107},
{"","",63,"Great dodecahemidodecacron",107},
{"","",64,"Stellation to infinity"," rhombic dodecahedron",115},
{"","",65,"Stellation to infinity"," five cubes",115},
{"","",66,"Stellation to infinity"," rhombic triacontahedron",115},
{110,"Small snub icosicosidodecahedron",67,"Small hexagonal hexecontahedron",119},
{111,"Snub dodecadodecahedron",68,"Medial pentagonal hexecontahedron",120},
{112,"Snub icosidodecadodecahedron",69,"Medial hexagonal hexecontahedron",121},
{113,"Great snub icosidodecahedron",70,"Great pentagonal hexecontahedron",123},
{114,"Inverted snub dodecadodecahedron",71,"Medial inverted pentagonal hexecontahedron",124},
{"","",72,"Truncation of the dual of 114",124},
{116,"Great inverted snub icosidodecahedron",73,"Great inverted pentagonal hexecontahedron",126},
{117,"Great retrosnub icosidodecahedron",74,"Great pentagrammic hexecontahedron",128},
{118,"Small retrosnub icosicosidodecahedron",75,"Small hexagrammic hexecontahedron",135},
{115,"Great snub dodecicosidodecahedron",76,"Great hexagonal hexecontahedron",136},
{119,"Great dirhombicosidodecahedron",77,"Great dirhombicosidodecacron",139},
{"","",78,"Compound of twenty cubes",140},
{"","",79,"Compound of two small stellated dodecahedra",144},
{"","",80,"Compound of two great stellated dodecahedra",144},
{"","",81,"Compound of five dodecahedra",145},
{"","",82,"Compound of five small stellated dodecahedra",146},
{"","",83,"Compound of five great stellated dodecahedra",147}
}

WythoffSymbol[n_]:=standard[[n,2]]

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     (1/4)*(1 + Sqrt[5])}, {-(1/2), (1/2)*(3 + Sqrt[5]), (1/4)*(1 + Sqrt[5])}}], 
  Polygon[{{(1/4)*(-5 - Sqrt[5]), (1/2)*(2 + Sqrt[5]), 0}, {-(1/2), (1/2)*(3 + Sqrt[5]), 
     (1/4)*(-1 - Sqrt[5])}, {-(1/2), (1/2)*(3 + Sqrt[5]), (1/4)*(1 + Sqrt[5])}}], 
  Polygon[{{(1/4)*(1 + Sqrt[5]), (1/2)*(2 + Sqrt[5]), (1/2)*(-1 - Sqrt[5])}, 
    {1/2, (1/2)*(3 + Sqrt[5]), (1/4)*(-1 - Sqrt[5])}, {-(1/2), (1/2)*(3 + Sqrt[5]), 
     (1/4)*(-1 - Sqrt[5])}, {(1/4)*(-1 - Sqrt[5]), (1/2)*(2 + Sqrt[5]), (1/2)*(-1 - Sqrt[5])}, 
    {0, (1/4)*(5 + Sqrt[5]), -1 - Sqrt[5]/2}}], 
  Polygon[{{(1/4)*(1 + Sqrt[5]), (1/2)*(2 + Sqrt[5]), (1/2)*(1 + Sqrt[5])}, 
    {1/2, (1/2)*(3 + Sqrt[5]), (1/4)*(1 + Sqrt[5])}, {-(1/2), (1/2)*(3 + Sqrt[5]), 
     (1/4)*(1 + Sqrt[5])}, {(1/4)*(-1 - Sqrt[5]), (1/2)*(2 + Sqrt[5]), (1/2)*(1 + Sqrt[5])}, 
    {0, (1/4)*(5 + Sqrt[5]), (1/2)*(2 + Sqrt[5])}}], 
  Polygon[{{(1/4)*(5 + Sqrt[5]), (1/2)*(2 + Sqrt[5]), 0}, {1/2, (1/2)*(3 + Sqrt[5]), 
     (1/4)*(-1 - Sqrt[5])}, {1/2, (1/2)*(3 + Sqrt[5]), (1/4)*(1 + Sqrt[5])}}], 
  Polygon[{{(1/4)*(-5 - Sqrt[5]), (1/2)*(2 + Sqrt[5]), 0}, {-(1/2), (1/2)*(3 + Sqrt[5]), 
     (1/4)*(-1 - Sqrt[5])}, {(1/4)*(-1 - Sqrt[5]), (1/2)*(2 + Sqrt[5]), (1/2)*(-1 - Sqrt[5])}, 
    {-1 - Sqrt[5]/2, (1/2)*(1 + Sqrt[5]), (1/4)*(-1 - Sqrt[5])}}], 
  Polygon[{{(1/4)*(5 + Sqrt[5]), (1/2)*(2 + Sqrt[5]), 0}, {1/2, (1/2)*(3 + Sqrt[5]), 
     (1/4)*(-1 - Sqrt[5])}, {(1/4)*(1 + Sqrt[5]), (1/2)*(2 + Sqrt[5]), (1/2)*(-1 - Sqrt[5])}, 
    {(1/2)*(2 + Sqrt[5]), (1/2)*(1 + Sqrt[5]), (1/4)*(-1 - Sqrt[5])}}], 
  Polygon[{{(1/4)*(-5 - Sqrt[5]), (1/2)*(2 + Sqrt[5]), 0}, {-(1/2), (1/2)*(3 + Sqrt[5]), 
     (1/4)*(1 + Sqrt[5])}, {(1/4)*(-1 - Sqrt[5]), (1/2)*(2 + Sqrt[5]), (1/2)*(1 + Sqrt[5])}, 
    {-1 - Sqrt[5]/2, (1/2)*(1 + Sqrt[5]), (1/4)*(1 + Sqrt[5])}}], 
  Polygon[{{(1/4)*(5 + Sqrt[5]), (1/2)*(2 + Sqrt[5]), 0}, {1/2, (1/2)*(3 + Sqrt[5]), 
     (1/4)*(1 + Sqrt[5])}, {(1/4)*(1 + Sqrt[5]), (1/2)*(2 + Sqrt[5]), (1/2)*(1 + Sqrt[5])}, 
    {(1/2)*(2 + Sqrt[5]), (1/2)*(1 + Sqrt[5]), (1/4)*(1 + Sqrt[5])}}], 
  Polygon[{{(-(1/4))*(1 + Sqrt[5])^2, (1/4)*(1 + Sqrt[5]), 1/2}, 
    {(-(1/4))*(1 + Sqrt[5])^2, (1/4)*(1 + Sqrt[5]), -(1/2)}, {(-(1/4))*(1 + Sqrt[5])^2, 
     (1/4)*(-1 - Sqrt[5]), -(1/2)}, {(-(1/4))*(1 + Sqrt[5])^2, (1/4)*(-1 - Sqrt[5]), 1/2}}], 
  Polygon[{{-1 - Sqrt[5]/2, 0, (1/4)*(5 + Sqrt[5])}, {(-(1/4))*(1 + Sqrt[5])^2, 
     (1/4)*(1 + Sqrt[5]), 1/2}, {(-(1/4))*(1 + Sqrt[5])^2, (1/4)*(-1 - Sqrt[5]), 1/2}}], 
  Polygon[{{-1 - Sqrt[5]/2, (1/2)*(1 + Sqrt[5]), (1/4)*(-1 - Sqrt[5])}, 
    {(-(1/4))*(1 + Sqrt[5])^2, (1/4)*(1 + Sqrt[5]), -(1/2)}, {(-(1/4))*(1 + Sqrt[5])^2, 
     (1/4)*(1 + Sqrt[5]), 1/2}, {-1 - Sqrt[5]/2, (1/2)*(1 + Sqrt[5]), (1/4)*(1 + Sqrt[5])}, 
    {(1/4)*(-5 - Sqrt[5]), (1/2)*(2 + Sqrt[5]), 0}}], 
  Polygon[{{-1 - Sqrt[5]/2, (1/2)*(-1 - Sqrt[5]), (1/4)*(-1 - Sqrt[5])}, 
    {(-(1/4))*(1 + Sqrt[5])^2, (1/4)*(-1 - Sqrt[5]), -(1/2)}, {(-(1/4))*(1 + Sqrt[5])^2, 
     (1/4)*(-1 - Sqrt[5]), 1/2}, {-1 - Sqrt[5]/2, (1/2)*(-1 - Sqrt[5]), (1/4)*(1 + Sqrt[5])}, 
    {(1/4)*(-5 - Sqrt[5]), -1 - Sqrt[5]/2, 0}}], 
  Polygon[{{-1 - Sqrt[5]/2, 0, (1/4)*(-5 - Sqrt[5])}, {(-(1/4))*(1 + Sqrt[5])^2, 
     (1/4)*(1 + Sqrt[5]), -(1/2)}, {(-(1/4))*(1 + Sqrt[5])^2, (1/4)*(-1 - Sqrt[5]), -(1/2)}}], 
  Polygon[{{-1 - Sqrt[5]/2, 0, (1/4)*(5 + Sqrt[5])}, {(-(1/4))*(1 + Sqrt[5])^2, 
     (1/4)*(1 + Sqrt[5]), 1/2}, {-1 - Sqrt[5]/2, (1/2)*(1 + Sqrt[5]), (1/4)*(1 + Sqrt[5])}, 
    {(1/2)*(-1 - Sqrt[5]), (1/4)*(1 + Sqrt[5]), (1/2)*(2 + Sqrt[5])}}], 
  Polygon[{{-1 - Sqrt[5]/2, 0, (1/4)*(-5 - Sqrt[5])}, {(-(1/4))*(1 + Sqrt[5])^2, 
     (1/4)*(1 + Sqrt[5]), -(1/2)}, {-1 - Sqrt[5]/2, (1/2)*(1 + Sqrt[5]), (1/4)*(-1 - Sqrt[5])}, 
    {(1/2)*(-1 - Sqrt[5]), (1/4)*(1 + Sqrt[5]), -1 - Sqrt[5]/2}}], 
  Polygon[{{-1 - Sqrt[5]/2, 0, (1/4)*(5 + Sqrt[5])}, {(-(1/4))*(1 + Sqrt[5])^2, 
     (1/4)*(-1 - Sqrt[5]), 1/2}, {-1 - Sqrt[5]/2, (1/2)*(-1 - Sqrt[5]), (1/4)*(1 + Sqrt[5])}, 
    {(1/2)*(-1 - Sqrt[5]), (1/4)*(-1 - Sqrt[5]), (1/2)*(2 + Sqrt[5])}}], 
  Polygon[{{-1 - Sqrt[5]/2, 0, (1/4)*(-5 - Sqrt[5])}, {(-(1/4))*(1 + Sqrt[5])^2, 
     (1/4)*(-1 - Sqrt[5]), -(1/2)}, {-1 - Sqrt[5]/2, (1/2)*(-1 - Sqrt[5]), (1/4)*(-1 - Sqrt[5])}, 
    {(1/2)*(-1 - Sqrt[5]), (1/4)*(-1 - Sqrt[5]), -1 - Sqrt[5]/2}}], 
  Polygon[{{(1/2)*(3 + Sqrt[5]), (1/4)*(-1 - Sqrt[5]), -(1/2)}, 
    {(1/2)*(3 + Sqrt[5]), (1/4)*(-1 - Sqrt[5]), 1/2}, {(1/2)*(3 + Sqrt[5]), (1/4)*(1 + Sqrt[5]), 
     1/2}, {(1/2)*(3 + Sqrt[5]), (1/4)*(1 + Sqrt[5]), -(1/2)}}], 
  Polygon[{{(1/2)*(2 + Sqrt[5]), 0, (1/4)*(-5 - Sqrt[5])}, {(1/2)*(3 + Sqrt[5]), 
     (1/4)*(-1 - Sqrt[5]), -(1/2)}, {(1/2)*(3 + Sqrt[5]), (1/4)*(1 + Sqrt[5]), -(1/2)}}], 
  Polygon[{{(1/2)*(2 + Sqrt[5]), (1/2)*(-1 - Sqrt[5]), (1/4)*(1 + Sqrt[5])}, 
    {(1/2)*(3 + Sqrt[5]), (1/4)*(-1 - Sqrt[5]), 1/2}, {(1/2)*(3 + Sqrt[5]), (1/4)*(-1 - Sqrt[5]), 
     -(1/2)}, {(1/2)*(2 + Sqrt[5]), (1/2)*(-1 - Sqrt[5]), (1/4)*(-1 - Sqrt[5])}, 
    {(1/4)*(5 + Sqrt[5]), -1 - Sqrt[5]/2, 0}}], 
  Polygon[{{(1/2)*(2 + Sqrt[5]), (1/2)*(1 + Sqrt[5]), (1/4)*(1 + Sqrt[5])}, 
    {(1/2)*(3 + Sqrt[5]), (1/4)*(1 + Sqrt[5]), 1/2}, {(1/2)*(3 + Sqrt[5]), (1/4)*(1 + Sqrt[5]), 
     -(1/2)}, {(1/2)*(2 + Sqrt[5]), (1/2)*(1 + Sqrt[5]), (1/4)*(-1 - Sqrt[5])}, 
    {(1/4)*(5 + Sqrt[5]), (1/2)*(2 + Sqrt[5]), 0}}], 
  Polygon[{{(1/2)*(2 + Sqrt[5]), 0, (1/4)*(5 + Sqrt[5])}, {(1/2)*(3 + Sqrt[5]), 
     (1/4)*(-1 - Sqrt[5]), 1/2}, {(1/2)*(3 + Sqrt[5]), (1/4)*(1 + Sqrt[5]), 1/2}}], 
  Polygon[{{(1/2)*(2 + Sqrt[5]), 0, (1/4)*(-5 - Sqrt[5])}, {(1/2)*(3 + Sqrt[5]), 
     (1/4)*(-1 - Sqrt[5]), -(1/2)}, {(1/2)*(2 + Sqrt[5]), (1/2)*(-1 - Sqrt[5]), 
     (1/4)*(-1 - Sqrt[5])}, {(1/2)*(1 + Sqrt[5]), (1/4)*(-1 - Sqrt[5]), -1 - Sqrt[5]/2}}], 
  Polygon[{{(1/2)*(2 + Sqrt[5]), 0, (1/4)*(5 + Sqrt[5])}, {(1/2)*(3 + Sqrt[5]), 
     (1/4)*(-1 - Sqrt[5]), 1/2}, {(1/2)*(2 + Sqrt[5]), (1/2)*(-1 - Sqrt[5]), 
     (1/4)*(1 + Sqrt[5])}, {(1/2)*(1 + Sqrt[5]), (1/4)*(-1 - Sqrt[5]), (1/2)*(2 + Sqrt[5])}}], 
  Polygon[{{(1/2)*(2 + Sqrt[5]), 0, (1/4)*(-5 - Sqrt[5])}, {(1/2)*(3 + Sqrt[5]), 
     (1/4)*(1 + Sqrt[5]), -(1/2)}, {(1/2)*(2 + Sqrt[5]), (1/2)*(1 + Sqrt[5]), 
     (1/4)*(-1 - Sqrt[5])}, {(1/2)*(1 + Sqrt[5]), (1/4)*(1 + Sqrt[5]), -1 - Sqrt[5]/2}}], 
  Polygon[{{(1/2)*(2 + Sqrt[5]), 0, (1/4)*(5 + Sqrt[5])}, {(1/2)*(3 + Sqrt[5]), 
     (1/4)*(1 + Sqrt[5]), 1/2}, {(1/2)*(2 + Sqrt[5]), (1/2)*(1 + Sqrt[5]), (1/4)*(1 + Sqrt[5])}, 
    {(1/2)*(1 + Sqrt[5]), (1/4)*(1 + Sqrt[5]), (1/2)*(2 + Sqrt[5])}}], 
  Polygon[{{(1/2)*(1 + Sqrt[5]), (1/4)*(1 + Sqrt[5]), (1/2)*(2 + Sqrt[5])}, 
    {(1/4)*(1 + Sqrt[5]), (1/2)*(2 + Sqrt[5]), (1/2)*(1 + Sqrt[5])}, 
    {(1/2)*(2 + Sqrt[5]), (1/2)*(1 + Sqrt[5]), (1/4)*(1 + Sqrt[5])}}], 
  Polygon[{{(1/2)*(1 + Sqrt[5]), (1/4)*(1 + Sqrt[5]), -1 - Sqrt[5]/2}, 
    {(1/4)*(1 + Sqrt[5]), (1/2)*(2 + Sqrt[5]), (1/2)*(-1 - Sqrt[5])}, 
    {(1/2)*(2 + Sqrt[5]), (1/2)*(1 + Sqrt[5]), (1/4)*(-1 - Sqrt[5])}}], 
  Polygon[{{(1/2)*(1 + Sqrt[5]), (1/4)*(-1 - Sqrt[5]), (1/2)*(2 + Sqrt[5])}, 
    {(1/4)*(1 + Sqrt[5]), -1 - Sqrt[5]/2, (1/2)*(1 + Sqrt[5])}, 
    {(1/2)*(2 + Sqrt[5]), (1/2)*(-1 - Sqrt[5]), (1/4)*(1 + Sqrt[5])}}], 
  Polygon[{{(1/2)*(-1 - Sqrt[5]), (1/4)*(1 + Sqrt[5]), (1/2)*(2 + Sqrt[5])}, 
    {(1/4)*(-1 - Sqrt[5]), (1/2)*(2 + Sqrt[5]), (1/2)*(1 + Sqrt[5])}, 
    {-1 - Sqrt[5]/2, (1/2)*(1 + Sqrt[5]), (1/4)*(1 + Sqrt[5])}}], 
  Polygon[{{(1/2)*(1 + Sqrt[5]), (1/4)*(-1 - Sqrt[5]), -1 - Sqrt[5]/2}, 
    {(1/4)*(1 + Sqrt[5]), -1 - Sqrt[5]/2, (1/2)*(-1 - Sqrt[5])}, 
    {(1/2)*(2 + Sqrt[5]), (1/2)*(-1 - Sqrt[5]), (1/4)*(-1 - Sqrt[5])}}], 
  Polygon[{{(1/2)*(-1 - Sqrt[5]), (1/4)*(-1 - Sqrt[5]), (1/2)*(2 + Sqrt[5])}, 
    {(1/4)*(-1 - Sqrt[5]), -1 - Sqrt[5]/2, (1/2)*(1 + Sqrt[5])}, 
    {-1 - Sqrt[5]/2, (1/2)*(-1 - Sqrt[5]), (1/4)*(1 + Sqrt[5])}}], 
  Polygon[{{(1/2)*(-1 - Sqrt[5]), (1/4)*(1 + Sqrt[5]), -1 - Sqrt[5]/2}, 
    {(1/4)*(-1 - Sqrt[5]), (1/2)*(2 + Sqrt[5]), (1/2)*(-1 - Sqrt[5])}, 
    {-1 - Sqrt[5]/2, (1/2)*(1 + Sqrt[5]), (1/4)*(-1 - Sqrt[5])}}], 
  Polygon[{{(1/2)*(-1 - Sqrt[5]), (1/4)*(-1 - Sqrt[5]), -1 - Sqrt[5]/2}, 
    {(1/4)*(-1 - Sqrt[5]), -1 - Sqrt[5]/2, (1/2)*(-1 - Sqrt[5])}, 
    {-1 - Sqrt[5]/2, (1/2)*(-1 - Sqrt[5]), (1/4)*(-1 - Sqrt[5])}}]};

(* Zonohedron *)
(* Code by David Eppstein:
http://www.ics.uci.edu/~eppstein/junkyard/ukraine/ukraine.ma
*)

eps=10^-16;

ZSignTest[vv_,f_] :=
	Det [ Part[vv,f] ~Join~
		  Table[(i+Pi)^j,
				{i,Length[vv[[1]]]-Length[f]},
				{j,Length[vv[[1]]]}]]

ZOldFaces[vv_,ff_,i_,d_] :=(ZAddToFace[vv, #, i, d])& /@ ff
ZAddToFace[vv_,f_,i_,d_] :=
	If[ Length[f] < (d-1) ||
		Abs[ZSignTest[vv, Append[Take[f,d-1],i]]] < eps,
		Append[f,i], f ]
		
ZTuples[i_,1,ff_] :=If[MemberQ[Flatten[ff],i], {}, {{i}}]
ZTuples[i_,d_,ff_] :=(Append[#,i]&) /@
	ZAllTuples[i-1,d-1,Select[ff,MemberQ[#,i]&]]
ZAllTuples[i_,1,ff_] :=List /@ Complement[Range[i], Flatten[ff]]
ZAllTuples[i_,d_,ff_] :=Join @@ Table[ZTuples[j,d,ff],{j,i}]
ZAddTuples[ff_,i_,d_] := ff ~Join~ ZTuples[i,d-1,ff]

ZNewFaces[vv_,ff_,i_,d_] := ZAddTuples[ZOldFaces[vv,ff,i,d], i, d]
ZAllFaces[vv_,d_] :=Fold[ZNewFaces[vv,#1,#2,d]&,
		 {{}}, Table[i,{i,Length[vv]}]]
ZLiftAll[vv_,ff_,d_] :=Join @@ Map[ZLiftFace[vv,#,d]&,ff]

ZOrigin[vv_] := Table[ 0, {Length[ vv[[1]] ]} ]
ZFace[vv_,f_,1] := ZOrigin[vv];
ZFace[vv_,{i_},2] := {ZOrigin[vv],vv[[i]]};
ZFace[vv_,f_,d_] := Zonotope[ Part[vv,f], d-1 ];

ZLiftVector[vv_,f_,i_,d_] :=
	If [ MemberQ[f, i], {ZOrigin[vv],ZOrigin[vv]},
		 If [ ZSignTest[vv, Append[Take[f,d-1],i]] < 0,
		 	  {vv[[i]], ZOrigin[vv]},
		 	  {ZOrigin[vv], vv[[i]]}]]
ZVecPlus[a_,v_] :=
	If[Head[First[a]] === List,
		ZVecPlus[#,v]& /@ a, a+v]
ZLiftFace[vv_,f_,d_] :=
	{#1 ~ZVecPlus~ #2[[1]], #1 ~ZVecPlus~ #2[[2]]}& [
		ZFace[vv,f,d],
		Sum[ZLiftVector[vv,f,j,d],{j,Length[vv]}]]

Zonotope[vv_,d_] :=
	ZLiftAll[vv,ZAllFaces[vv,d],d]

ZvEq[u_,v_] := Abs[Plus @@ (Abs /@ (u-v))] < eps

ZNext[f_,e_] :=
	Join @@ Map[
		If[ e[[2]] ~ZvEq~ #[[1]] &&
					Not[e[[1]] ~ZvEq~ #[[2]]], #,
			If[ e[[2]] ~ZvEq~ #[[2]] &&
					Not[e[[1]] ~ZvEq~ #[[1]]],
				Reverse[#], {}]]&, f]

ZMakePolygon[f_] :=
	First /@ NestList[ ZNext[f,#]&, f[[1]], Length[f]-1 ]

Zonohedron[vv_] := Polygon /@ ZMakePolygon /@ Zonotope[vv,3]

(* End David Eppstein code *)


End[]

(* Protect[  ] *)

EndPackage[]

Johnson Solids

(*:Mathematica Version: 5.2 *)

(*:Package Version: 1.54 *)

(*:Name: MathWorld`JohnsonSolids` *)

(*:Author: Eric W. Weisstein *)

(*:URL:
	http://mathworld.wolfram.com/packages/JohnsonSolids.m
*)

(*:Summary:
  This package uses _corrected_ versions of the Bell Labs Netlib database to 
  plot the 92 Johnson solids (together with the Platonics, Archimdeans, and
  a few other miscellaneous solids) and their nets.
*)

(*:History:
	v1.00 (Mar 28, 1996)
	      (Nov 24, 1996): Written by Eric W. Weisstein
	v1.10 (May 19, 1997): Ability to handle lumped file (for easier www distribution) 
	v1.20 (Feb 23, 1998): Fixed problems with definitions of Netlist options
	v1.21 (Feb 25, 1998): Added LabelVertices option to Netlist for nets
	v1.22 (Feb 28, 1998): Added LabelVertices option for solids
	v1.23 (Mar  5, 1998): Documentation changed slightly, JohnsonSolids.dat
	                      corrections completed!
	v1.24 (May 19, 1998): JohnsonFaces
	v1.25 (Oct 30, 1998): Updated docs on missing :solids for Archimedean Solids,
	                      Fixed Johnson.dat file
	v1.26 (Nov  9, 1998): Fixed J64 solid, fixed typo in Octahedron
	v1.27 (Dec 12, 1998): Fixed error in one N11 (cuboctahedron) triangle
	v1.28 (Jun 19, 1999): Updated documentation
	v1.29 (Jul  9, 1999): Added ability to read in analytic net vertices with StringParse.
	                      Corrcted analytic net vertices in JohnsonSolids.dat for
	                       N81, N137, N138
	v1.30 (Jul 27, 1999): Renamed ArchimedeanDuals` to Duals`
	v1.31 (Jul 30, 1999): Fixed truncated octahedron solid in JohnsonSolids.dat
	v1.32 (Aug  2, 1999): Added NetVertices, NetConnectivity for Steve C.
	v1.33 (Nov 22, 1999): Added some JohnsonSolid[n]
	v1.34 (Jan  1, 2000): Updated URL.  Added 3-, 4-, and 5-cupola
	                      Johnson solids.  Removed Graphics and Graphics3D
	                      wrappers.
	v1.35 (Feb  6, 2000): Added elongated cupolas, gyroelongated pyramids,
	                      and 5-rotunda
	v1.36 (Feb 10, 2000): Added a bunch of augmented prisms and dodecahedra.
	v1.37 (Feb 22, 2000): JohnsonSolid[84] orientations fixed
	v1.38 (May 23, 2000): Added missing vertex for J64 in JohnsonSolid.dat.
	                      Added backslashes to usage messages.
	v1.39 (Nov 13, 2000): Included package renamed MiscPolyhedra
	v1.40 (Feb 16, 2002): Fixed context for JohnsonSolid[17]
	v1.50 (Sep  8, 2002): Completely rewritten to contain all data directly in
	                      this package instead of needing to load Netlib data.
	                      As a result, most of the old ugly I/O routines were
	                      now simply ripped out.  Scaled net and solid vertices for
	                      Johnson solids 59, 60, 61, 95, 133, 134 so they now have 
	                      unit edge lengths.
	v1.51 (Nov 10, 2002): Switched to new Polyhedron[] syntax, broke out construction
	                      routines to PolyhedronOperations where they belong.  Moved
	                      non-Johnson solids into Archimedean.m and Polyehdron.m.
	v1.52 (Oct 18, 2003): updated context
	v1.53 (Nov 21, 2003): Fixed JohnsonSolid[17]
	v1.54 (Dec 25, 2003): JohnsonSolids, consolidated JohnsonExact
	
	(c) 1996-2005 Eric W. Weisstein
*)

(*:Keywords:
  Johnson solids, uniform polyhedra, Netlib database
*)

(*:Requirements:

  http://mathworld.wolfram.com/packages/MiscPolyhedra.m
	is required for exact Johnson solid constructions
  http://mathworld.wolfram.com/packages/PolyhedronOperations.m
	is required to actaully build the polyhedra and nets based on vertex and face 
	information contained in this package

*)

(*:Discussion:
  The original Bell Labs netlib database is accessible at:  
  
    http://netlib.bell-labs.com/netlib/polyhedra/
    
  The solids have been renumbered to start at 1 instead of 0, so in this
  package, Solid[1] is the tetrahedron.  To get J#, subtract 45 from Netlib 
  number.

  Several of the :solid entires have been corrected by George Hart:
  
    http://www.li.net/~george/virtual-polyhedra/netlib-info.html
  
  However, Hart removed the :net entries in the process, so I've added
  them back in. I've also corrected the erroneous nets (which Hart did not
  consider).  Here are the modifications made to the netlib entries:
  
  Hart      (george@li.net)      corrected :solids 66, 67, 68, 69, 81, 115
  Weisstein (eww@wolfram.com)    corrected :solids 70
                                 added     :solids 0, 9, 10, 11, 12,
                                                   (15, 16, 17, 18, 19, 20, 21)
                                 corrected :nets   66, 67, 68, 69, 81, 115
                                 
  1999-07-09: corrected analytic value for N81/J37 :net vertex
               -3.098076212[_] to [(-1/2-(3/2)Sqrt(3))]
              Added and corrected analytic net vertices for N137: tetrahemihexahedron
              Corrected analaytic net vertices for N138: octahemioctahedron
  1999-07-30: Fixed truncated octahedron solid in JohnsonSolids.dat
  2000-05-23: Added missing vertex 
              -3.2672194889332209, 1.3461181078064268, -1.5554725849995636
              and corrected last three faces of solid after notification by Jean Denel,
              JeanDenel@aol.com
  2002-09-09: Fixed NetFaces[1] (tetrahedron)
*)

(*:Limitations:

  The analytic vertices of the nets are not necessarily in FullSimplified form.
  
  The author hasn't finished computing all Johnsons analytically yet
  
  What I should do is insert the explicit coordinates of Johnson solids
  instead of relying on routines in Prism.m
*)


BeginPackage["MathWorld`JohnsonSolids`",
	{
	"Graphics`Colors`",
	"MathWorld`PolyhedronOperations`",
	"MathWorld`Prisms`"
	}
]

JohnsonSolid::usage =
"Polyhedron[JohnsonSolid[n]] gives a list of Polygons \
representing the nth Johnson solid.  Similarly, PolyhedronFaces[JohnsonSolid[n]], \
PolyhedronVertices[JohnsonSolid[n]], PolyhedronName[JohnsonSolid[n]], \
Net[JohnsonSolid[n]], NetFaces[JohnsonSolid[n]], and \
NetVertices[JohnsonSolid[n]] can be used.  Not all Johnson solids have \
exact vertices yet."

JohnsonSolids::usage =
"JohnsonSolids gives a list of the 92 Johnson solids."


Begin["`Private`"]

(* JohnsonSolids *)

JohnsonSolids:=JohnsonSolid/@Range[92]

(* Solid Names *)

PolyhedronName[JohnsonSolid[ 1]]:="square pyramid (J1)"
PolyhedronName[JohnsonSolid[ 2]]:="pentagonal pyramid (J2)"
PolyhedronName[JohnsonSolid[ 3]]:="triangular cupola (J3)"
PolyhedronName[JohnsonSolid[ 4]]:="square cupola (J4)"
PolyhedronName[JohnsonSolid[ 5]]:="pentagonal cupola (J5)"
PolyhedronName[JohnsonSolid[ 6]]:="pentagonal rotunda (J6)"
PolyhedronName[JohnsonSolid[ 7]]:="elongated triangular pyramid (J7)"
PolyhedronName[JohnsonSolid[ 8]]:="elongated square pyramid (J8)"
PolyhedronName[JohnsonSolid[ 9]]:="elongated pentagonal pyramid (J9)"
PolyhedronName[JohnsonSolid[10]]:="gyroelongated square pyramid (J10)"
PolyhedronName[JohnsonSolid[11]]:="gyroelongated pentagonal pyramid (J11)"
PolyhedronName[JohnsonSolid[12]]:="triangular dipyramid (J12)"
PolyhedronName[JohnsonSolid[13]]:="pentagonal dipyramid (J13)"
PolyhedronName[JohnsonSolid[14]]:="elongated triangular dipyramid (J14)"
PolyhedronName[JohnsonSolid[15]]:="elongated square dipyramid (J15)"
PolyhedronName[JohnsonSolid[16]]:="elongated pentagonal dipyramid (J16)"
PolyhedronName[JohnsonSolid[17]]:="gyroelongated square dipyramid (J17)"
PolyhedronName[JohnsonSolid[18]]:="elongated triangular cupola (J18)"
PolyhedronName[JohnsonSolid[19]]:="elongated square cupola (J19)"
PolyhedronName[JohnsonSolid[20]]:="elongated pentagonal cupola (J20)"
PolyhedronName[JohnsonSolid[21]]:="elongated pentagonal rotunds (J21)"
PolyhedronName[JohnsonSolid[22]]:="gyroelongated triangular cupola (J22)"
PolyhedronName[JohnsonSolid[23]]:="gyroelongated square cupola (J23)"
PolyhedronName[JohnsonSolid[24]]:="gyroelongated pentagonal cupola (J24)"
PolyhedronName[JohnsonSolid[25]]:="gyroelongated pentagonal rotunda (J25)"
PolyhedronName[JohnsonSolid[26]]:="gyrobifastigium (J26)"
PolyhedronName[JohnsonSolid[27]]:="triangular orthobicupola (J27)"
PolyhedronName[JohnsonSolid[28]]:="square orthobicupola (J28)"
PolyhedronName[JohnsonSolid[29]]:="square gyrobicupola (J29)"
PolyhedronName[JohnsonSolid[30]]:="pentagonal orthobicupola (J30)"
PolyhedronName[JohnsonSolid[31]]:="pentagonal gyrobicupola (J31)"
PolyhedronName[JohnsonSolid[32]]:="pentagonal orthocupolarontunda (J32)"
PolyhedronName[JohnsonSolid[33]]:="pentagonal gyrocupolarotunda (J33)"
PolyhedronName[JohnsonSolid[34]]:="pentagonal orthobirotunda (J34)"
PolyhedronName[JohnsonSolid[35]]:="elongated triangular orthobicupola (J35)"
PolyhedronName[JohnsonSolid[36]]:="elongated triangular gyrobicupola (J36)"
PolyhedronName[JohnsonSolid[37]]:="elongated square gyrobicupola (J37)"
PolyhedronName[JohnsonSolid[38]]:="elongated pentagonal orthobicupola (J38)"
PolyhedronName[JohnsonSolid[39]]:="elongated pentagonal gyrobicupola (J39)"
PolyhedronName[JohnsonSolid[40]]:="elongated pentagonal orthocupolarotunda (J40)"
PolyhedronName[JohnsonSolid[41]]:="elongated pentagonal gyrocupolarotunda (J41)"
PolyhedronName[JohnsonSolid[42]]:="elongated pentagonal orthobirotunda (J42)"
PolyhedronName[JohnsonSolid[43]]:="elongated pentagonal gyrobirotunda (J43)"
PolyhedronName[JohnsonSolid[44]]:="gyroelongated triangular bicupola (J44)"
PolyhedronName[JohnsonSolid[45]]:="gyroelongated square bicupola (J45)"
PolyhedronName[JohnsonSolid[46]]:="gyroelongated pentagonal bicupola (J46)"
PolyhedronName[JohnsonSolid[47]]:="gyroelongated pentagonal cupolarotunda (J47)"
PolyhedronName[JohnsonSolid[48]]:="gyroelongated pentagonal birotunda (J48)"
PolyhedronName[JohnsonSolid[49]]:="augmented triangular prism (J49)"
PolyhedronName[JohnsonSolid[50]]:="biaugmented triangular prism (J50)"
PolyhedronName[JohnsonSolid[51]]:="triaugmented triangular prism (J51)"
PolyhedronName[JohnsonSolid[52]]:="augmented pentagonal prism (J52)"
PolyhedronName[JohnsonSolid[53]]:="biaugmented pentagonal prism (J53)"
PolyhedronName[JohnsonSolid[54]]:="augmented hexagonal prism (J54)"
PolyhedronName[JohnsonSolid[55]]:="parabiaugmented hexagonal prism (J55)"
PolyhedronName[JohnsonSolid[56]]:="metabiaugmented hexagonal prism (J56)"
PolyhedronName[JohnsonSolid[57]]:="triaugmented hexagonal prism (J57)"
PolyhedronName[JohnsonSolid[58]]:="augmented dodecahedron (J58)"
PolyhedronName[JohnsonSolid[59]]:="parabiaugmented dodecahedron (J59)"
PolyhedronName[JohnsonSolid[60]]:="metabiaugmented dodecahedron (J60)"
PolyhedronName[JohnsonSolid[61]]:="triaugmented dodecahedron (J61)"
PolyhedronName[JohnsonSolid[62]]:="metabidiminished icosahedron (J62)"
PolyhedronName[JohnsonSolid[63]]:="tridiminished icosahedron (J63)"
PolyhedronName[JohnsonSolid[64]]:="augmented tridiminished icosahedron (J64)"
PolyhedronName[JohnsonSolid[65]]:="augmented truncated tetrahedron (J65)"
PolyhedronName[JohnsonSolid[66]]:="augmented truncated cube (J66)"
PolyhedronName[JohnsonSolid[67]]:="biaugmented truncated cube (J67)"
PolyhedronName[JohnsonSolid[68]]:="augmented truncated dodecahedron (J68)"
PolyhedronName[JohnsonSolid[69]]:="parabiaugmented truncated dodecahedron (J69)"
PolyhedronName[JohnsonSolid[70]]:="metabiaugmented truncated dodecahedron (J70)"
PolyhedronName[JohnsonSolid[71]]:="triaugmented truncated dodecahedron (J71)"
PolyhedronName[JohnsonSolid[72]]:="gyrate rhombicosidodecahedron (J72)"
PolyhedronName[JohnsonSolid[73]]:="parabigyrate rhombicosidodecahedron (J73)"
PolyhedronName[JohnsonSolid[74]]:="metabigyrate rhombicosidodecahedron (J74)"
PolyhedronName[JohnsonSolid[75]]:="trigyrate rhombicosidodecahedron (J75)"
PolyhedronName[JohnsonSolid[76]]:="diminished rhombicosidodecahedron (J76)"
PolyhedronName[JohnsonSolid[77]]:="paragyrate diminished rhombicosidodecahedron (J77)"
PolyhedronName[JohnsonSolid[78]]:="metagyrate diminished rhombicosidodecahedron (J78)"
PolyhedronName[JohnsonSolid[79]]:="bigyrate diminished rhombicosidodecahedron (J79)"
PolyhedronName[JohnsonSolid[80]]:="parabidiminished rhombicosidodecahedron (J80)"
PolyhedronName[JohnsonSolid[81]]:="metabidiminished rhombicosidodecahedron (J81)"
PolyhedronName[JohnsonSolid[82]]:="gyrate bidiminished rhombicosidodecahedron (J82)"
PolyhedronName[JohnsonSolid[83]]:="tridiminished rhombicosidodecahedron (J83)"
PolyhedronName[JohnsonSolid[84]]:="snub disphenoid (J84)"
PolyhedronName[JohnsonSolid[85]]:="snub square antiprism (J85)"
PolyhedronName[JohnsonSolid[86]]:="sphenocorona (J86)"
PolyhedronName[JohnsonSolid[87]]:="augmented sphenocorona (J87)"
PolyhedronName[JohnsonSolid[88]]:="sphenomegacorona (J88)"
PolyhedronName[JohnsonSolid[89]]:="hebesphenomegacorona (J89)"
PolyhedronName[JohnsonSolid[90]]:="disphenocingulum (J90)"
PolyhedronName[JohnsonSolid[91]]:="bilunabirotunda (J91)"
PolyhedronName[JohnsonSolid[92]]:="triangular hebesphenorotunda (J92)"

(* Net Faces *)

NetFaces[JohnsonSolid[ 1]]:={{3,2,6,7},{2,3,1},{6,2,4},{7,6,8},{3,7,5}}
NetFaces[JohnsonSolid[ 2]]:={{3,4,8,9,5},{4,3,1},{8,4,6},{9,8,10},{5,9,7},{3,5,2}}
NetFaces[JohnsonSolid[ 3]]:={{3,5,10,14,11,6},{7,12,8},{12,7,6,11},{12,11,15},{8,12,16,13},{8,13,9},{7,8,4,2},{7,2,1}}
NetFaces[JohnsonSolid[ 4]]:={{6,4,2,1,3,5,7,8},{13,12,16,17},{12,13,8,7},{12,7,10},{16,12,11,15},{16,15,20},{17,16,21,22},{17,22,19},{13,17,18,14},{13,14,9}}
NetFaces[JohnsonSolid[ 5]]:={{8,6,4,2,1,3,5,7,9,10},{16,15,20,22,21},{15,16,10,9},{15,9,12},{20,15,13,17},{20,17,23},{22,20,24,26},{22,26,28},{21,22,27,25},{21,25,19},{16,21,18,14},{16,14,11}}
NetFaces[JohnsonSolid[ 6]]:={{4,8,15,21,28,32,29,22,16,9},{18,26,25,17,13},{18,12,16,22,24},{18,24,26},{24,22,31},{26,27,36,37,33},{26,33,25},{33,37,38},{25,34,35,30,23},{25,23,17},{23,30,20},{17,19,14,7,10},{17,10,13},{10,7,2},{13,6,1,3,11},{13,11,18},{11,3,5}}
NetFaces[JohnsonSolid[ 7]]:={{4,6,8},{2,1,4,5},{3,2,5},{5,4,8,9},{7,5,9},{9,8,11,12},{10,9,12}}
NetFaces[JohnsonSolid[ 8]]:={{5,4,8,9},{2,1,5,6},{3,2,6},{6,5,9,10},{7,6,10},{10,9,12,13},{11,10,13},{13,12,15,16},{14,13,16}}
NetFaces[JohnsonSolid[ 9]]:={{8,7,10,14,12},{2,1,4,5},{3,2,5},{5,4,8,9},{6,5,9},{9,8,12,13},{11,9,13},{13,12,16,17},{15,13,17},{17,16,19,20},{18,17,20}}
NetFaces[JohnsonSolid[10]]:={{1,2,4},{4,2,6},{4,6,8},{8,6,10},{8,10,12},{12,10,13},{12,13,15},{15,13,16},{1,4,3},{4,8,7},{8,12,11},{12,15,14},{10,6,5,9}}
NetFaces[JohnsonSolid[11]]:={{1,2,4},{4,2,5},{4,5,7},{7,5,9},{7,9,12},{12,9,13},{12,13,16},{16,13,17},{16,17,19},{19,17,20},{1,4,3},{4,7,6},{7,12,10},{12,16,14},{16,19,18},{13,9,8,11,15}}
NetFaces[JohnsonSolid[12]]:={{3,2,5},{2,4,5},{2,1,4},{5,7,8},{5,4,7},{7,4,6}}
NetFaces[JohnsonSolid[13]]:={{5,4,7},{4,6,7},{4,3,6},{7,9,10},{7,6,9},{9,6,8},{4,5,2},{10,9,12},{3,4,1},{9,8,11}}
NetFaces[JohnsonSolid[14]]:={{3,7,4,1},{7,3,5},{1,4,2},{7,11,8,4},{11,7,9},{4,8,6},{11,14,12,8},{14,11,13},{8,12,10}}
NetFaces[JohnsonSolid[15]]:={{3,7,4,1},{7,3,5},{1,4,2},{7,11,8,4},{11,7,9},{4,8,6},{11,15,12,8},{15,11,13},{8,12,10},{15,18,16,12},{18,15,17},{12,16,14}}
NetFaces[JohnsonSolid[16]]:={{3,7,4,1},{7,3,5},{1,4,2},{7,11,8,4},{11,7,9},{4,8,6},{11,15,12,8},{15,11,13},{8,12,10},{15,19,16,12},{19,15,17},{12,16,14},{19,22,20,16},{22,19,21},{16,20,18}}
NetFaces[JohnsonSolid[17]]:={{1,2,5},{5,2,6},{5,6,9},{9,6,10},{9,10,13},{13,10,14},{13,14,17},{17,14,18},{1,5,3},{5,9,7},{9,13,11},{13,17,15},{6,2,4},{10,6,8},{14,10,12},{18,14,16}}
NetFaces[JohnsonSolid[18]]:={{16,10,7,9,15,20},{2,1,3,4},{4,3,10,11},{11,10,16,17},{17,16,23,24},{24,23,25,26},{26,25,27,28},{12,18,13},{18,12,11,17},{18,17,21},{13,18,22,19},{13,19,14},{12,13,8,6},{12,6,5}}
NetFaces[JohnsonSolid[19]]:={{20,14,11,10,13,19,26,27},{2,1,3,4},{4,3,5,6},{6,5,14,15},{15,14,20,21},{21,20,29,30},{30,29,33,34},{34,33,35,36},{36,35,37,38},{16,22,23,17},{22,16,15,21},{22,21,28},{23,22,31,32},{23,32,25},{17,23,24,18},{17,18,9},{16,17,8,7},{16,7,12}}
NetFaces[JohnsonSolid[20]]:={{26,20,15,9,14,19,25,31,39,32},{2,1,3,4},{4,3,5,6},{6,5,10,11},{11,10,20,21},{21,20,26,27},{27,26,36,37},{37,36,41,42},{42,41,43,44},{44,43,45,46},{46,45,47,48},{22,28,30,24,16},{28,22,21,27},{28,27,33},{30,28,35,40},{30,40,38},{24,30,34,29},{24,29,23},{16,24,18,13},{16,13,7},{22,16,8,12},{22,12,17}}
NetFaces[JohnsonSolid[21]]:={{31,24,17,10,16,23,30,38,44,39},{27,36,35,26,21},{27,20,25,32,34},{27,34,36},{34,32,41},{36,37,48,51,45},{36,45,35},{45,51,52},{35,46,47,40,33},{35,33,26},{33,40,29},{26,28,22,15,18},{26,18,21},{18,15,8},{21,14,7,9,19},{21,19,27},{19,9,13},{2,1,3,4},{4,3,5,6},{6,5,11,12},{12,11,24,25},{25,24,31,32},{32,31,42,43},{43,42,49,50},{50,49,53,54},{54,53,55,56},{56,55,57,58}}
NetFaces[JohnsonSolid[22]]:={{17,10,6,9,16,20},{1,4,7},{7,4,10},{7,10,13},{13,10,17},{13,17,21},{21,17,22},{8,14,11},{14,8,7,13},{14,13,18},{11,14,19,15},{11,15,12},{8,11,5,3},{8,3,2},{21,22,23},{22,23,24},{23,24,25},{24,25,26},{25,26,27},{26,27,28}}
NetFaces[JohnsonSolid[23]]:={{17,10,4,3,9,16,22,23},{1,2,5},{5,2,10},{5,10,12},{12,10,17},{12,17,18},{18,17,24},{18,24,27},{27,24,30},{13,19,20,14},{19,13,12,18},{19,18,26},{20,19,28,29},{20,29,25},{14,20,21,15},{14,15,8},{13,14,7,6},{13,6,11},{27,30,31},{30,31,32},{31,32,33},{32,33,34},{33,34,35},{34,35,36},{35,36,37},{36,37,38}}
NetFaces[JohnsonSolid[24]]:={{26,19,11,8,10,18,25,33,36,34},{1,2,5},{5,2,9},{5,9,15},{15,9,19},{15,19,21},{21,19,26},{21,26,30},{30,26,35},{30,35,37},{37,35,38},{16,22,24,20,12},{22,16,15,21},{22,21,27},{24,22,29,32},{24,32,31},{20,24,28,23},{20,23,17},{12,20,14,7},{12,7,3},{16,12,4,6},{16,6,13},{37,38,39},{38,39,40},{39,40,41},{40,41,42},{41,42,43},{42,43,44},{43,44,45},{44,45,46},{45,46,47},{46,47,48}}
NetFaces[JohnsonSolid[25]]:={{32,22,15,11,14,21,31,38,42,39},{1,4,6},{6,4,12},{6,12,19},{19,12,22},{19,22,26},{26,22,32},{26,32,36},{36,32,41},{36,41,45},{45,41,48},{23,30,29,20,17},{23,16,19,26,28},{23,28,30},{28,26,35},{30,33,44,46,37},{30,37,29},{37,46,47},{29,40,43,34,27},{29,27,20},{27,34,25},{20,24,18,9,10},{20,10,17},{10,9,3},{17,8,2,5,13},{17,13,23},{13,5,7},{45,48,49},{48,49,50},{49,50,51},{50,51,52},{51,52,53},{52,53,54},{53,54,55},{54,55,56},{55,56,57},{56,57,58}}
NetFaces[JohnsonSolid[26]]:={{2,1,3,4},{6,4,8},{4,3,7,8},{3,5,7},{10,13,14},{10,8,11,13},{8,7,11},{11,7,9,12}}
NetFaces[JohnsonSolid[27]]:={{7,8,4},{8,7,11,12},{8,12,10},{4,8,6,3},{4,3,1},{7,4,2,5},{7,5,9},{16,15,19},{15,16,12,11},{15,11,13},{19,15,17,20},{19,20,22},{16,19,21,18},{16,18,14}}
NetFaces[JohnsonSolid[28]]:={{10,11,7,6},{11,10,15,16},{11,16,13},{7,11,12,8},{7,8,3},{6,7,2,1},{6,1,4},{10,6,5,9},{10,9,14},{21,20,24,25},{20,21,16,15},{20,15,18},{24,20,19,23},{24,23,28},{25,24,29,30},{25,30,27},{21,25,26,22},{21,22,17}}
NetFaces[JohnsonSolid[29]]:={{10,11,7,6},{11,10,15,16},{11,16,13},{7,11,12,8},{7,8,3},{6,7,2,1},{6,1,4},{10,6,5,9},{10,9,14},{22,19,23,26},{19,22,18,16},{19,16,15},{23,19,17,21},{23,21,25},{26,23,27,29},{26,29,30},{22,26,28,24},{22,24,20}}
NetFaces[JohnsonSolid[30]]:={{13,14,9,7,8},{14,13,19,20},{14,20,17},{9,14,16,12},{9,12,6},{7,9,5,3},{7,3,1},{8,7,2,4},{8,4,10},{13,8,11,15},{13,15,18},{26,25,30,32,31},{25,26,20,19},{25,19,22},{30,25,23,27},{30,27,33},{32,30,34,36},{32,36,38},{31,32,37,35},{31,35,29},{26,31,28,24},{26,24,21}}
NetFaces[JohnsonSolid[31]]:={{13,14,9,7,8},{14,13,19,20},{14,20,17},{9,14,16,12},{9,12,6},{7,9,5,3},{7,3,1},{8,7,2,4},{8,4,10},{13,8,11,15},{13,15,18},{26,25,27,32,31},{25,26,22,20},{25,20,19},{27,25,21,24},{27,24,29},{32,27,30,35},{32,35,37},{31,32,38,36},{31,36,34},{26,31,33,28},{26,28,23}}
NetFaces[JohnsonSolid[32]]:={{9,7,8,13,14},{7,9,5,3},{7,3,1},{8,7,2,4},{8,4,10},{13,8,11,15},{13,15,20},{14,13,21,22},{14,22,18},{9,14,16,12},{9,12,6},{28,34,39,36,30},{28,23,18,22,27},{28,27,34},{27,22,26},{34,29,33,38,40},{34,40,39},{40,38,45},{39,41,46,48,42},{39,42,36},{42,48,47},{36,43,44,37,35},{36,35,30},{35,37,32},{30,31,25,19,24},{30,24,28},{24,19,17}}
NetFaces[JohnsonSolid[33]]:={{9,7,8,13,14},{7,9,5,3},{7,3,1},{8,7,2,4},{8,4,10},{13,8,11,15},{13,15,18},{14,13,19,20},{14,20,17},{9,14,16,12},{9,12,6},{29,31,38,39,32},{29,24,20,19,23},{29,23,31},{23,19,21},{31,26,25,30,34},{31,34,38},{34,30,36},{38,37,42,46,43},{38,43,39},{43,46,48},{39,44,47,45,40},{39,40,32},{40,45,41},{32,35,33,28,27},{32,27,29},{27,28,22}}
NetFaces[JohnsonSolid[34]]:={{20,18,11,10,17},{20,26,30,29,24},{20,24,18},{24,29,27},{18,22,21,16,14},{18,14,11},{14,16,8},{11,9,4,2,6},{11,6,10},{6,2,1},{10,5,3,7,13},{10,13,17},{13,7,12},{17,15,19,25,23},{17,23,20},{23,25,28},{39,41,48,49,42},{39,33,29,30,35},{39,35,41},{35,30,32},{41,37,38,43,45},{41,45,48},{45,43,51},{48,50,55,57,53},{48,53,49},{53,57,58},{49,54,56,52,46},{49,46,42},{46,52,47},{42,44,40,34,36},{42,36,39},{36,34,31}}
NetFaces[JohnsonSolid[35]]:={{14,22,17},{22,14,13,21},{22,21,27},{17,22,28,24},{17,24,18},{14,17,10,8},{14,8,7},{19,11,16},{11,19,20,12},{11,12,6},{16,11,5,9},{16,9,15},{19,16,23,25},{19,25,26},{2,1,3,4},{4,3,12,13},{13,12,20,21},{21,20,29,30},{30,29,31,32},{32,31,33,34}}
NetFaces[JohnsonSolid[36]]:={{10,19,13},{19,10,9,18},{19,18,23},{13,19,24,20},{13,20,14},{10,13,7,6},{10,6,5},{25,16,22},{16,25,26,17},{16,17,12},{22,16,11,15},{22,15,21},{25,22,28,29},{25,29,30},{2,1,3,4},{4,3,8,9},{9,8,17,18},{18,17,26,27},{27,26,31,32},{32,31,33,34}}
NetFaces[JohnsonSolid[37]]:={{13,18,19,14},{18,13,12,17},{18,17,22},{19,18,25,26},{19,26,21},{14,19,20,15},{14,15,9},{13,14,8,7},{13,7,10},{2,1,3,4},{4,3,5,6},{6,5,11,12},{12,11,16,17},{17,16,23,24},{24,23,27,28},{28,27,29,30},{30,29,31,32},{33,37,39,35},{37,41,39},{39,43,45,42},{45,46,42},{35,39,42,38},{42,44,16,38},{11,16,38},{40,35,36},{35,38,34,40}}
NetFaces[JohnsonSolid[38]]:={{26,34,39,29,19},{34,26,25,33},{34,33,40},{39,34,44,49},{39,49,47},{29,39,42,36},{29,36,27},{19,29,22,16},{19,16,7},{26,19,9,14},{26,14,20},{31,23,18,28,38},{23,31,32,24},{23,24,17},{18,23,13,8},{18,8,10},{28,18,15,21},{28,21,30},{38,28,35,41},{38,41,50},{31,38,48,43},{31,43,37},{2,1,3,4},{4,3,5,6},{6,5,11,12},{12,11,24,25},{25,24,32,33},{33,32,45,46},{46,45,51,52},{52,51,53,54},{54,53,55,56},{56,55,57,58}}
NetFaces[JohnsonSolid[39]]:={{20,31,33,25,13},{31,20,19,30},{31,30,35},{33,31,38,45},{33,45,42},{25,33,36,32},{25,32,22},{13,25,16,12},{13,12,7},{20,13,8,11},{20,11,15},{39,28,26,34,46},{28,39,40,29},{28,29,24},{26,28,21,14},{26,14,17},{34,26,23,27},{34,27,37},{46,34,43,47},{46,47,52},{39,46,51,48},{39,48,44},{2,1,3,4},{4,3,5,6},{6,5,9,10},{10,9,18,19},{19,18,29,30},{30,29,40,41},{41,40,49,50},{50,49,53,54},{54,53,55,56},{56,55,57,58}}
NetFaces[JohnsonSolid[40]]:={{24,36,39,29,18},{36,24,23,35},{36,35,43},{39,36,45,51},{39,51,48},{29,39,44,37},{29,37,26},{18,29,21,14},{18,14,7},{24,18,8,13},{24,13,20},{41,30,31,42,50},{41,52,46,34,32},{41,32,30},{32,34,25},{30,28,15,10,19},{30,19,31},{19,10,9},{31,17,16,27,33},{31,33,42},{33,27,38},{42,40,49,55,54},{42,54,50},{54,55,61},{50,56,62,60,53},{50,53,41},{53,60,57},{2,1,3,4},{4,3,5,6},{6,5,11,12},{12,11,22,23},{23,22,34,35},{35,34,46,47},{47,46,58,59},{59,58,63,64},{64,63,65,66},{66,65,67,68}}
NetFaces[JohnsonSolid[41]]:={{31,40,45,36,25},{40,31,30,39},{40,39,48},{45,40,50,57},{45,57,56},{36,45,49,41},{36,41,32},{25,36,27,19},{25,19,12},{31,25,13,18},{31,18,26},{35,22,23,37,43},{35,44,38,29,24},{35,24,22},{24,29,17},{22,21,9,6,14},{22,14,23},{14,6,5},{23,11,10,20,28},{23,28,37},{28,20,33},{37,34,42,51,47},{37,47,43},{47,51,59},{43,52,60,58,46},{43,46,35},{46,58,53},{2,1,3,4},{4,3,7,8},{8,7,15,16},{16,15,29,30},{30,29,38,39},{39,38,54,55},{55,54,61,62},{62,61,63,64},{64,63,65,66},{66,65,67,68}}
NetFaces[JohnsonSolid[42]]:={{39,51,50,36,30},{39,29,34,44,49},{39,49,51},{49,44,59},{51,54,67,71,62},{51,62,50},{62,71,72},{50,63,65,55,45},{50,45,36},{45,55,42},{36,40,31,21,24},{36,24,30},{24,21,11},{30,20,9,13,25},{30,25,39},{25,13,19},{38,26,27,41,47},{38,48,43,33,28},{38,28,26},{28,33,18},{26,23,10,6,15},{26,15,27},{15,6,5},{27,14,12,22,32},{27,32,41},{32,22,35},{41,37,46,56,53},{41,53,47},{53,56,66},{47,57,68,64,52},{47,52,38},{52,64,58},{2,1,3,4},{4,3,7,8},{8,7,16,17},{17,16,33,34},{34,33,43,44},{44,43,60,61},{61,60,69,70},{70,69,73,74},{74,73,75,76},{76,75,77,78}}
NetFaces[JohnsonSolid[43]]:={{32,46,44,31,24},{32,22,27,40,43},{32,43,46},{43,40,51},{46,49,62,68,56},{46,56,44},{56,68,69},{44,58,60,50,41},{44,41,31},{41,50,37},{31,34,25,16,18},{31,18,24},{18,16,8},{24,15,7,9,20},{24,20,32},{20,9,14},{47,33,35,48,55},{47,57,52,39,36},{47,36,33},{36,39,28},{33,30,17,11,23},{33,23,35},{23,11,10},{35,21,19,29,38},{35,38,48},{38,29,42},{48,45,54,63,61},{48,61,55},{61,63,71},{55,64,72,70,59},{55,59,47},{59,70,65},{2,1,3,4},{4,3,5,6},{6,5,12,13},{13,12,26,27},{27,26,39,40},{40,39,52,53},{53,52,66,67},{67,66,73,74},{74,73,75,76},{76,75,77,78}}
NetFaces[JohnsonSolid[44]]:={{13,22,17},{22,13,12,21},{22,21,27},{17,22,28,23},{17,23,18},{13,17,11,7},{13,7,6},{24,15,20},{15,24,25,16},{15,16,10},{20,15,9,14},{20,14,19},{24,20,26,30},{24,30,31},{1,2,3},{3,2,4},{3,4,5},{5,4,8},{5,8,12},{12,8,16},{12,16,21},{21,16,25},{21,25,29},{29,25,32},{29,32,33},{33,32,34}}
NetFaces[JohnsonSolid[45]]:={{18,26,27,19},{26,18,17,25},{26,25,34},{27,26,37,38},{27,38,33},{19,27,28,20},{19,20,10},{18,19,9,8},{18,8,14},{31,23,22,30},{23,31,32,24},{23,24,15},{22,23,12,11},{22,11,16},{30,22,21,29},{30,29,39},{31,30,40,41},{31,41,35},{1,2,3},{3,2,4},{3,4,5},{5,4,6},{5,6,7},{7,6,13},{7,13,17},{17,13,24},{17,24,25},{25,24,32},{25,32,36},{36,32,42},{36,42,43},{43,42,44},{43,44,45},{45,44,46}}
NetFaces[JohnsonSolid[46]]:={{24,33,35,30,20},{33,24,23,32},{33,32,42},{35,33,44,47},{35,47,46},{30,35,43,34},{30,34,25},{20,30,22,13},{20,13,9},{24,20,10,12},{24,12,21},{37,28,26,31,41},{28,37,38,29},{28,29,19},{26,28,17,14},{26,14,15},{31,26,18,27},{31,27,36},{41,31,39,48},{41,48,52},{37,41,51,49},{37,49,40},{1,2,3},{3,2,4},{3,4,5},{5,4,6},{5,6,7},{7,6,8},{7,8,11},{11,8,16},{11,16,23},{23,16,29},{23,29,32},{32,29,38},{32,38,45},{45,38,50},{45,50,53},{53,50,54},{53,54,55},{55,54,56},{55,56,57},{57,56,58}}
NetFaces[JohnsonSolid[47]]:={{28,40,43,35,23},{40,28,27,39},{40,39,48},{43,40,50,55},{43,55,53},{35,43,49,41},{35,41,30},{23,35,26,17},{23,17,11},{28,23,12,16},{28,16,25},{38,29,31,42,46},{38,47,44,34,32},{38,32,29},{32,34,21},{29,24,13,9,19},{29,19,31},{19,9,8},{31,18,14,22,33},{31,33,42},{33,22,36},{42,37,45,56,54},{42,54,46},{54,56,62},{46,57,63,60,51},{46,51,38},{51,60,58},{1,2,3},{3,2,4},{3,4,5},{5,4,6},{5,6,7},{7,6,10},{7,10,15},{15,10,20},{15,20,27},{27,20,34},{27,34,39},{39,34,44},{39,44,52},{52,44,59},{52,59,61},{61,59,64},{61,64,65},{65,64,66},{65,66,67},{67,66,68}}
NetFaces[JohnsonSolid[48]]:={{38,49,48,35,29},{38,27,31,44,47},{38,47,49},{47,44,57},{49,53,67,70,60},{49,60,48},{60,70,71},{48,61,66,55,45},{48,45,35},{45,55,42},{35,40,30,19,22},{35,22,29},{22,19,9},{29,18,8,13,25},{29,25,38},{25,13,17},{43,32,33,46,52},{43,54,50,37,34},{43,34,32},{34,37,24},{32,28,14,11,21},{32,21,33},{21,11,10},{33,20,15,26,36},{33,36,46},{36,26,39},{46,41,51,62,59},{46,59,52},{59,62,72},{52,63,73,68,56},{52,56,43},{56,68,64},{1,2,3},{3,2,4},{3,4,5},{5,4,6},{5,6,7},{7,6,12},{7,12,16},{16,12,23},{16,23,31},{31,23,37},{31,37,44},{44,37,50},{44,50,58},{58,50,65},{58,65,69},{69,65,74},{69,74,75},{75,74,76},{75,76,77},{77,76,78}}
NetFaces[JohnsonSolid[49]]:={{5,4,8,9},{9,8,10,11},{3,2,5},{2,4,5},{4,2,1},{5,9,7},{8,4,6},{11,10,12}}
NetFaces[JohnsonSolid[50]]:={{7,10,8,5},{4,7,3},{3,7,5},{3,5,2},{10,7,9},{5,8,6},{10,13,12},{10,12,8},{8,12,11},{12,13,14},{3,2,1}}
NetFaces[JohnsonSolid[51]]:={{13,15,16},{12,9,14},{9,11,14},{9,7,11},{5,7,9},{1,2,3},{3,2,4},{3,4,7},{7,4,8},{4,6,8},{7,8,11},{11,8,13},{8,10,13},{13,10,15}}
NetFaces[JohnsonSolid[52]]:={{5,6,11,12,7},{9,14,19,15,10},{6,5,1,3},{11,6,8,13},{7,12,14,9},{5,7,4,2},{11,17,12},{11,16,17},{12,17,18},{18,17,20}}
NetFaces[JohnsonSolid[53]]:={{6,8,13,14,9},{11,16,21,17,12},{8,6,3,5},{13,8,10,15},{9,14,16,11},{13,19,14},{13,18,19},{14,19,20},{20,19,22},{9,4,6},{9,7,4},{6,4,2},{2,4,1}}
NetFaces[JohnsonSolid[54]]:={{6,5,10,14,15,11},{20,19,21,23,24,22},{5,6,2,1},{10,5,3,7},{15,14,19,20},{11,15,16,12},{6,11,8,4},{10,13,14},{10,9,13},{14,13,18},{18,13,17}}
NetFaces[JohnsonSolid[55]]:={{7,6,11,16,17,12},{22,21,23,25,26,24},{6,7,2,1},{11,6,5,9},{17,16,21,22},{12,17,18,14},{11,15,16},{11,10,15},{16,15,20},{20,15,19},{12,8,7},{12,13,8},{7,8,3},{3,8,4}}
NetFaces[JohnsonSolid[56]]:={{6,5,10,14,15,11},{22,21,23,25,26,24},{5,6,2,1},{10,5,3,7},{15,14,21,22},{6,11,8,4},{10,13,14},{10,9,13},{14,13,19},{19,13,18},{15,16,11},{15,20,16},{11,16,12},{12,16,17}}
NetFaces[JohnsonSolid[57]]:={{8,7,12,16,17,13},{24,23,25,27,28,26},{12,7,5,9},{17,16,23,24},{8,13,10,6},{12,15,16},{12,11,15},{16,15,21},{21,15,20},{17,18,13},{17,22,18},{13,18,14},{14,18,19},{8,3,7},{8,4,3},{7,3,2},{2,3,1}}
NetFaces[JohnsonSolid[58]]:={{10,14,13,8,6},{11,7,8,13,15},{1,2,6,8,5},{4,9,10,6,3},{16,18,14,10,12},{20,17,13,14,19},{19,22,26,25,20},{28,32,31,26,24},{35,30,29,33,37},{23,21,25,29,27},{29,25,26,31,33},{36,33,31},{36,31,34},{36,38,33},{36,40,38},{40,36,39}}
NetFaces[JohnsonSolid[59]]:={{11,16,14,8,7},{1,2,7,8,5},{4,10,11,7,3},{18,20,16,11,13},{22,19,14,16,21},{21,24,29,27,22},{42,41,36,35,38},{39,33,32,36,40},{25,23,27,32,30},{32,27,29,35,36},{31,35,29},{31,29,26},{31,37,35},{31,34,37},{34,31,28},{12,8,14},{12,14,17},{12,6,8},{12,9,6},{9,12,15}}
NetFaces[JohnsonSolid[60]]:={{12,16,15,10,8},{13,9,10,15,17},{6,11,12,8,4},{18,20,16,12,14},{22,19,15,16,21},{21,24,29,27,22},{42,41,36,35,38},{39,33,32,36,40},{25,23,27,32,30},{32,27,29,35,36},{31,35,29},{31,29,26},{31,37,35},{31,34,37},{34,31,28},{5,8,10},{5,10,7},{5,3,8},{5,1,3},{1,5,2}}
NetFaces[JohnsonSolid[61]]:={{12,16,15,10,8},{13,9,10,15,17},{6,11,12,8,4},{18,20,16,12,14},{23,19,15,16,21},{21,25,31,29,23},{44,43,38,37,40},{41,35,34,38,42},{34,29,31,37,38},{33,37,31},{33,31,27},{33,39,37},{33,36,39},{36,33,30},{5,8,10},{5,10,7},{5,3,8},{5,1,3},{1,5,2},{28,29,34},{28,34,32},{28,24,29},{28,22,24},{22,28,26}}
NetFaces[JohnsonSolid[62]]:={{4,6,11,12,7},{8,7,12,15,13},{5,6,1},{1,6,4},{4,7,2},{3,8,9},{8,13,9},{13,15,18},{13,18,14},{12,11,17},{11,16,17},{11,10,16}}
NetFaces[JohnsonSolid[63]]:={{4,6,11,12,7},{8,7,12,15,13},{1,2,4},{2,6,4},{4,7,3},{8,13,9},{11,6,5,10,14},{12,11,16}}
NetFaces[JohnsonSolid[64]]:={{4,6,11,12,7},{8,7,12,15,13},{1,2,4},{2,6,4},{4,7,3},{8,13,9},{11,6,5,10,14},{12,11,16},{11,14,17},{15,12,18}}
NetFaces[JohnsonSolid[65]]:={{3,1,2,4},{3,4,6},{6,4,5,9,12,10},{10,13,7},{13,15,11,8},{15,13,17},{13,10,12,16,19,17},{17,19,20},{18,16,14},{19,16,18,21,24,22},{24,21,25},{27,23,22,26},{22,24,26},{27,26,28}}
NetFaces[JohnsonSolid[66]]:={{23,28,22,17},{28,23,29,35},{28,35,36},{22,28,30,24},{22,24,18},{17,22,12,8},{17,8,7},{23,17,11,21},{23,21,27},{4,2,1,3,5,15,16,6},{21,16,15,20,26,33,34,27},{40,34,33,39,41,43,44,42},{48,44,43,47,49,51,52,50},{20,14,13,19,25,31,32,26},{50,52,54},{49,53,51},{42,44,46},{41,45,43},{27,34,38},{26,37,33},{10,16,21},{9,20,15}}
NetFaces[JohnsonSolid[67]]:={{27,32,24,19},{32,27,35,41},{32,41,42},{24,32,36,28},{24,28,20},{19,24,14,8},{19,8,7},{27,19,13,23},{27,23,31},{26,21,29,34},{21,26,18,12},{21,12,11},{29,21,17,25},{29,25,33},{34,29,39,45},{34,45,46},{26,34,40,30},{26,30,22},{4,2,1,3,5,15,16,6},{23,16,15,22,30,37,38,31},{48,38,37,47,49,51,52,50},{56,52,51,55,57,59,60,58},{58,60,62},{57,61,59},{50,52,54},{49,53,51},{31,38,44},{30,43,37},{10,16,23},{9,22,15}}
NetFaces[JohnsonSolid[68]]:={{27,22,16,21,26,34,38,43,39,35},{42,35,39},{28,24,18,23,27,35,40,45,41,36},{47,41,45},{19,22,27},{1,2,6,11,16,22,17,12,7,3},{10,7,12},{15,21,16},{13,20,25,29,26,21,14,9,5,8},{4,5,9},{30,34,26},{52,54,53,49,38,34,31,33,37,48},{32,37,33},{46,43,38},{58,56,51,44,39,43,50,55,57,62},{59,57,55},{81,76,80,84,94,101,108,102,95,85},{89,85,95},{67,65,66,70,81,85,88,86,82,71},{87,82,86},{73,76,81},{61,63,68,75,80,76,69,64,62,57},{60,62,64},{77,84,80},{93,99,106,100,94,84,79,74,78,83},{72,78,74},{103,101,94},{128,126,123,114,108,101,107,113,122,125},{117,122,113},{111,102,108},{127,123,126},{104,112,116,109,96},{112,104,105,115},{112,115,120},{116,112,121,124},{116,124,119},{109,116,118,110},{109,110,98},{96,109,97,92},{90,96,91},{104,96,90,95},{104,95,102}}
NetFaces[JohnsonSolid[69]]:={{27,22,16,21,26,34,40,47,41,35},{45,35,41},{28,24,18,23,27,35,42,49,43,36},{52,43,49},{19,22,27},{1,2,6,11,16,22,17,12,7,3},{10,7,12},{15,21,16},{13,20,25,30,26,21,14,9,5,8},{4,5,9},{31,34,26},{103,109,116,110,104,94,89,84,88,93},{29,25,20},{51,47,40},{68,63,56,48,41,47,55,62,67,72},{69,67,62},{91,86,90,94,104,111,118,112,105,95},{99,95,105},{77,75,76,81,91,95,98,96,92,82},{97,92,96},{83,86,91},{71,73,79,85,90,86,80,74,72,67},{70,72,74},{87,94,90},{78,79,73},{113,111,104},{138,136,133,124,118,111,117,123,132,135},{127,132,123},{121,112,118},{137,133,136},{114,122,126,119,106},{122,114,115,125},{122,125,130},{126,122,131,134},{126,134,129},{119,126,128,120},{119,120,108},{106,119,107,102},{100,106,101},{114,106,100,105},{114,105,112},{44,39,50,59,54},{39,44,33,32},{39,32,37},{50,39,38,46},{50,46,57},{59,50,58,64},{59,64,66},{54,59,65,61},{53,54,60},{44,54,53,40},{44,40,34}}
NetFaces[JohnsonSolid[70]]:={{27,22,16,21,26,34,38,43,39,35},{42,35,39},{28,24,18,23,27,35,40,45,41,36},{47,41,45},{19,22,27},{1,2,6,11,16,22,17,12,7,3},{10,7,12},{15,21,16},{13,20,25,29,26,21,14,9,5,8},{4,5,9},{30,34,26},{52,54,53,49,38,34,31,33,37,48},{32,37,33},{46,43,38},{58,56,51,44,39,43,50,55,57,62},{59,57,55},{81,77,80,86,99,107,115,108,100,87},{95,87,100},{67,65,66,71,81,87,93,88,82,72},{90,82,88},{73,77,81},{61,63,69,76,80,77,70,64,62,57},{60,62,64},{78,86,80},{68,69,63},{110,107,99},{138,136,133,124,115,107,114,123,132,135},{127,132,123},{121,108,115},{137,133,136},{111,122,126,117,104},{122,111,112,125},{122,125,130},{126,122,131,134},{126,134,129},{117,126,128,120},{117,120,106},{104,117,105,98},{96,104,97},{111,104,96,100},{111,100,108},{94,84,91,101,103},{84,94,85,79},{84,79,74},{91,84,75,83},{91,83,89},{101,91,92,102},{101,102,113},{103,101,116,118},{109,103,119},{94,103,109,99},{94,99,86}}
NetFaces[JohnsonSolid[71]]:={{27,22,16,21,26,34,38,44,39,35},{42,35,39},{28,24,18,23,27,35,40,45,41,36},{50,41,45},{19,22,27},{1,2,6,11,16,22,17,12,7,3},{10,7,12},{15,21,16},{13,20,25,29,26,21,14,9,5,8},{4,5,9},{30,34,26},{56,60,57,52,38,34,31,33,37,51},{32,37,33},{48,44,38},{66,57,60},{130,99,107,118,124,93,98,106,117,123},{127,123,117},{148,147,143,138,130,123,129,137,142,146},{139,142,137},{134,99,130},{136,125,119,115,118,124,135,140,144,141},{143,145,147},{114,107,118},{116,119,125},{88,93,124},{71,76,84,92,98,93,85,77,72,68},{69,72,77},{95,106,98},{80,84,76},{113,102,110,120,122},{102,113,104,97},{102,97,90},{110,102,91,100},{110,100,108},{120,110,111,121},{120,121,128},{122,120,131,132},{126,122,133},{113,122,126,117},{113,117,106},{96,101,89,81,86},{101,96,109,112},{101,112,105},{89,101,103,94},{89,94,83},{81,89,82,75},{81,75,73},{86,81,74,78},{87,86,79},{96,86,87,99},{96,99,107},{49,58,65,61,54},{58,49,46,55},{58,55,63},{65,58,64,68},{65,68,72},{61,65,70,67},{61,67,62},{54,61,59,53},{43,54,47},{49,54,43,39},{49,39,44}}
NetFaces[JohnsonSolid[72]]:={{10,8,16,18},{18,16,25},{16,15,24,25},{5,1,4,15,16},{7,14,15},{15,14,23,24},{3,9,14,7},{3,2,9},{2,6,13,9},{9,13,19,26,17},{6,12,13},{13,12,21,22},{35,32,38,40},{40,38,45},{38,37,44,45},{30,27,29,37,38},{37,39,44},{44,39,50,54},{31,34,39,37},{24,23,31},{23,28,34,31},{34,33,41,46,42},{28,33,34},{21,12,11,20},{58,55,61,64},{64,61,68},{61,60,67,68},{52,47,51,60,61},{60,63,67},{67,63,72,76},{54,57,63,60},{54,50,57},{50,49,56,57},{57,56,65,69,66},{49,53,56},{56,53,59,62},{80,77,83,86},{86,83,90},{83,82,89,90},{74,70,73,82,83},{82,85,89},{89,85,95,100},{76,79,85,82},{76,72,79},{72,71,78,79},{79,78,87,92,88},{71,75,78},{78,75,81,84},{104,101,108,111},{111,108,115},{108,107,114,115},{98,93,97,107,108},{107,110,114},{114,110,117,118},{100,103,110,107},{100,95,103},{95,94,102,103},{103,102,112,116,113},{94,99,102},{102,99,106,109},{36,35,40,48,43},{99,91,96,105,106}}
NetFaces[JohnsonSolid[73]]:={{10,8,16,18},{18,16,25},{16,15,24,25},{5,1,4,15,16},{7,14,15},{15,14,23,24},{3,9,14,7},{3,2,9},{2,6,13,9},{9,13,19,26,17},{6,12,13},{13,12,21,22},{35,32,38,40},{40,38,45},{38,37,44,45},{30,27,29,37,38},{37,39,44},{44,39,50,54},{31,34,39,37},{24,23,31},{23,28,34,31},{34,33,41,46,42},{28,33,34},{21,12,11,20},{81,72,71,80},{59,58,64},{58,61,69,64},{52,47,51,58,59},{61,68,69},{69,68,77,78},{54,56,61,58},{54,50,56},{50,49,55,56},{56,55,62,65,63},{49,53,55},{55,53,57,60},{71,70,79,80},{80,79,86},{79,83,90,86},{73,66,75,83,79},{83,89,90},{89,85,95,100},{78,85,89,83},{78,77,85},{68,67,76,77},{77,76,87,92,88},{67,74,76},{76,74,82,84},{104,101,108,111},{111,108,115},{108,107,114,115},{98,93,97,107,108},{107,110,114},{114,110,117,118},{100,103,110,107},{100,95,103},{95,94,102,103},{103,102,112,116,113},{94,99,102},{102,99,106,109},{36,35,40,48,43},{99,91,96,105,106}}
NetFaces[JohnsonSolid[74]]:={{10,8,16,18},{18,16,25},{16,15,24,25},{5,1,4,15,16},{7,14,15},{15,14,23,24},{3,9,14,7},{3,2,9},{2,6,13,9},{9,13,19,26,17},{6,12,13},{13,12,21,22},{56,47,46,55},{35,34,39},{34,36,44,39},{30,27,29,34,35},{36,43,44},{44,43,52,53},{31,33,36,34},{24,23,31},{23,28,33,31},{33,32,37,40,38},{28,32,33},{21,12,11,20},{46,45,54,55},{55,54,61},{54,58,67,61},{48,41,50,58,54},{58,66,67},{66,60,72,76},{53,60,66,58},{53,52,60},{43,42,51,52},{52,51,62,68,63},{42,49,51},{51,49,57,59},{80,77,83,86},{86,83,90},{83,82,89,90},{74,70,73,82,83},{82,85,89},{89,85,95,100},{76,79,85,82},{76,72,79},{72,71,78,79},{79,78,87,92,88},{71,75,78},{78,75,81,84},{104,101,108,111},{111,108,115},{108,107,114,115},{98,93,97,107,108},{107,110,114},{114,110,117,118},{100,103,110,107},{100,95,103},{95,94,102,103},{103,102,112,116,113},{94,99,102},{102,99,106,109},{65,56,55,64,69},{99,91,96,105,106}}
NetFaces[JohnsonSolid[75]]:={{10,8,16,18},{18,16,25},{16,15,24,25},{5,1,4,15,16},{7,14,15},{15,14,23,24},{3,9,14,7},{3,2,9},{2,6,13,9},{9,13,19,26,17},{6,12,13},{13,12,21,22},{56,47,46,55},{35,34,39},{34,36,44,39},{30,27,29,34,35},{36,43,44},{44,43,52,53},{31,33,36,34},{24,23,31},{23,28,33,31},{33,32,37,40,38},{28,32,33},{21,12,11,20},{46,45,54,55},{55,54,61},{54,58,67,61},{48,41,50,58,54},{58,66,67},{66,60,73,79},{53,60,66,58},{53,52,60},{43,42,51,52},{52,51,62,68,63},{42,49,51},{51,49,57,59},{82,80,88,90},{90,88,97},{88,87,96,97},{77,71,76,87,88},{79,86,87},{87,86,95,96},{73,81,86,79},{73,72,81},{72,78,85,81},{81,85,91,98,89},{78,84,85},{85,84,93,94},{107,104,109,111},{111,109,115},{109,108,114,115},{102,99,101,108,109},{108,110,114},{114,110,117,118},{103,106,110,108},{96,95,103},{95,100,106,103},{106,105,112,116,113},{100,105,106},{93,84,83,92},{65,56,55,64,69},{84,75,70,74,83}}
NetFaces[JohnsonSolid[76]]:={{11,8,14,17},{17,14,21},{14,13,20,21},{5,1,4,13,14},{13,16,20},{20,16,25,29},{7,10,16,13},{7,3,10},{3,2,9,10},{10,9,18,22,19},{2,6,9},{9,6,12,15},{33,30,37,40},{40,37,45},{37,36,44,45},{27,23,26,36,37},{36,39,44},{44,39,50,54},{29,32,39,36},{29,25,32},{25,24,31,32},{32,31,41,46,42},{24,28,31},{31,28,35,38},{58,55,61,64},{64,61,68},{61,60,67,68},{52,47,51,60,61},{60,63,67},{67,63,72,75},{54,57,63,60},{54,50,57},{50,49,56,57},{57,56,65,69,66},{49,53,56},{56,53,59,62},{82,77,83,88},{88,83,81,85,90,96,99,100,98,93},{76,70,73,81,83},{75,79,85,81},{75,72,79},{72,71,78,79},{79,78,86,89,87},{71,74,78},{78,74,80,84},{100,99,107,108},{90,91,97,96},{96,97,105,106,103},{91,95,97},{97,95,102,104},{34,33,40,48,43},{95,92,94,101,102}}
NetFaces[JohnsonSolid[77]]:={{11,8,14,17},{17,14,21},{14,13,20,21},{5,1,4,13,14},{13,16,20},{20,16,25,29},{7,10,16,13},{7,3,10},{3,2,9,10},{10,9,18,22,19},{2,6,9},{9,6,12,15},{32,30,39,41},{41,39,49},{39,38,48,49},{27,23,26,38,39},{29,37,38},{38,37,47,48},{25,31,37,29},{25,24,31},{24,28,36,31},{31,36,43,50,40},{28,35,36},{36,35,45,46},{60,57,62,64},{64,62,68},{62,61,67,68},{55,51,54,61,62},{61,63,67},{67,63,72,75},{56,59,63,61},{48,47,56},{47,53,59,56},{59,58,65,69,66},{53,58,59},{45,35,34,44},{82,77,83,88},{88,83,81,85,90,96,99,100,98,93},{76,70,73,81,83},{75,79,85,81},{75,72,79},{72,71,78,79},{79,78,86,89,87},{71,74,78},{78,74,80,84},{100,99,107,108},{90,91,97,96},{96,97,105,106,103},{91,95,97},{97,95,102,104},{33,32,41,52,42},{95,92,94,101,102}}
NetFaces[JohnsonSolid[78]]:={{11,8,14,17},{17,14,21},{14,13,20,21},{5,1,4,13,14},{13,16,20},{20,16,25,29},{7,10,16,13},{7,3,10},{3,2,9,10},{10,9,18,22,19},{2,6,9},{9,6,12,15},{33,30,37,40},{40,37,45},{37,36,44,45},{27,23,26,36,37},{36,39,44},{44,39,50,54},{29,32,39,36},{29,25,32},{25,24,31,32},{32,31,41,46,42},{24,28,31},{31,28,35,38},{57,55,63,65},{65,63,72},{63,62,71,72},{52,47,51,62,63},{54,61,62},{62,61,70,71},{50,56,61,54},{50,49,56},{49,53,60,56},{56,60,66,73,64},{53,59,60},{60,59,68,69},{83,79,84,88},{88,84,82,85,90,96,99,100,98,93},{78,74,76,82,84},{77,81,85,82},{71,70,77},{70,75,81,77},{81,80,86,89,87},{75,80,81},{68,59,58,67},{100,99,107,108},{90,91,97,96},{96,97,105,106,103},{91,95,97},{97,95,102,104},{34,33,40,48,43},{95,92,94,101,102}}
NetFaces[JohnsonSolid[79]]:={{34,25,24,33},{12,11,17},{11,14,22,17},{5,1,4,11,12},{14,21,22},{22,21,30,31},{7,9,14,11},{7,3,9},{3,2,8,9},{9,8,15,18,16},{2,6,8},{8,6,10,13},{24,23,32,33},{33,32,40},{32,37,45,40},{26,19,28,37,32},{37,44,45},{44,39,50,54},{31,39,44,37},{31,30,39},{21,20,29,30},{30,29,41,46,42},{20,27,29},{29,27,36,38},{57,55,63,65},{65,63,72},{63,62,71,72},{52,47,51,62,63},{54,61,62},{62,61,70,71},{50,56,61,54},{50,49,56},{49,53,60,56},{56,60,66,73,64},{53,59,60},{60,59,68,69},{83,79,84,88},{88,84,82,85,90,96,99,100,98,93},{78,74,76,82,84},{77,81,85,82},{71,70,77},{70,75,81,77},{81,80,86,89,87},{75,80,81},{68,59,58,67},{100,99,107,108},{90,91,97,96},{96,97,105,106,103},{91,95,97},{97,95,102,104},{35,33,40,48,43},{95,92,94,101,102}}
NetFaces[JohnsonSolid[80]]:={{4,3,8,9},{9,8,13},{8,11,17,13},{2,1,5,11,8},{11,16,17},{17,16,28,30},{7,12,16,11},{7,6,12},{6,10,15,12},{12,15,24,29,18},{10,14,15},{15,14,22,23},{26,21,32,34},{34,32,38},{32,33,39,38},{20,19,25,33,32},{48,44,39,33,30,28,31,36,41,46},{49,45,51,55},{55,51,58},{51,50,57,58},{43,40,42,50,51},{50,54,57},{57,54,62,65},{44,48,54,50},{48,46,53,59,56},{46,41,47,52},{72,67,73,78},{78,73,71,75,80,86,89,90,88,83},{66,60,63,71,73},{65,69,75,71},{65,62,69},{62,61,68,69},{69,68,76,79,77},{61,64,68},{68,64,70,74},{90,89,97,98},{80,81,87,86},{86,87,95,96,93},{81,85,87},{87,85,92,94},{27,26,34,37,35},{85,82,84,91,92}}
NetFaces[JohnsonSolid[81]]:={{11,8,14,17},{17,14,21},{14,13,20,21},{5,1,4,13,14},{13,16,20},{20,16,25,29},{7,10,16,13},{7,3,10},{3,2,9,10},{10,9,18,22,19},{2,6,9},{9,6,12,15},{33,30,37,40},{40,37,45},{37,36,44,45},{27,23,26,36,37},{36,39,44},{44,39,48,52},{29,32,39,36},{29,25,32},{25,24,31,32},{32,31,41,46,42},{24,28,31},{31,28,35,38},{57,55,58,61},{61,58,63},{58,56,62,63},{53,49,51,56,58},{69,66,62,56,52,48,50,54,59,64},{83,75,79,85},{85,79,72,76,81,88,92,96,94,90},{71,65,68,72,79},{66,69,76,72},{69,64,70,77,74},{64,59,60,67},{96,92,97,98},{81,80,86,88},{88,86,91,95,93},{80,82,86},{86,82,87,89},{34,33,40,47,43},{82,73,78,84,87}}
NetFaces[JohnsonSolid[82]]:={{6,3,9,13},{13,9,17},{9,12,19,17},{2,1,5,12,9},{8,14,12},{12,14,22,19},{4,11,14,8},{4,7,11},{7,10,16,11},{11,16,25,23,15},{10,18,16},{16,18,28,27},{30,26,35,38},{38,35,43},{35,37,45,43},{24,21,29,37,35},{37,41,45},{45,41,49,54},{33,36,41,37},{19,22,33},{22,32,36,33},{36,40,46,48,44},{32,40,36},{28,18,20,34},{55,52,58,59},{59,58,62},{58,57,61,62},{51,47,50,57,58},{68,67,61,57,54,49,53,56,60,66},{77,71,75,82},{82,75,73,74,81,86,90,95,91,87},{69,63,65,73,75},{67,68,74,73},{68,66,72,78,76},{66,60,64,70},{95,90,97,98},{81,80,85,86},{86,85,93,96,94},{80,84,85},{85,84,89,92},{31,30,38,42,39},{84,79,83,88,89}}
NetFaces[JohnsonSolid[83]]:={{4,3,7,9},{9,7,12},{7,10,13,12},{2,1,5,10,7},{24,21,13,10,6,8,11,15,22,26},{19,17,23,27},{27,23,32},{23,25,35,32},{16,14,18,25,23},{25,30,35},{35,30,39,44},{21,24,30,25},{24,26,36,38,33},{26,22,29,34},{45,42,48,49},{49,48,52},{48,47,51,52},{41,37,40,47,48},{58,57,51,47,44,39,43,46,50,56},{67,61,65,72},{72,65,63,64,71,76,80,85,81,77},{59,53,55,63,65},{57,58,64,63},{58,56,62,68,66},{56,50,54,60},{85,80,87,88},{71,70,75,76},{76,75,83,86,84},{70,74,75},{75,74,79,82},{20,19,27,31,28},{74,69,73,78,79}}
NetFaces[JohnsonSolid[84]]:={{7,6,12},{12,6,11},{12,11,14},{5,6,2},{6,5,11},{11,5,10},{5,4,10},{10,4,9},{10,9,13},{3,4,1},{4,3,9},{9,3,8}}
NetFaces[JohnsonSolid[85]]:={{7,6,10,11},{7,9,5},{9,7,11},{9,11,13},{6,2,1},{2,6,7},{2,7,3},{10,8,12},{8,10,6},{8,6,4},{11,16,17},{16,11,10},{16,10,15},{21,20,24,25},{21,23,19},{23,21,25},{23,25,27},{20,15,14},{15,20,21},{15,21,16},{24,22,26},{22,24,20},{22,20,18},{25,29,30},{29,25,24},{29,24,28}}
NetFaces[JohnsonSolid[86]]:={{6,5,9,10},{10,9,13,14},{1,2,3},{3,6,4},{3,2,5},{3,5,6},{8,10,12},{12,10,14},{11,9,7},{9,11,13},{14,16,17},{14,13,16},{16,13,15},{17,16,18}}
NetFaces[JohnsonSolid[87]]:={{12,6,11},{12,11,15,16},{2,1,5},{5,7,8},{5,1,4},{5,4,7},{10,12,14},{14,12,16},{13,11,9},{11,13,15},{16,18,19},{16,15,18},{18,15,17},{19,18,20},{7,6,12},{7,4,6},{6,4,3}}
NetFaces[JohnsonSolid[88]]:={{8,12,9,5},{3,8,5},{10,8,6},{10,12,8},{5,7,2},{5,9,7},{4,8,3},{3,5,1},{13,9,12,16},{18,13,16},{11,13,15},{11,9,13},{16,14,19},{16,12,14},{17,18,21},{17,13,18},{18,20,22},{18,16,20}}
NetFaces[JohnsonSolid[89]]:={{8,12,9,5},{3,8,5},{10,8,6},{10,12,8},{5,7,2},{5,9,7},{4,8,3},{3,5,1},{16,13,9,12},{9,13,11},{16,12,14},{17,13,16,20},{22,17,20},{15,17,19},{15,13,17},{20,18,23},{20,16,18},{21,22,25},{21,17,22},{22,24,26},{22,20,24}}
NetFaces[JohnsonSolid[90]]:={{5,2,1},{3,2,5,9},{3,9,10},{12,5,4},{9,5,12,15},{9,15,16},{12,18,15},{8,7,14},{12,14,18},{18,14,20},{14,13,19,20},{7,6,13},{14,7,13},{13,11,17},{13,17,19},{20,19,24,25},{22,20,25},{22,25,27},{25,29,30},{25,24,29},{29,24,28},{26,21,23},{24,21,26},{19,21,24}}
NetFaces[JohnsonSolid[91]]:={{10,13,18,14},{13,10,5},{18,13,21},{14,18,22},{10,14,6},{5,10,3,1,2},{13,5,4,12,17},{22,18,24,26,25},{14,22,23,15,11},{12,4,8},{17,12,20},{15,23,19},{11,15,7},{7,15,16,9}}
NetFaces[JohnsonSolid[92]]:={{27,30,33,34,31},{30,27,25},{27,21,25},{21,27,24},{21,20,14,15},{14,20,17},{7,9,5,1,2},{15,9,7},{12,15,7},{12,7,4},{8,6,3},{12,6,8,16},{18,11,10},{18,22,19,13,11},{19,16,13},{19,23,16},{23,19,26},{29,25,23,28},{29,28,32},{23,25,21,15,12,16}}

(* Net Vertices *)

NetVertices[JohnsonSolid[ 1]]:={{(-1-Sqrt[3])/2,0},{-1/2,-1/2},{-1/2,1/2},{0,(-1-Sqrt[3])/2},{0,(1+Sqrt[3])/2},{1/2,-1/2},{1/2,1/2},{(1+Sqrt[3])/2,0}}
NetVertices[JohnsonSolid[ 2]]:={{(-5Sqrt[3]-5Sqrt[15]-5Sqrt[2(5-Sqrt[5])]-5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,(-1-Sqrt[5]-Sqrt[6(5-Sqrt[5])])/8},{(-5Sqrt[3]-5Sqrt[15]+5Sqrt[2(5-Sqrt[5])]-4Sqrt[10(5+Sqrt[5])])/40,(1+Sqrt[5]+Sqrt[6(5-Sqrt[5])])/8},{-Sqrt[(5+Sqrt[5])/10],0},{((-5+Sqrt[5])Sqrt[(5+Sqrt[5])/2])/20,(-1-Sqrt[5])/4},{((-5+Sqrt[5])Sqrt[(5+Sqrt[5])/2])/20,(1+Sqrt[5])/4},{(-10Sqrt[3]+10Sqrt[15]-5Sqrt[2(5-Sqrt[5])]-5Sqrt[10(5-Sqrt[5])]+10Sqrt[2(5+Sqrt[5])]+2Sqrt[10(5+Sqrt[5])])/80,(-6-2Sqrt[5]-Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])])/16},{(-10Sqrt[3]+10Sqrt[15]+5Sqrt[2(5-Sqrt[5])]+5Sqrt[10(5-Sqrt[5])]-10Sqrt[2(5+Sqrt[5])]+2Sqrt[10(5+Sqrt[5])])/80,(6+2Sqrt[5]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])])/16},{(5+Sqrt[5])^(3/2)/(20Sqrt[2]),-1/2},{(5+Sqrt[5])^(3/2)/(20Sqrt[2]),1/2},{(20Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,0}}
NetVertices[JohnsonSolid[ 3]]:={{(-1-Sqrt[3])/2,(1+Sqrt[3])/2},{(-1-Sqrt[3])/2,(3+Sqrt[3])/2},{-1,0},{-Sqrt[3]/2,(3+2Sqrt[3])/2},{-1/2,-Sqrt[3]/2},{-1/2,Sqrt[3]/2},{-1/2,(2+Sqrt[3])/2},{0,1+Sqrt[3]},{0,2+Sqrt[3]},{1/2,-Sqrt[3]/2},{1/2,Sqrt[3]/2},{1/2,(2+Sqrt[3])/2},{Sqrt[3]/2,(3+2Sqrt[3])/2},{1,0},{(1+Sqrt[3])/2,(1+Sqrt[3])/2},{(1+Sqrt[3])/2,(3+Sqrt[3])/2}}
NetVertices[JohnsonSolid[ 4]]:={{(-5-2Sqrt[2])/2,-1/2},{(-5-2Sqrt[2])/2,1/2},{(-5-Sqrt[2])/2,(-1-Sqrt[2])/2},{(-5-Sqrt[2])/2,(1+Sqrt[2])/2},{(-3-Sqrt[2])/2,(-1-Sqrt[2])/2},{(-3-Sqrt[2])/2,(1+Sqrt[2])/2},{-3/2,-1/2},{-3/2,1/2},{(-1-Sqrt[3])/2,1},{-1,(-1-Sqrt[3])/2},{-1/2,-3/2},{-1/2,-1/2},{-1/2,1/2},{-1/2,3/2},{1/2,-3/2},{1/2,-1/2},{1/2,1/2},{1/2,3/2},{1,(1+Sqrt[3])/2},{(1+Sqrt[3])/2,-1},{3/2,-1/2},{3/2,1/2}}
NetVertices[JohnsonSolid[ 5]]:={{(-2-Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/4,-1/2},{(-2-Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/4,1/2},{(-4-3Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8,(-3-Sqrt[5])/4},{(-4-3Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8,(3+Sqrt[5])/4},{(-4-Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8,(-1-Sqrt[5])/2},{(-4-Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8,(1+Sqrt[5])/2},{(-4+Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8,(-3-Sqrt[5])/4},{(-4+Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8,(3+Sqrt[5])/4},{-1/2,-1/2},{-1/2,1/2},{(10-2Sqrt[5]-Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])])/16,(8+2Sqrt[3]-2Sqrt[15]+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])])/16},{0,(-1-Sqrt[3])/2},{(12-4Sqrt[5]-4Sqrt[2(5-Sqrt[5])]-2Sqrt[10(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/16,(-4-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])])/8},{(3-Sqrt[5])/4,(4+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])])/8},{1/2,-1/2},{1/2,1/2},{(12-4Sqrt[5]-2Sqrt[2(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/16,(-2-2Sqrt[5]-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])])/8},{(6-2Sqrt[5]+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])])/8,(2+2Sqrt[5]+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])])/8},{(10+2Sqrt[5]+Sqrt[2(5-Sqrt[5])]-2Sqrt[6(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])]+2Sqrt[2(5+Sqrt[5])])/16,(2+Sqrt[3]+2Sqrt[5]+Sqrt[15]+Sqrt[2(5-Sqrt[5])])/8},{(8-2Sqrt[2(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/16,(-1-Sqrt[5])/4},{(4+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])])/8,(1+Sqrt[5])/4},{(8+2Sqrt[2(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/16,0},{(10-2Sqrt[5]-2Sqrt[2(5-Sqrt[5])]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/16,(-4+2Sqrt[3]-4Sqrt[5]-2Sqrt[15]-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])])/16},{(12+4Sqrt[5]-2Sqrt[2(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/16,(-1-Sqrt[5]-Sqrt[2(5-Sqrt[5])])/4},{(12+4Sqrt[5]+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])]+2Sqrt[2(5+Sqrt[5])])/16,(1+Sqrt[5]+Sqrt[2(5-Sqrt[5])])/4},{(12+4Sqrt[5]+2Sqrt[2(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/16,-Sqrt[(5-Sqrt[5])/2]/2},{(12+4Sqrt[5]+5Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])]+2Sqrt[2(5+Sqrt[5])])/16,Sqrt[(5-Sqrt[5])/2]/2},{(10+2Sqrt[5]+2Sqrt[2(5-Sqrt[5])]+2Sqrt[6(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/16,(Sqrt[3]+Sqrt[15]-Sqrt[2(5-Sqrt[5])])/8}}
NetVertices[JohnsonSolid[ 6]]:={{(-10-6Sqrt[5]-Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])])/16,(-2Sqrt[3]+2Sqrt[15]+3Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])]+5Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/16},{(-16-5Sqrt[6(5-Sqrt[5])]-3Sqrt[30(5-Sqrt[5])]-3Sqrt[6(5+Sqrt[5])]+Sqrt[30(5+Sqrt[5])])/32,(16Sqrt[3]+5Sqrt[2(5-Sqrt[5])]+3Sqrt[10(5-Sqrt[5])]+11Sqrt[2(5+Sqrt[5])]+7Sqrt[10(5+Sqrt[5])])/32},{(-14-2Sqrt[5]-Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])])/16,(-2Sqrt[3]+2Sqrt[15]-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])]+4Sqrt[2(5+Sqrt[5])]+4Sqrt[10(5+Sqrt[5])])/16},{(-1-Sqrt[5])/2,0},{(-6-2Sqrt[5]-Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])])/16,(-10Sqrt[3]+2Sqrt[15]-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])]+4Sqrt[2(5+Sqrt[5])]+4Sqrt[10(5+Sqrt[5])])/16},{(-6-2Sqrt[5]-Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])])/16,(-2Sqrt[3]+2Sqrt[15]+5Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])]+4Sqrt[2(5+Sqrt[5])]+4Sqrt[10(5+Sqrt[5])])/16},{(-12-4Sqrt[5]-3Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])]-3Sqrt[6(5+Sqrt[5])]+Sqrt[30(5+Sqrt[5])])/32,(12Sqrt[3]+4Sqrt[15]+7Sqrt[2(5-Sqrt[5])]+5Sqrt[10(5-Sqrt[5])]+11Sqrt[2(5+Sqrt[5])]+7Sqrt[10(5+Sqrt[5])])/32},{(-3-Sqrt[5])/4,-Sqrt[(5+Sqrt[5])/2]/2},{(-3-Sqrt[5])/4,Sqrt[(5+Sqrt[5])/2]/2},{(-20+4Sqrt[5]-3Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])]-3Sqrt[6(5+Sqrt[5])]+Sqrt[30(5+Sqrt[5])])/32,(12Sqrt[3]+4Sqrt[15]+3Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])]+11Sqrt[2(5+Sqrt[5])]+7Sqrt[10(5+Sqrt[5])])/32},{(2-2Sqrt[5]-Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])])/16,(-2Sqrt[3]+2Sqrt[15]-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])]+4Sqrt[2(5+Sqrt[5])]+4Sqrt[10(5+Sqrt[5])])/16},{(-1-Sqrt[5])/4,((3+Sqrt[5])Sqrt[(5+Sqrt[5])/2])/4},{(-2+2Sqrt[5]-Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])])/16,(-2Sqrt[3]+2Sqrt[15]+2Sqrt[2(5-Sqrt[5])]+7Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/16},{(-4+4Sqrt[5]-3Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])]-3Sqrt[6(5+Sqrt[5])]+Sqrt[30(5+Sqrt[5])])/32,(12Sqrt[3]+4Sqrt[15]+9Sqrt[2(5-Sqrt[5])]+3Sqrt[10(5-Sqrt[5])]+13Sqrt[2(5+Sqrt[5])]+9Sqrt[10(5+Sqrt[5])])/32},{-1/2,-((1+Sqrt[5])Sqrt[(5+Sqrt[5])/2])/4},{-1/2,((1+Sqrt[5])Sqrt[(5+Sqrt[5])/2])/4},{(12+4Sqrt[5]-Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(12Sqrt[3]+4Sqrt[15]+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])]+10Sqrt[2(5+Sqrt[5])]+8Sqrt[10(5+Sqrt[5])])/32},{0,((1+Sqrt[5])Sqrt[(5+Sqrt[5])/2])/2},{(4+12Sqrt[5]-3Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])]-3Sqrt[6(5+Sqrt[5])]+Sqrt[30(5+Sqrt[5])])/32,(12Sqrt[3]+4Sqrt[15]+7Sqrt[2(5-Sqrt[5])]+5Sqrt[10(5-Sqrt[5])]+11Sqrt[2(5+Sqrt[5])]+7Sqrt[10(5+Sqrt[5])])/32},{(20+4Sqrt[5]-3Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(12Sqrt[3]+12Sqrt[15]+7Sqrt[2(5-Sqrt[5])]+3Sqrt[10(5-Sqrt[5])]+10Sqrt[2(5+Sqrt[5])]+8Sqrt[10(5+Sqrt[5])])/32},{1/2,-((1+Sqrt[5])Sqrt[(5+Sqrt[5])/2])/4},{1/2,((1+Sqrt[5])Sqrt[(5+Sqrt[5])/2])/4},{(16+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(8Sqrt[3]+8Sqrt[15]+3Sqrt[2(5-Sqrt[5])]+3Sqrt[10(5-Sqrt[5])]+10Sqrt[2(5+Sqrt[5])]+8Sqrt[10(5+Sqrt[5])])/32},{(1+Sqrt[5])/4,((3+Sqrt[5])Sqrt[(5+Sqrt[5])/2])/4},{(8+8Sqrt[5]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(8Sqrt[3]+8Sqrt[15]+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])]+6Sqrt[2(5+Sqrt[5])]+8Sqrt[10(5+Sqrt[5])])/32},{(1+Sqrt[5]+Sqrt[6(5-Sqrt[5])])/8,(Sqrt[3]+Sqrt[15]-Sqrt[2(5-Sqrt[5])]+2Sqrt[2(5+Sqrt[5])]+2Sqrt[10(5+Sqrt[5])])/8},{(-4+12Sqrt[5]+5Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(4Sqrt[3]+4Sqrt[15]-9Sqrt[2(5-Sqrt[5])]-5Sqrt[10(5-Sqrt[5])]+10Sqrt[2(5+Sqrt[5])]+8Sqrt[10(5+Sqrt[5])])/32},{(3+Sqrt[5])/4,-Sqrt[(5+Sqrt[5])/2]/2},{(3+Sqrt[5])/4,Sqrt[(5+Sqrt[5])/2]/2},{(24+8Sqrt[5]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(8Sqrt[3]+8Sqrt[15]+5Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])]+12Sqrt[2(5+Sqrt[5])]+10Sqrt[10(5+Sqrt[5])])/32},{(6+2Sqrt[5]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])])/16,(2Sqrt[3]-2Sqrt[15]-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])]+6Sqrt[2(5+Sqrt[5])]+2Sqrt[10(5+Sqrt[5])])/16},{(1+Sqrt[5])/2,0},{(12+12Sqrt[5]+5Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(4Sqrt[3]+4Sqrt[15]+3Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])]+10Sqrt[2(5+Sqrt[5])]+8Sqrt[10(5+Sqrt[5])])/32},{(40+8Sqrt[5]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(8Sqrt[3]+8Sqrt[15]-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])]+10Sqrt[2(5+Sqrt[5])]+8Sqrt[10(5+Sqrt[5])])/32},{(32+16Sqrt[5]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(8Sqrt[3]+8Sqrt[15]-3Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])]+12Sqrt[2(5+Sqrt[5])]+10Sqrt[10(5+Sqrt[5])])/32},{(28+12Sqrt[5]+5Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(4Sqrt[3]+4Sqrt[15]-5Sqrt[2(5-Sqrt[5])]-5Sqrt[10(5-Sqrt[5])]+12Sqrt[2(5+Sqrt[5])]+6Sqrt[10(5+Sqrt[5])])/32},{(20+20Sqrt[5]+5Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(4Sqrt[3]+4Sqrt[15]-5Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])]+10Sqrt[2(5+Sqrt[5])]+8Sqrt[10(5+Sqrt[5])])/32},{(16+16Sqrt[5]+9Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(8Sqrt[3]+8Sqrt[15]-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])]+10Sqrt[2(5+Sqrt[5])]+8Sqrt[10(5+Sqrt[5])])/32}}
NetVertices[JohnsonSolid[ 7]]:={{-3/2,1/(2Sqrt[3])},{-3/2,(6+Sqrt[3])/6},{-1,(3+2Sqrt[3])/3},{-1/2,1/(2Sqrt[3])},{-1/2,(6+Sqrt[3])/6},{0,-(1/Sqrt[3])},{0,(3+2Sqrt[3])/3},{1/2,1/(2Sqrt[3])},{1/2,(6+Sqrt[3])/6},{1,(3+2Sqrt[3])/3},{3/2,1/(2Sqrt[3])},{3/2,(6+Sqrt[3])/6}}
NetVertices[JohnsonSolid[ 8]]:={{-3/2,1/2},{-3/2,3/2},{-1,(3+Sqrt[3])/2},{-1/2,-1/2},{-1/2,1/2},{-1/2,3/2},{0,(3+Sqrt[3])/2},{1/2,-1/2},{1/2,1/2},{1/2,3/2},{1,(3+Sqrt[3])/2},{3/2,1/2},{3/2,3/2},{2,(3+Sqrt[3])/2},{5/2,1/2},{5/2,3/2}}
NetVertices[JohnsonSolid[ 9]]:={{-5/2,(5+Sqrt[5])^(3/2)/(20Sqrt[2])},{-5/2,(40+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40},{-2,(40+20Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40},{-3/2,(5+Sqrt[5])^(3/2)/(20Sqrt[2])},{-3/2,(40+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40},{-1,(40+20Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40},{(-1-Sqrt[5])/4,(-5Sqrt[2(5-Sqrt[5])]-5Sqrt[10(5-Sqrt[5])]+2Sqrt[10(5+Sqrt[5])])/80},{-1/2,(5+Sqrt[5])^(3/2)/(20Sqrt[2])},{-1/2,(40+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40},{0,(-10Sqrt[2(5-Sqrt[5])]-5Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/80},{0,(40+20Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40},{1/2,(5+Sqrt[5])^(3/2)/(20Sqrt[2])},{1/2,(40+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40},{(1+Sqrt[5])/4,(10Sqrt[2(5-Sqrt[5])]-5Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/80},{1,(40+20Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40},{3/2,(5+Sqrt[5])^(3/2)/(20Sqrt[2])},{3/2,(40+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40},{2,(40+20Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40},{5/2,(5+Sqrt[5])^(3/2)/(20Sqrt[2])},{5/2,(40+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40}}
NetVertices[JohnsonSolid[10]]:={{-1/2,1/(2Sqrt[3])},{0,-(1/Sqrt[3])},{0,2/Sqrt[3]},{1/2,1/(2Sqrt[3])},{1,(-3-Sqrt[3])/3},{1,-(1/Sqrt[3])},{1,2/Sqrt[3]},{3/2,1/(2Sqrt[3])},{2,(-3-Sqrt[3])/3},{2,-(1/Sqrt[3])},{2,2/Sqrt[3]},{5/2,1/(2Sqrt[3])},{3,-(1/Sqrt[3])},{3,2/Sqrt[3]},{7/2,1/(2Sqrt[3])},{4,-(1/Sqrt[3])}}
NetVertices[JohnsonSolid[11]]:={{-1/2,1/(2Sqrt[3])},{0,-(1/Sqrt[3])},{0,2/Sqrt[3]},{1/2,1/(2Sqrt[3])},{1,-(1/Sqrt[3])},{1,2/Sqrt[3]},{3/2,1/(2Sqrt[3])},{(9-Sqrt[5])/4,(-16Sqrt[3]+6Sqrt[2(5-Sqrt[5])]-9Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/48},{2,-(1/Sqrt[3])},{2,2/Sqrt[3]},{5/2,(-16Sqrt[3]-6Sqrt[2(5-Sqrt[5])]-9Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/48},{5/2,1/(2Sqrt[3])},{3,-(1/Sqrt[3])},{3,2/Sqrt[3]},{(11+Sqrt[5])/4,(-16Sqrt[3]-3Sqrt[2(5-Sqrt[5])]-3Sqrt[10(5-Sqrt[5])]-6Sqrt[2(5+Sqrt[5])])/48},{7/2,1/(2Sqrt[3])},{4,-(1/Sqrt[3])},{4,2/Sqrt[3]},{9/2,1/(2Sqrt[3])},{5,-(1/Sqrt[3])}}
NetVertices[JohnsonSolid[12]]:={{-1/(2Sqrt[3]),-3/2},{-1/(2Sqrt[3]),-1/2},{-1/(2Sqrt[3]),1/2},{1/Sqrt[3],-1},{1/Sqrt[3],0},{5/(2Sqrt[3]),-3/2},{5/(2Sqrt[3]),-1/2},{5/(2Sqrt[3]),1/2}}
NetVertices[JohnsonSolid[13]]:={{-2/Sqrt[3],-1},{-2/Sqrt[3],0},{-1/(2Sqrt[3]),-3/2},{-1/(2Sqrt[3]),-1/2},{-1/(2Sqrt[3]),1/2},{1/Sqrt[3],-1},{1/Sqrt[3],0},{5/(2Sqrt[3]),-3/2},{5/(2Sqrt[3]),-1/2},{5/(2Sqrt[3]),1/2},{4/Sqrt[3],-1},{4/Sqrt[3],0}}
NetVertices[JohnsonSolid[14]]:=Sqrt[2/5]{{-1,1/2},{(-1-Sqrt[3])/4,(3(1+Sqrt[3]))/4},{-1/2,-1},{1/2,1},{(1+Sqrt[3])/4,(-3(1+Sqrt[3]))/4},{(5-Sqrt[3])/4,(5+3Sqrt[3])/4},{1,-1/2},{2,3/2},{(7+Sqrt[3])/4,(-1-3Sqrt[3])/4},{(11-Sqrt[3])/4,(7+3Sqrt[3])/4},{5/2,0},{7/2,2},{(13+Sqrt[3])/4,(1-3Sqrt[3])/4},{4,1/2}}
NetVertices[JohnsonSolid[15]]:=Sqrt[2/5]{{-1,1/2},{(-1-Sqrt[3])/4,(3(1+Sqrt[3]))/4},{-1/2,-1},{1/2,1},{(1+Sqrt[3])/4,(-3(1+Sqrt[3]))/4},{(5-Sqrt[3])/4,(5+3Sqrt[3])/4},{1,-1/2},{2,3/2},{(7+Sqrt[3])/4,(-1-3Sqrt[3])/4},{(11-Sqrt[3])/4,(7+3Sqrt[3])/4},{5/2,0},{7/2,2},{(13+Sqrt[3])/4,(1-3Sqrt[3])/4},{(17-Sqrt[3])/4,(3(3+Sqrt[3]))/4},{4,1/2},{5,5/2},{(19+Sqrt[3])/4,(-3(-1+Sqrt[3]))/4},{11/2,1}}
NetVertices[JohnsonSolid[16]]:=Sqrt[2/5]{{-1,1/2},{(-1-Sqrt[3])/4,(3(1+Sqrt[3]))/4},{-1/2,-1},{1/2,1},{(1+Sqrt[3])/4,(-3(1+Sqrt[3]))/4},{(5-Sqrt[3])/4,(5+3Sqrt[3])/4},{1,-1/2},{2,3/2},{(7+Sqrt[3])/4,(-1-3Sqrt[3])/4},{(11-Sqrt[3])/4,(7+3Sqrt[3])/4},{5/2,0},{7/2,2},{(13+Sqrt[3])/4,(1-3Sqrt[3])/4},{(17-Sqrt[3])/4,(3(3+Sqrt[3]))/4},{4,1/2},{5,5/2},{(19+Sqrt[3])/4,(-3(-1+Sqrt[3]))/4},{(23-Sqrt[3])/4,(11+3Sqrt[3])/4},{11/2,1},{13/2,3},{(25+Sqrt[3])/4,(5-3Sqrt[3])/4},{7,3/2}}
NetVertices[JohnsonSolid[17]]:={{-1/2,1/(2Sqrt[3])},{0,-(1/Sqrt[3])},{0,2/Sqrt[3]},{1/2,-5/(2Sqrt[3])},{1/2,1/(2Sqrt[3])},{1,-(1/Sqrt[3])},{1,2/Sqrt[3]},{3/2,-5/(2Sqrt[3])},{3/2,1/(2Sqrt[3])},{2,-(1/Sqrt[3])},{2,2/Sqrt[3]},{5/2,-5/(2Sqrt[3])},{5/2,1/(2Sqrt[3])},{3,-(1/Sqrt[3])},{3,2/Sqrt[3]},{7/2,-5/(2Sqrt[3])},{7/2,1/(2Sqrt[3])},{4,-(1/Sqrt[3])}}
NetVertices[JohnsonSolid[18]]:={{-1/2,-1/2},{-1/2,1/2},{1/2,-1/2},{1/2,1/2},{(3-Sqrt[3])/2,1},{(3-Sqrt[3])/2,2},{1,(-1-Sqrt[3])/2},{(4-Sqrt[3])/2,(4+Sqrt[3])/2},{3/2,(-1-2Sqrt[3])/2},{3/2,-1/2},{3/2,1/2},{3/2,3/2},{2,(3+Sqrt[3])/2},{2,(5+Sqrt[3])/2},{5/2,(-1-2Sqrt[3])/2},{5/2,-1/2},{5/2,1/2},{5/2,3/2},{(4+Sqrt[3])/2,(4+Sqrt[3])/2},{3,(-1-Sqrt[3])/2},{(5+Sqrt[3])/2,1},{(5+Sqrt[3])/2,2},{7/2,-1/2},{7/2,1/2},{9/2,-1/2},{9/2,1/2},{11/2,-1/2},{11/2,1/2}}
NetVertices[JohnsonSolid[19]]:={{-1/2,-1/2},{-1/2,1/2},{1/2,-1/2},{1/2,1/2},{3/2,-1/2},{3/2,1/2},{3/2,3/2},{3/2,5/2},{(5-Sqrt[3])/2,3},{(5-Sqrt[2])/2,(-3-Sqrt[2])/2},{(5-Sqrt[2])/2,(-1-Sqrt[2])/2},{2,(3-Sqrt[3])/2},{5/2,(-3-2Sqrt[2])/2},{5/2,-1/2},{5/2,1/2},{5/2,3/2},{5/2,5/2},{5/2,7/2},{7/2,(-3-2Sqrt[2])/2},{7/2,-1/2},{7/2,1/2},{7/2,3/2},{7/2,5/2},{7/2,7/2},{4,(5+Sqrt[3])/2},{(7+Sqrt[2])/2,(-3-Sqrt[2])/2},{(7+Sqrt[2])/2,(-1-Sqrt[2])/2},{(7+Sqrt[3])/2,1},{9/2,-1/2},{9/2,1/2},{9/2,3/2},{9/2,5/2},{11/2,-1/2},{11/2,1/2},{13/2,-1/2},{13/2,1/2},{15/2,-1/2},{15/2,1/2}}
NetVertices[JohnsonSolid[20]]:={{-1/2,-1/2},{-1/2,1/2},{1/2,-1/2},{1/2,1/2},{3/2,-1/2},{3/2,1/2},{(30-Sqrt[3]-2Sqrt[5]-Sqrt[15]-Sqrt[2(5-Sqrt[5])])/8,(26+2Sqrt[5]+Sqrt[2(5-Sqrt[5])]-2Sqrt[6(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])]+2Sqrt[2(5+Sqrt[5])])/16},{(30-2Sqrt[5]-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])])/8,(14-2Sqrt[5]+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])])/8},{(7-Sqrt[5])/2,(-4-Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{5/2,-1/2},{5/2,1/2},{(28-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])])/8,(7-Sqrt[5])/4},{(15-Sqrt[5]-Sqrt[2(5-Sqrt[5])])/4,(28+4Sqrt[5]+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])]+2Sqrt[2(5+Sqrt[5])])/16},{(13-Sqrt[5])/4,(-4-3Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(13-Sqrt[5])/4,(-4+Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(15-Sqrt[5])/4,(12+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])])/8},{(56-2Sqrt[3]+2Sqrt[15]-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])])/16,(26-2Sqrt[5]-Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])])/16},{(16-Sqrt[2(5-Sqrt[5])])/4,(28+4Sqrt[5]+5Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])]+2Sqrt[2(5+Sqrt[5])])/16},{7/2,(-2-Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/4},{7/2,-1/2},{7/2,1/2},{7/2,3/2},{(32-Sqrt[3]-Sqrt[15]+Sqrt[2(5-Sqrt[5])])/8,(26+2Sqrt[5]+2Sqrt[2(5-Sqrt[5])]+2Sqrt[6(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/16},{4,(24+2Sqrt[2(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/16},{9/2,(-2-Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/4},{9/2,-1/2},{9/2,1/2},{9/2,3/2},{(16+Sqrt[2(5-Sqrt[5])])/4,(28+4Sqrt[5]+2Sqrt[2(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/16},{(17+Sqrt[5])/4,(24-2Sqrt[2(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/16},{(19+Sqrt[5])/4,(-4-3Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(19+Sqrt[5])/4,(-4+Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(9+Sqrt[3])/2,1},{(17+Sqrt[5]+Sqrt[2(5-Sqrt[5])])/4,(28+4Sqrt[5]-2Sqrt[2(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/16},{(36+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])])/8,(28-4Sqrt[5]-4Sqrt[2(5-Sqrt[5])]-2Sqrt[10(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/16},{11/2,-1/2},{11/2,1/2},{(68-2Sqrt[3]+4Sqrt[5]+2Sqrt[15]+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])])/16,(26-2Sqrt[5]-2Sqrt[2(5-Sqrt[5])]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/16},{(9+Sqrt[5])/2,(-4-Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(34+2Sqrt[5]+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])])/8,(28-4Sqrt[5]-2Sqrt[2(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/16},{13/2,-1/2},{13/2,1/2},{15/2,-1/2},{15/2,1/2},{17/2,-1/2},{17/2,1/2},{19/2,-1/2},{19/2,1/2}}
NetVertices[JohnsonSolid[21]]:={{-1/2,-1/2},{-1/2,1/2},{1/2,-1/2},{1/2,1/2},{3/2,-1/2},{3/2,1/2},{(54-6Sqrt[5]-Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])])/16,(8-2Sqrt[3]+2Sqrt[15]+3Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/16},{(112-5Sqrt[6(5-Sqrt[5])]-3Sqrt[30(5-Sqrt[5])]-3Sqrt[6(5+Sqrt[5])]+Sqrt[30(5+Sqrt[5])])/32,(16+16Sqrt[3]+5Sqrt[2(5-Sqrt[5])]+3Sqrt[10(5-Sqrt[5])]+7Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/32},{(50-2Sqrt[5]-Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])])/16,(8-2Sqrt[3]+2Sqrt[15]-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])]+2Sqrt[2(5+Sqrt[5])]+2Sqrt[10(5+Sqrt[5])])/16},{(7-Sqrt[5])/2,(-4-Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{5/2,-1/2},{5/2,1/2},{(58-2Sqrt[5]-Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])])/16,(8-10Sqrt[3]+2Sqrt[15]-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])]+2Sqrt[2(5+Sqrt[5])]+2Sqrt[10(5+Sqrt[5])])/16},{(58-2Sqrt[5]-Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])])/16,(8-2Sqrt[3]+2Sqrt[15]+5Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])]+2Sqrt[2(5+Sqrt[5])]+2Sqrt[10(5+Sqrt[5])])/16},{(116-4Sqrt[5]-3Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])]-3Sqrt[6(5+Sqrt[5])]+Sqrt[30(5+Sqrt[5])])/32,(16+12Sqrt[3]+4Sqrt[15]+Sqrt[2(5-Sqrt[5])]+3Sqrt[10(5-Sqrt[5])]+9Sqrt[2(5+Sqrt[5])]+5Sqrt[10(5+Sqrt[5])])/32},{(13-Sqrt[5])/4,(-4-3Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(13-Sqrt[5])/4,(-4+Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(108+4Sqrt[5]-3Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])]-3Sqrt[6(5+Sqrt[5])]+Sqrt[30(5+Sqrt[5])])/32,(16+12Sqrt[3]+4Sqrt[15]+3Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])]+7Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/32},{(66-2Sqrt[5]-Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])])/16,(8-2Sqrt[3]+2Sqrt[15]-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])]+2Sqrt[2(5+Sqrt[5])]+2Sqrt[10(5+Sqrt[5])])/16},{(15-Sqrt[5])/4,(2+Sqrt[2(5+Sqrt[5])])/4},{(62+2Sqrt[5]-Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])])/16,(8-2Sqrt[3]+2Sqrt[15]+2Sqrt[2(5-Sqrt[5])]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/16},{(124+4Sqrt[5]-3Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])]-3Sqrt[6(5+Sqrt[5])]+Sqrt[30(5+Sqrt[5])])/32,(16+12Sqrt[3]+4Sqrt[15]+9Sqrt[2(5-Sqrt[5])]+3Sqrt[10(5-Sqrt[5])]+9Sqrt[2(5+Sqrt[5])]+5Sqrt[10(5+Sqrt[5])])/32},{7/2,(-2-Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/4},{7/2,-1/2},{7/2,1/2},{(140+4Sqrt[5]-Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(16+12Sqrt[3]+4Sqrt[15]+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])]+6Sqrt[2(5+Sqrt[5])]+4Sqrt[10(5+Sqrt[5])])/32},{4,(4+Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/8},{(132+12Sqrt[5]-3Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])]-3Sqrt[6(5+Sqrt[5])]+Sqrt[30(5+Sqrt[5])])/32,(16+12Sqrt[3]+4Sqrt[15]+7Sqrt[2(5-Sqrt[5])]+5Sqrt[10(5-Sqrt[5])]+7Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/32},{(148+4Sqrt[5]-3Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(16+12Sqrt[3]+12Sqrt[15]+7Sqrt[2(5-Sqrt[5])]+3Sqrt[10(5-Sqrt[5])]+6Sqrt[2(5+Sqrt[5])]+4Sqrt[10(5+Sqrt[5])])/32},{9/2,(-2-Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/4},{9/2,-1/2},{9/2,1/2},{(144+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(16+8Sqrt[3]+8Sqrt[15]+3Sqrt[2(5-Sqrt[5])]+3Sqrt[10(5-Sqrt[5])]+6Sqrt[2(5+Sqrt[5])]+4Sqrt[10(5+Sqrt[5])])/32},{(17+Sqrt[5])/4,(2+Sqrt[2(5+Sqrt[5])])/4},{(136+8Sqrt[5]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(16+8Sqrt[3]+8Sqrt[15]+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])]+2Sqrt[2(5+Sqrt[5])]+4Sqrt[10(5+Sqrt[5])])/32},{(33+Sqrt[5]+Sqrt[6(5-Sqrt[5])])/8,(4+Sqrt[3]+Sqrt[15]-Sqrt[2(5-Sqrt[5])]+Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/8},{(124+12Sqrt[5]+5Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(16+4Sqrt[3]+4Sqrt[15]-9Sqrt[2(5-Sqrt[5])]-5Sqrt[10(5-Sqrt[5])]+6Sqrt[2(5+Sqrt[5])]+4Sqrt[10(5+Sqrt[5])])/32},{(19+Sqrt[5])/4,(-4-3Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(19+Sqrt[5])/4,(-4+Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(152+8Sqrt[5]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(16+8Sqrt[3]+8Sqrt[15]+5Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])]+8Sqrt[2(5+Sqrt[5])]+6Sqrt[10(5+Sqrt[5])])/32},{(70+2Sqrt[5]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])])/16,(8+2Sqrt[3]-2Sqrt[15]-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])]+4Sqrt[2(5+Sqrt[5])])/16},{11/2,-1/2},{11/2,1/2},{(9+Sqrt[5])/2,(-4-Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(140+12Sqrt[5]+5Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(16+4Sqrt[3]+4Sqrt[15]+3Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])]+6Sqrt[2(5+Sqrt[5])]+4Sqrt[10(5+Sqrt[5])])/32},{(168+8Sqrt[5]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(16+8Sqrt[3]+8Sqrt[15]-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])]+6Sqrt[2(5+Sqrt[5])]+4Sqrt[10(5+Sqrt[5])])/32},{(160+16Sqrt[5]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(16+8Sqrt[3]+8Sqrt[15]-3Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])]+8Sqrt[2(5+Sqrt[5])]+6Sqrt[10(5+Sqrt[5])])/32},{(156+12Sqrt[5]+5Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(16+4Sqrt[3]+4Sqrt[15]-5Sqrt[2(5-Sqrt[5])]-5Sqrt[10(5-Sqrt[5])]+8Sqrt[2(5+Sqrt[5])]+2Sqrt[10(5+Sqrt[5])])/32},{13/2,-1/2},{13/2,1/2},{(148+20Sqrt[5]+5Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(16+4Sqrt[3]+4Sqrt[15]-5Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])]+6Sqrt[2(5+Sqrt[5])]+4Sqrt[10(5+Sqrt[5])])/32},{(144+16Sqrt[5]+9Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(16+8Sqrt[3]+8Sqrt[15]-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])]+6Sqrt[2(5+Sqrt[5])]+4Sqrt[10(5+Sqrt[5])])/32},{15/2,-1/2},{15/2,1/2},{17/2,-1/2},{17/2,1/2},{19/2,-1/2},{19/2,1/2}}
NetVertices[JohnsonSolid[22]]:={{-1/2,1/(2Sqrt[3])},{(1-Sqrt[3])/2,(3+Sqrt[3])/6},{(1-Sqrt[3])/2,(9+Sqrt[3])/6},{0,-(1/Sqrt[3])},{(2-Sqrt[3])/2,(9+4Sqrt[3])/6},{1/2,-5/(2Sqrt[3])},{1/2,1/(2Sqrt[3])},{1/2,(6+Sqrt[3])/6},{1,-4/Sqrt[3]},{1,-(1/Sqrt[3])},{1,(3+2Sqrt[3])/3},{1,(2(3+Sqrt[3]))/3},{3/2,1/(2Sqrt[3])},{3/2,(6+Sqrt[3])/6},{(2+Sqrt[3])/2,(9+4Sqrt[3])/6},{2,-4/Sqrt[3]},{2,-(1/Sqrt[3])},{(3+Sqrt[3])/2,(3+Sqrt[3])/6},{(3+Sqrt[3])/2,(9+Sqrt[3])/6},{5/2,-5/(2Sqrt[3])},{5/2,1/(2Sqrt[3])},{3,-(1/Sqrt[3])},{7/2,1/(2Sqrt[3])},{4,-(1/Sqrt[3])},{9/2,1/(2Sqrt[3])},{5,-(1/Sqrt[3])},{11/2,1/(2Sqrt[3])},{6,-(1/Sqrt[3])}}
NetVertices[JohnsonSolid[23]]:={{-1/2,1/(2Sqrt[3])},{0,-(1/Sqrt[3])},{(2-Sqrt[2])/2,(-6-3Sqrt[2]-2Sqrt[3])/6},{(2-Sqrt[2])/2,(-3Sqrt[2]-2Sqrt[3])/6},{1/2,1/(2Sqrt[3])},{1/2,(6+Sqrt[3])/6},{1/2,(12+Sqrt[3])/6},{(3-Sqrt[3])/2,(15+Sqrt[3])/6},{1,(-3-3Sqrt[2]-Sqrt[3])/3},{1,-(1/Sqrt[3])},{1,(3-Sqrt[3])/3},{3/2,1/(2Sqrt[3])},{3/2,(6+Sqrt[3])/6},{3/2,(12+Sqrt[3])/6},{3/2,(18+Sqrt[3])/6},{2,(-3-3Sqrt[2]-Sqrt[3])/3},{2,-(1/Sqrt[3])},{5/2,1/(2Sqrt[3])},{5/2,(6+Sqrt[3])/6},{5/2,(12+Sqrt[3])/6},{5/2,(18+Sqrt[3])/6},{(4+Sqrt[2])/2,(-6-3Sqrt[2]-2Sqrt[3])/6},{(4+Sqrt[2])/2,(-3Sqrt[2]-2Sqrt[3])/6},{3,-(1/Sqrt[3])},{3,(2(3+Sqrt[3]))/3},{(5+Sqrt[3])/2,(3+Sqrt[3])/6},{7/2,1/(2Sqrt[3])},{7/2,(6+Sqrt[3])/6},{7/2,(12+Sqrt[3])/6},{4,-(1/Sqrt[3])},{9/2,1/(2Sqrt[3])},{5,-(1/Sqrt[3])},{11/2,1/(2Sqrt[3])},{6,-(1/Sqrt[3])},{13/2,1/(2Sqrt[3])},{7,-(1/Sqrt[3])},{15/2,1/(2Sqrt[3])},{8,-(1/Sqrt[3])}}
NetVertices[JohnsonSolid[24]]:={{-1/2,1/(2Sqrt[3])},{0,-(1/Sqrt[3])},{(14-Sqrt[3]-2Sqrt[5]-Sqrt[15]-Sqrt[2(5-Sqrt[5])])/8,(54+8Sqrt[3]+6Sqrt[5]+3Sqrt[2(5-Sqrt[5])]-6Sqrt[6(5-Sqrt[5])]+3Sqrt[10(5-Sqrt[5])]+6Sqrt[2(5+Sqrt[5])])/48},{(14-2Sqrt[5]-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])])/8,(30+4Sqrt[3]-6Sqrt[5]+3Sqrt[2(5-Sqrt[5])]+3Sqrt[10(5-Sqrt[5])])/24},{1/2,1/(2Sqrt[3])},{(12-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])])/8,(15+2Sqrt[3]-3Sqrt[5])/12},{(7-Sqrt[5]-Sqrt[2(5-Sqrt[5])])/4,(60+8Sqrt[3]+12Sqrt[5]+3Sqrt[2(5-Sqrt[5])]+3Sqrt[10(5-Sqrt[5])]+6Sqrt[2(5+Sqrt[5])])/48},{(4-Sqrt[5])/2,(-8Sqrt[3]-3Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/24},{1,-(1/Sqrt[3])},{(7-Sqrt[5])/4,(-8Sqrt[3]-9Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/24},{(7-Sqrt[5])/4,(-8Sqrt[3]+3Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/24},{(7-Sqrt[5])/4,(24+4Sqrt[3]+3Sqrt[2(5-Sqrt[5])]+3Sqrt[10(5-Sqrt[5])])/24},{(24-2Sqrt[3]+2Sqrt[15]-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])])/16,(54+8Sqrt[3]-6Sqrt[5]-3Sqrt[6(5-Sqrt[5])]-3Sqrt[30(5-Sqrt[5])])/48},{(8-Sqrt[2(5-Sqrt[5])])/4,(60+8Sqrt[3]+12Sqrt[5]+15Sqrt[2(5-Sqrt[5])]+3Sqrt[10(5-Sqrt[5])]+6Sqrt[2(5+Sqrt[5])])/48},{3/2,1/(2Sqrt[3])},{3/2,(6+Sqrt[3])/6},{(16-Sqrt[3]-Sqrt[15]+Sqrt[2(5-Sqrt[5])])/8,(54+8Sqrt[3]+6Sqrt[5]+6Sqrt[2(5-Sqrt[5])]+6Sqrt[6(5-Sqrt[5])]+9Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/48},{2,(-4Sqrt[3]-3Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/12},{2,-(1/Sqrt[3])},{2,(48+8Sqrt[3]+6Sqrt[2(5-Sqrt[5])]+9Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/48},{5/2,1/(2Sqrt[3])},{5/2,(6+Sqrt[3])/6},{(8+Sqrt[2(5-Sqrt[5])])/4,(60+8Sqrt[3]+12Sqrt[5]+6Sqrt[2(5-Sqrt[5])]+9Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/48},{(9+Sqrt[5])/4,(48+8Sqrt[3]-6Sqrt[2(5-Sqrt[5])]+9Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/48},{3,(-4Sqrt[3]-3Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/12},{3,-(1/Sqrt[3])},{(5+Sqrt[3])/2,(3+Sqrt[3])/6},{(9+Sqrt[5]+Sqrt[2(5-Sqrt[5])])/4,(60+8Sqrt[3]+12Sqrt[5]-6Sqrt[2(5-Sqrt[5])]+9Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/48},{(20+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])])/8,(60+8Sqrt[3]-12Sqrt[5]-12Sqrt[2(5-Sqrt[5])]-6Sqrt[10(5-Sqrt[5])]+9Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/48},{7/2,1/(2Sqrt[3])},{(36-2Sqrt[3]+4Sqrt[5]+2Sqrt[15]+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])])/16,(54+8Sqrt[3]-6Sqrt[5]-6Sqrt[2(5-Sqrt[5])]+3Sqrt[6(5-Sqrt[5])]+3Sqrt[30(5-Sqrt[5])]+9Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/48},{(18+2Sqrt[5]+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])])/8,(60+8Sqrt[3]-12Sqrt[5]-6Sqrt[2(5-Sqrt[5])]+9Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/48},{(13+Sqrt[5])/4,(-8Sqrt[3]-9Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/24},{(13+Sqrt[5])/4,(-8Sqrt[3]+3Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/24},{4,-(1/Sqrt[3])},{(6+Sqrt[5])/2,(-8Sqrt[3]-3Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/24},{9/2,1/(2Sqrt[3])},{5,-(1/Sqrt[3])},{11/2,1/(2Sqrt[3])},{6,-(1/Sqrt[3])},{13/2,1/(2Sqrt[3])},{7,-(1/Sqrt[3])},{15/2,1/(2Sqrt[3])},{8,-(1/Sqrt[3])},{17/2,1/(2Sqrt[3])},{9,-(1/Sqrt[3])},{19/2,1/(2Sqrt[3])},{10,-(1/Sqrt[3])}}
NetVertices[JohnsonSolid[25]]:={{-1/2,1/(2Sqrt[3])},{(22-6Sqrt[5]-Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])])/16,(2Sqrt[3]+6Sqrt[15]+9Sqrt[2(5-Sqrt[5])]+3Sqrt[10(5-Sqrt[5])]+9Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/48},{(48-5Sqrt[6(5-Sqrt[5])]-3Sqrt[30(5-Sqrt[5])]-3Sqrt[6(5+Sqrt[5])]+Sqrt[30(5+Sqrt[5])])/32,(64Sqrt[3]+15Sqrt[2(5-Sqrt[5])]+9Sqrt[10(5-Sqrt[5])]+21Sqrt[2(5+Sqrt[5])]+9Sqrt[10(5+Sqrt[5])])/96},{0,-(1/Sqrt[3])},{(18-2Sqrt[5]-Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])])/16,(2Sqrt[3]+6Sqrt[15]-3Sqrt[2(5-Sqrt[5])]-3Sqrt[10(5-Sqrt[5])]+6Sqrt[2(5+Sqrt[5])]+6Sqrt[10(5+Sqrt[5])])/48},{1/2,1/(2Sqrt[3])},{(26-2Sqrt[5]-Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])])/16,(-22Sqrt[3]+6Sqrt[15]-3Sqrt[2(5-Sqrt[5])]-3Sqrt[10(5-Sqrt[5])]+6Sqrt[2(5+Sqrt[5])]+6Sqrt[10(5+Sqrt[5])])/48},{(26-2Sqrt[5]-Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])])/16,(2Sqrt[3]+6Sqrt[15]+15Sqrt[2(5-Sqrt[5])]+3Sqrt[10(5-Sqrt[5])]+6Sqrt[2(5+Sqrt[5])]+6Sqrt[10(5+Sqrt[5])])/48},{(52-4Sqrt[5]-3Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])]-3Sqrt[6(5+Sqrt[5])]+Sqrt[30(5+Sqrt[5])])/32,(52Sqrt[3]+12Sqrt[15]+3Sqrt[2(5-Sqrt[5])]+9Sqrt[10(5-Sqrt[5])]+27Sqrt[2(5+Sqrt[5])]+15Sqrt[10(5+Sqrt[5])])/96},{(44+4Sqrt[5]-3Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])]-3Sqrt[6(5+Sqrt[5])]+Sqrt[30(5+Sqrt[5])])/32,(52Sqrt[3]+12Sqrt[15]+9Sqrt[2(5-Sqrt[5])]+3Sqrt[10(5-Sqrt[5])]+21Sqrt[2(5+Sqrt[5])]+9Sqrt[10(5+Sqrt[5])])/96},{(4-Sqrt[5])/2,(-8Sqrt[3]-3Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/24},{1,-(1/Sqrt[3])},{(34-2Sqrt[5]-Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])])/16,(2Sqrt[3]+6Sqrt[15]-3Sqrt[2(5-Sqrt[5])]-3Sqrt[10(5-Sqrt[5])]+6Sqrt[2(5+Sqrt[5])]+6Sqrt[10(5+Sqrt[5])])/48},{(7-Sqrt[5])/4,(-8Sqrt[3]-9Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/24},{(7-Sqrt[5])/4,(-8Sqrt[3]+3Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/24},{(7-Sqrt[5])/4,(2Sqrt[3]+3Sqrt[2(5+Sqrt[5])])/12},{(30+2Sqrt[5]-Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])])/16,(2Sqrt[3]+6Sqrt[15]+6Sqrt[2(5-Sqrt[5])]+15Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/48},{(60+4Sqrt[5]-3Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])]-3Sqrt[6(5+Sqrt[5])]+Sqrt[30(5+Sqrt[5])])/32,(52Sqrt[3]+12Sqrt[15]+27Sqrt[2(5-Sqrt[5])]+9Sqrt[10(5-Sqrt[5])]+27Sqrt[2(5+Sqrt[5])]+15Sqrt[10(5+Sqrt[5])])/96},{3/2,1/(2Sqrt[3])},{(76+4Sqrt[5]-Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(52Sqrt[3]+12Sqrt[15]+3Sqrt[2(5-Sqrt[5])]+3Sqrt[10(5-Sqrt[5])]+18Sqrt[2(5+Sqrt[5])]+12Sqrt[10(5+Sqrt[5])])/96},{2,(-4Sqrt[3]-3Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/12},{2,-(1/Sqrt[3])},{2,(4Sqrt[3]+3Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/24},{(68+12Sqrt[5]-3Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])]-3Sqrt[6(5+Sqrt[5])]+Sqrt[30(5+Sqrt[5])])/32,(52Sqrt[3]+12Sqrt[15]+21Sqrt[2(5-Sqrt[5])]+15Sqrt[10(5-Sqrt[5])]+21Sqrt[2(5+Sqrt[5])]+9Sqrt[10(5+Sqrt[5])])/96},{(84+4Sqrt[5]-3Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(52Sqrt[3]+36Sqrt[15]+21Sqrt[2(5-Sqrt[5])]+9Sqrt[10(5-Sqrt[5])]+18Sqrt[2(5+Sqrt[5])]+12Sqrt[10(5+Sqrt[5])])/96},{5/2,1/(2Sqrt[3])},{(80+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(40Sqrt[3]+24Sqrt[15]+9Sqrt[2(5-Sqrt[5])]+9Sqrt[10(5-Sqrt[5])]+18Sqrt[2(5+Sqrt[5])]+12Sqrt[10(5+Sqrt[5])])/96},{(9+Sqrt[5])/4,(2Sqrt[3]+3Sqrt[2(5+Sqrt[5])])/12},{(72+8Sqrt[5]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(40Sqrt[3]+24Sqrt[15]+3Sqrt[2(5-Sqrt[5])]+3Sqrt[10(5-Sqrt[5])]+6Sqrt[2(5+Sqrt[5])]+12Sqrt[10(5+Sqrt[5])])/96},{(17+Sqrt[5]+Sqrt[6(5-Sqrt[5])])/8,(7Sqrt[3]+3Sqrt[15]-3Sqrt[2(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+3Sqrt[10(5+Sqrt[5])])/24},{3,(-4Sqrt[3]-3Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/12},{3,-(1/Sqrt[3])},{(60+12Sqrt[5]+5Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(28Sqrt[3]+12Sqrt[15]-27Sqrt[2(5-Sqrt[5])]-15Sqrt[10(5-Sqrt[5])]+18Sqrt[2(5+Sqrt[5])]+12Sqrt[10(5+Sqrt[5])])/96},{(88+8Sqrt[5]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(40Sqrt[3]+24Sqrt[15]+15Sqrt[2(5-Sqrt[5])]+3Sqrt[10(5-Sqrt[5])]+24Sqrt[2(5+Sqrt[5])]+18Sqrt[10(5+Sqrt[5])])/96},{(38+2Sqrt[5]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])])/16,(14Sqrt[3]-6Sqrt[15]-3Sqrt[2(5-Sqrt[5])]-3Sqrt[10(5-Sqrt[5])]+12Sqrt[2(5+Sqrt[5])])/48},{7/2,1/(2Sqrt[3])},{(76+12Sqrt[5]+5Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(28Sqrt[3]+12Sqrt[15]+9Sqrt[2(5-Sqrt[5])]-3Sqrt[10(5-Sqrt[5])]+18Sqrt[2(5+Sqrt[5])]+12Sqrt[10(5+Sqrt[5])])/96},{(13+Sqrt[5])/4,(-8Sqrt[3]-9Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/24},{(13+Sqrt[5])/4,(-8Sqrt[3]+3Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/24},{(104+8Sqrt[5]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(40Sqrt[3]+24Sqrt[15]-3Sqrt[2(5-Sqrt[5])]-3Sqrt[10(5-Sqrt[5])]+18Sqrt[2(5+Sqrt[5])]+12Sqrt[10(5+Sqrt[5])])/96},{4,-(1/Sqrt[3])},{(6+Sqrt[5])/2,(-8Sqrt[3]-3Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/24},{(96+16Sqrt[5]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(40Sqrt[3]+24Sqrt[15]-9Sqrt[2(5-Sqrt[5])]+3Sqrt[10(5-Sqrt[5])]+24Sqrt[2(5+Sqrt[5])]+18Sqrt[10(5+Sqrt[5])])/96},{(92+12Sqrt[5]+5Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(28Sqrt[3]+12Sqrt[15]-15Sqrt[2(5-Sqrt[5])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NetVertices[JohnsonSolid[28]]:={{-5/2,-1/2},{-5/2,1/2},{(-3-Sqrt[3])/2,1},{-2,(-1-Sqrt[3])/2},{-3/2,-3/2},{-3/2,-1/2},{-3/2,1/2},{-3/2,3/2},{-1/2,-3/2},{-1/2,-1/2},{-1/2,1/2},{-1/2,3/2},{0,(1+Sqrt[3])/2},{(-1+Sqrt[3])/2,-1},{1/2,-1/2},{1/2,1/2},{(3-Sqrt[3])/2,1},{1,(-1-Sqrt[3])/2},{3/2,-3/2},{3/2,-1/2},{3/2,1/2},{3/2,3/2},{5/2,-3/2},{5/2,-1/2},{5/2,1/2},{5/2,3/2},{3,(1+Sqrt[3])/2},{(5+Sqrt[3])/2,-1},{7/2,-1/2},{7/2,1/2}}
NetVertices[JohnsonSolid[29]]:={{-5/2,-1/2},{-5/2,1/2},{(-3-Sqrt[3])/2,1},{-2,(-1-Sqrt[3])/2},{-3/2,-3/2},{-3/2,-1/2},{-3/2,1/2},{-3/2,3/2},{-1/2,-3/2},{-1/2,-1/2},{-1/2,1/2},{-1/2,3/2},{0,(1+Sqrt[3])/2},{(-1+Sqrt[3])/2,-1},{1/2,-1/2},{1/2,1/2},{Sqrt[3]/2,-Sqrt[3]/2},{1,(1+Sqrt[3])/2},{(1+Sqrt[3])/2,0},{(1+Sqrt[3])/2,Sqrt[3]},{Sqrt[3],(-1-Sqrt[3])/2},{(2+Sqrt[3])/2,Sqrt[3]/2},{(1+2Sqrt[3])/2,-1/2},{(3+Sqrt[3])/2,Sqrt[3]},{1+Sqrt[3],(-1-Sqrt[3])/2},{1+Sqrt[3],(-1+Sqrt[3])/2},{(1+3Sqrt[3])/2,-1},{(3+2Sqrt[3])/2,(-1+2Sqrt[3])/2},{(2+3Sqrt[3])/2,(-2+Sqrt[3])/2},{(2+3Sqrt[3])/2,Sqrt[3]/2}}
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NetVertices[JohnsonSolid[37]]:={{-1/2,-1/2},{-1/2,1/2},{1/2,-1/2},{1/2,1/2},{3/2,-1/2},{3/2,1/2},{3/2,3/2},{3/2,5/2},{(5-Sqrt[3])/2,3},{2,(3-Sqrt[3])/2},{5/2,-1/2},{5/2,1/2},{5/2,3/2},{5/2,5/2},{5/2,7/2},{7/2,-1/2},{7/2,1/2},{7/2,3/2},{7/2,5/2},{7/2,7/2},{4,(5+Sqrt[3])/2},{(7+Sqrt[3])/2,1},{9/2,-1/2},{9/2,1/2},{9/2,3/2},{9/2,5/2},{11/2,-1/2},{11/2,1/2},{13/2,-1/2},{13/2,1/2},{15/2,-1/2},{15/2,1/2},{2,(-1-3Sqrt[3])/2},{(6-Sqrt[3])/2,-Sqrt[3]/2},{5/2,(-1-2Sqrt[3])/2},{(5-Sqrt[3])/2,-1-Sqrt[3]},{(4+Sqrt[3])/2,(-2-3Sqrt[3])/2},{3,(-1-Sqrt[3])/2},{(5+Sqrt[3])/2,-1-Sqrt[3]},{(5-Sqrt[3])/2,-Sqrt[3]},{(6+Sqrt[3])/2,(-2-3Sqrt[3])/2},{(6+Sqrt[3])/2,(-2-Sqrt[3])/2},{(5+2Sqrt[3])/2,(-3-2Sqrt[3])/2},{(7+Sqrt[3])/2,-1},{3+Sqrt[3],(-3-Sqrt[3])/2},{3+Sqrt[3],(-1-Sqrt[3])/2}}
NetVertices[JohnsonSolid[38]]:={{-1/2,-1/2},{-1/2,1/2},{1/2,-1/2},{1/2,1/2},{3/2,-1/2},{3/2,1/2},{(30-Sqrt[3]-2Sqrt[5]-Sqrt[15]-Sqrt[2(5-Sqrt[5])])/8,(26+2Sqrt[5]+Sqrt[2(5-Sqrt[5])]-2Sqrt[6(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])]+2Sqrt[2(5+Sqrt[5])])/16},{(30-2Sqrt[5]-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])])/8,(-28+4Sqrt[5]+2Sqrt[2(5-Sqrt[5])]-3Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/16},{(30-2Sqrt[5]-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])])/8,(14-2Sqrt[5]+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])])/8},{(60+2Sqrt[3]-4Sqrt[5]-2Sqrt[15]-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])])/16,(-26+2Sqrt[5]+2Sqrt[2(5-Sqrt[5])]-Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])]-3Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/16},{5/2,-1/2},{5/2,1/2},{(28-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])])/8,(-28+4Sqrt[5]+4Sqrt[2(5-Sqrt[5])]+2Sqrt[10(5-Sqrt[5])]-3Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/16},{(28-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])])/8,(7-Sqrt[5])/4},{(15-Sqrt[5]-Sqrt[2(5-Sqrt[5])])/4,(-28-4Sqrt[5]+2Sqrt[2(5-Sqrt[5])]-3Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/16},{(15-Sqrt[5]-Sqrt[2(5-Sqrt[5])])/4,(28+4Sqrt[5]+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])]+2Sqrt[2(5+Sqrt[5])])/16},{(7-Sqrt[3])/2,-1},{(15-Sqrt[5])/4,(-24+2Sqrt[2(5-Sqrt[5])]-3Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/16},{(15-Sqrt[5])/4,(12+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])])/8},{(56-2Sqrt[3]+2Sqrt[15]-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])])/16,(26-2Sqrt[5]-Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])])/16},{(16-Sqrt[2(5-Sqrt[5])])/4,(-28-4Sqrt[5]-2Sqrt[2(5-Sqrt[5])]-3Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/16},{(16-Sqrt[2(5-Sqrt[5])])/4,(28+4Sqrt[5]+5Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])]+2Sqrt[2(5+Sqrt[5])])/16},{7/2,-3/2},{7/2,-1/2},{7/2,1/2},{7/2,3/2},{(32-Sqrt[3]-Sqrt[15]+Sqrt[2(5-Sqrt[5])])/8,(26+2Sqrt[5]+2Sqrt[2(5-Sqrt[5])]+2Sqrt[6(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/16},{4,(-24-2Sqrt[2(5-Sqrt[5])]-3Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/16},{4,(24+2Sqrt[2(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/16},{(32+Sqrt[3]+Sqrt[15]-Sqrt[2(5-Sqrt[5])])/8,(-26-2Sqrt[5]-2Sqrt[2(5-Sqrt[5])]-2Sqrt[6(5-Sqrt[5])]-3Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/16},{9/2,-3/2},{9/2,-1/2},{9/2,1/2},{9/2,3/2},{(16+Sqrt[2(5-Sqrt[5])])/4,(-28-4Sqrt[5]-5Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])]-2Sqrt[2(5+Sqrt[5])])/16},{(16+Sqrt[2(5-Sqrt[5])])/4,(28+4Sqrt[5]+2Sqrt[2(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/16},{(72+2Sqrt[3]-2Sqrt[15]+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])])/16,(-26+2Sqrt[5]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])])/16},{(17+Sqrt[5])/4,(-12-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])])/8},{(17+Sqrt[5])/4,(24-2Sqrt[2(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/16},{(9+Sqrt[3])/2,1},{(17+Sqrt[5]+Sqrt[2(5-Sqrt[5])])/4,(-28-4Sqrt[5]-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])]-2Sqrt[2(5+Sqrt[5])])/16},{(17+Sqrt[5]+Sqrt[2(5-Sqrt[5])])/4,(28+4Sqrt[5]-2Sqrt[2(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/16},{(36+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])])/8,(-7+Sqrt[5])/4},{(36+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])])/8,(28-4Sqrt[5]-4Sqrt[2(5-Sqrt[5])]-2Sqrt[10(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/16},{11/2,-1/2},{11/2,1/2},{(68-2Sqrt[3]+4Sqrt[5]+2Sqrt[15]+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])])/16,(26-2Sqrt[5]-2Sqrt[2(5-Sqrt[5])]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/16},{(34+2Sqrt[5]+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])])/8,(-14+2Sqrt[5]-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])])/8},{(34+2Sqrt[5]+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])])/8,(28-4Sqrt[5]-2Sqrt[2(5-Sqrt[5])]+3Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/16},{(34+Sqrt[3]+2Sqrt[5]+Sqrt[15]+Sqrt[2(5-Sqrt[5])])/8,(-26-2Sqrt[5]-Sqrt[2(5-Sqrt[5])]+2Sqrt[6(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])]-2Sqrt[2(5+Sqrt[5])])/16},{13/2,-1/2},{13/2,1/2},{15/2,-1/2},{15/2,1/2},{17/2,-1/2},{17/2,1/2},{19/2,-1/2},{19/2,1/2}}
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Sqrt[5]+3Sqrt[6(5-Sqrt[5])]-Sqrt[30(5-Sqrt[5])]+2Sqrt[6(5+Sqrt[5])])/32,(-16-12Sqrt[3]-12Sqrt[15]-7Sqrt[2(5-Sqrt[5])]-3Sqrt[10(5-Sqrt[5])]-6Sqrt[2(5+Sqrt[5])]-4Sqrt[10(5+Sqrt[5])])/32},{(17+Sqrt[5])/4,(2+Sqrt[2(5+Sqrt[5])])/4},{(136+8Sqrt[5]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(16+8Sqrt[3]+8Sqrt[15]+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])]+2Sqrt[2(5+Sqrt[5])]+4Sqrt[10(5+Sqrt[5])])/32},{(156-12Sqrt[5]+3Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]+3Sqrt[6(5+Sqrt[5])]-Sqrt[30(5+Sqrt[5])])/32,(-16-12Sqrt[3]-4Sqrt[15]-7Sqrt[2(5-Sqrt[5])]-5Sqrt[10(5-Sqrt[5])]-7Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/32},{(33+Sqrt[5]+Sqrt[6(5-Sqrt[5])])/8,(4+Sqrt[3]+Sqrt[15]-Sqrt[2(5-Sqrt[5])]+Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/8},{5,(-4-Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/8},{(148-4Sqrt[5]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]+2Sqrt[6(5+Sqrt[5])])/32,(-16-12Sqrt[3]-4Sqrt[15]-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])]-6Sqrt[2(5+Sqrt[5])]-4Sqrt[10(5+Sqrt[5])])/32},{(124+12Sqrt[5]+5Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(16+4Sqrt[3]+4Sqrt[15]-9Sqrt[2(5-Sqrt[5])]-5Sqrt[10(5-Sqrt[5])]+6Sqrt[2(5+Sqrt[5])]+4Sqrt[10(5+Sqrt[5])])/32},{(152+8Sqrt[5]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(16+8Sqrt[3]+8Sqrt[15]+5Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])]+8Sqrt[2(5+Sqrt[5])]+6Sqrt[10(5+Sqrt[5])])/32},{(70+2Sqrt[5]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])])/16,(8+2Sqrt[3]-2Sqrt[15]-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])]+4Sqrt[2(5+Sqrt[5])])/16},{11/2,-1/2},{11/2,1/2},{(164-4Sqrt[5]+3Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]+3Sqrt[6(5+Sqrt[5])]-Sqrt[30(5+Sqrt[5])])/32,(-16-12Sqrt[3]-4Sqrt[15]-9Sqrt[2(5-Sqrt[5])]-3Sqrt[10(5-Sqrt[5])]-9Sqrt[2(5+Sqrt[5])]-5Sqrt[10(5+Sqrt[5])])/32},{(82-2Sqrt[5]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])])/16,(-8+2Sqrt[3]-2Sqrt[15]-2Sqrt[2(5-Sqrt[5])]-5Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/16},{(140+12Sqrt[5]+5Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(16+4Sqrt[3]+4Sqrt[15]+3Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])]+6Sqrt[2(5+Sqrt[5])]+4Sqrt[10(5+Sqrt[5])])/32},{(21+Sqrt[5])/4,(-2-Sqrt[2(5+Sqrt[5])])/4},{(168+8Sqrt[5]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(16+8Sqrt[3]+8Sqrt[15]-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])]+6Sqrt[2(5+Sqrt[5])]+4Sqrt[10(5+Sqrt[5])])/32},{(78+2Sqrt[5]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])])/16,(-8+2Sqrt[3]-2Sqrt[15]+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])]-2Sqrt[2(5+Sqrt[5])]-2Sqrt[10(5+Sqrt[5])])/16},{(160+16Sqrt[5]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(16+8Sqrt[3]+8Sqrt[15]-3Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])]+8Sqrt[2(5+Sqrt[5])]+6Sqrt[10(5+Sqrt[5])])/32},{(180-4Sqrt[5]+3Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]+3Sqrt[6(5+Sqrt[5])]-Sqrt[30(5+Sqrt[5])])/32,(-16-12Sqrt[3]-4Sqrt[15]-3Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])]-7Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/32},{(156+12Sqrt[5]+5Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(16+4Sqrt[3]+4Sqrt[15]-5Sqrt[2(5-Sqrt[5])]-5Sqrt[10(5-Sqrt[5])]+8Sqrt[2(5+Sqrt[5])]+2Sqrt[10(5+Sqrt[5])])/32},{(172+4Sqrt[5]+3Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]+3Sqrt[6(5+Sqrt[5])]-Sqrt[30(5+Sqrt[5])])/32,(-16-12Sqrt[3]-4Sqrt[15]-Sqrt[2(5-Sqrt[5])]-3Sqrt[10(5-Sqrt[5])]-9Sqrt[2(5+Sqrt[5])]-5Sqrt[10(5+Sqrt[5])])/32},{(86+2Sqrt[5]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])])/16,(-8+2Sqrt[3]-2Sqrt[15]-5Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])]-2Sqrt[2(5+Sqrt[5])]-2Sqrt[10(5+Sqrt[5])])/16},{(86+2Sqrt[5]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])])/16,(-8+10Sqrt[3]-2Sqrt[15]+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])]-2Sqrt[2(5+Sqrt[5])]-2Sqrt[10(5+Sqrt[5])])/16},{13/2,-1/2},{13/2,1/2},{(148+20Sqrt[5]+5Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(16+4Sqrt[3]+4Sqrt[15]-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])]+8Sqrt[2(5+Sqrt[5])]+2Sqrt[10(5+Sqrt[5])])/32},{(144+16Sqrt[5]+9Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-2Sqrt[6(5+Sqrt[5])])/32,(16+8Sqrt[3]+8Sqrt[15]-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])]+6Sqrt[2(5+Sqrt[5])]+4Sqrt[10(5+Sqrt[5])])/32},{(94+2Sqrt[5]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])])/16,(-8+2Sqrt[3]-2Sqrt[15]+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])]-2Sqrt[2(5+Sqrt[5])]-2Sqrt[10(5+Sqrt[5])])/16},{(176+5Sqrt[6(5-Sqrt[5])]+3Sqrt[30(5-Sqrt[5])]+3Sqrt[6(5+Sqrt[5])]-Sqrt[30(5+Sqrt[5])])/32,(-16-16Sqrt[3]-5Sqrt[2(5-Sqrt[5])]-3Sqrt[10(5-Sqrt[5])]-7Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/32},{(90+6Sqrt[5]+Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])])/16,(-8+2Sqrt[3]-2Sqrt[15]-3Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])]-3Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/16},{15/2,-1/2},{15/2,1/2},{17/2,-1/2},{17/2,1/2},{19/2,-1/2},{19/2,1/2}}
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NetVertices[JohnsonSolid[50]]:=Sqrt[2/5]{{(-2-3Sqrt[3])/2,(1-Sqrt[3])/2},{(-5-3Sqrt[3])/4,(5-Sqrt[3])/4},{(-3(1+Sqrt[3]))/4,(-1-Sqrt[3])/4},{(-1-3Sqrt[3])/4,(-7-Sqrt[3])/4},{-1,1/2},{(-1-Sqrt[3])/4,(3(1+Sqrt[3]))/4},{-1/2,-1},{1/2,1},{(1+Sqrt[3])/4,(-3(1+Sqrt[3]))/4},{1,-1/2},{(1+3Sqrt[3])/4,(7+Sqrt[3])/4},{(3(1+Sqrt[3]))/4,(1+Sqrt[3])/4},{(5+3Sqrt[3])/4,(-5+Sqrt[3])/4},{(2+3Sqrt[3])/2,(-1+Sqrt[3])/2}}
NetVertices[JohnsonSolid[51]]:={{-4,2/Sqrt[3]},{-7/2,1/(2Sqrt[3])},{-3,2/Sqrt[3]},{-5/2,1/(2Sqrt[3])},{-5/2,7/(2Sqrt[3])},{-2,-(1/Sqrt[3])},{-2,2/Sqrt[3]},{-3/2,1/(2Sqrt[3])},{-3/2,7/(2Sqrt[3])},{-1,-(1/Sqrt[3])},{-1,2/Sqrt[3]},{-1,5/Sqrt[3]},{-1/2,1/(2Sqrt[3])},{-1/2,7/(2Sqrt[3])},{0,-(1/Sqrt[3])},{1/2,1/(2Sqrt[3])}}
NetVertices[JohnsonSolid[52]]:={{(-10-10Sqrt[5]-10Sqrt[2(5-Sqrt[5])]-5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,-Sqrt[(5-Sqrt[5])/2]/2},{(-5-5Sqrt[5]-2Sqrt[10(5+Sqrt[5])])/20,Sqrt[(5-Sqrt[5])/2]/2},{(-10-10Sqrt[5]-5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,(-1-Sqrt[5]-Sqrt[2(5-Sqrt[5])])/4},{(-5-5Sqrt[5]+5Sqrt[2(5-Sqrt[5])]-2Sqrt[10(5+Sqrt[5])])/20,(1+Sqrt[5]+Sqrt[2(5-Sqrt[5])])/4},{-Sqrt[(5+Sqrt[5])/10],0},{((-5+Sqrt[5])Sqrt[(5+Sqrt[5])/2])/20,(-1-Sqrt[5])/4},{((-5+Sqrt[5])Sqrt[(5+Sqrt[5])/2])/20,(1+Sqrt[5])/4},{(-10+10Sqrt[5]-5Sqrt[2(5-Sqrt[5])]-5Sqrt[10(5-Sqrt[5])]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,(-2-2Sqrt[5]-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])])/8},{(-10+10Sqrt[5]-5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,(2+2Sqrt[5]+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])])/8},{(-20+20Sqrt[5]-5Sqrt[2(5-Sqrt[5])]-5Sqrt[10(5-Sqrt[5])]+2Sqrt[10(5+Sqrt[5])])/80,(10+2Sqrt[5]+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])])/8},{(5+Sqrt[5])^(3/2)/(20Sqrt[2]),-1/2},{(5+Sqrt[5])^(3/2)/(20Sqrt[2]),1/2},{(-10+10Sqrt[5]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,(-4-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])])/8},{(-20+20Sqrt[5]+5Sqrt[2(5-Sqrt[5])]+5Sqrt[10(5-Sqrt[5])]+2Sqrt[10(5+Sqrt[5])])/80,(4+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])])/8},{(-20+20Sqrt[5]-10Sqrt[2(5-Sqrt[5])]+5Sqrt[2(5+Sqrt[5])]+7Sqrt[10(5+Sqrt[5])])/80,(8+4Sqrt[5]+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])])/8},{(20Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,-1},{(20Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,0},{(20Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,1},{(-20+20Sqrt[5]+10Sqrt[2(5-Sqrt[5])]+5Sqrt[2(5+Sqrt[5])]+7Sqrt[10(5+Sqrt[5])])/80,(6+2Sqrt[5]+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])])/8},{(40Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,1/2}}
NetVertices[JohnsonSolid[53]]:={{(-5Sqrt[3]-5Sqrt[15]-2Sqrt[10(5+Sqrt[5])])/20,Sqrt[(3(5-Sqrt[5]))/2]/2},{(-5Sqrt[3]-5Sqrt[15]-5Sqrt[2(5-Sqrt[5])]-4Sqrt[10(5+Sqrt[5])])/40,(-1-Sqrt[5]+Sqrt[6(5-Sqrt[5])])/8},{(-10-10Sqrt[5]-10Sqrt[2(5-Sqrt[5])]-5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,-Sqrt[(5-Sqrt[5])/2]/2},{(-5Sqrt[3]-5Sqrt[15]+5Sqrt[2(5-Sqrt[5])]-4Sqrt[10(5+Sqrt[5])])/40,(1+Sqrt[5]+Sqrt[6(5-Sqrt[5])])/8},{(-10-10Sqrt[5]-5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,(-1-Sqrt[5]-Sqrt[2(5-Sqrt[5])])/4},{-Sqrt[(5+Sqrt[5])/10],0},{(-5Sqrt[3]-5Sqrt[15]+15Sqrt[2(5-Sqrt[5])]-4Sqrt[10(5+Sqrt[5])])/40,(3+3Sqrt[5]+Sqrt[6(5-Sqrt[5])])/8},{((-5+Sqrt[5])Sqrt[(5+Sqrt[5])/2])/20,(-1-Sqrt[5])/4},{((-5+Sqrt[5])Sqrt[(5+Sqrt[5])/2])/20,(1+Sqrt[5])/4},{(-10+10Sqrt[5]-5Sqrt[2(5-Sqrt[5])]-5Sqrt[10(5-Sqrt[5])]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,(-2-2Sqrt[5]-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])])/8},{(-10+10Sqrt[5]-5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,(2+2Sqrt[5]+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])])/8},{(-20+20Sqrt[5]-5Sqrt[2(5-Sqrt[5])]-5Sqrt[10(5-Sqrt[5])]+2Sqrt[10(5+Sqrt[5])])/80,(10+2Sqrt[5]+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])])/8},{(5+Sqrt[5])^(3/2)/(20Sqrt[2]),-1/2},{(5+Sqrt[5])^(3/2)/(20Sqrt[2]),1/2},{(-10+10Sqrt[5]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,(-4-Sqrt[2(5-Sqrt[5])]-Sqrt[10(5-Sqrt[5])])/8},{(-20+20Sqrt[5]+5Sqrt[2(5-Sqrt[5])]+5Sqrt[10(5-Sqrt[5])]+2Sqrt[10(5+Sqrt[5])])/80,(4+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])])/8},{(-20+20Sqrt[5]-10Sqrt[2(5-Sqrt[5])]+5Sqrt[2(5+Sqrt[5])]+7Sqrt[10(5+Sqrt[5])])/80,(8+4Sqrt[5]+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])])/8},{(20Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,-1},{(20Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,0},{(20Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,1},{(-20+20Sqrt[5]+10Sqrt[2(5-Sqrt[5])]+5Sqrt[2(5+Sqrt[5])]+7Sqrt[10(5+Sqrt[5])])/80,(6+2Sqrt[5]+Sqrt[2(5-Sqrt[5])]+Sqrt[10(5-Sqrt[5])])/8},{(40Sqrt[3]+5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,1/2}}
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NetVertices[JohnsonSolid[55]]:={{(-2-Sqrt[3])/2,-1/2},{(-2-Sqrt[3])/2,1/2},{-Sqrt[3],1},{-Sqrt[3],2},{(-1-Sqrt[3])/2,(-1-Sqrt[3])/2},{-Sqrt[3]/2,-1/2},{-Sqrt[3]/2,1/2},{-Sqrt[3]/2,3/2},{-1/2,(-2-Sqrt[3])/2},{0,-2},{0,-1},{0,1},{0,2},{1/2,(2+Sqrt[3])/2},{Sqrt[3]/2,-3/2},{Sqrt[3]/2,-1/2},{Sqrt[3]/2,1/2},{(1+Sqrt[3])/2,(1+Sqrt[3])/2},{Sqrt[3],-2},{Sqrt[3],-1},{(2+Sqrt[3])/2,-1/2},{(2+Sqrt[3])/2,1/2},{1+Sqrt[3],-1},{1+Sqrt[3],1},{(2+3Sqrt[3])/2,-1/2},{(2+3Sqrt[3])/2,1/2}}
NetVertices[JohnsonSolid[56]]:={{(-2-Sqrt[3])/2,-1/2},{(-2-Sqrt[3])/2,1/2},{(-1-Sqrt[3])/2,(-1-Sqrt[3])/2},{(-1-Sqrt[3])/2,(1+Sqrt[3])/2},{-Sqrt[3]/2,-1/2},{-Sqrt[3]/2,1/2},{-1/2,(-2-Sqrt[3])/2},{-1/2,(2+Sqrt[3])/2},{0,-2},{0,-1},{0,1},{0,2},{Sqrt[3]/2,-3/2},{Sqrt[3]/2,-1/2},{Sqrt[3]/2,1/2},{Sqrt[3]/2,3/2},{Sqrt[3]/2,5/2},{Sqrt[3],-2},{Sqrt[3],-1},{Sqrt[3],1},{(2+Sqrt[3])/2,-1/2},{(2+Sqrt[3])/2,1/2},{1+Sqrt[3],-1},{1+Sqrt[3],1},{(2+3Sqrt[3])/2,-1/2},{(2+3Sqrt[3])/2,1/2}}
NetVertices[JohnsonSolid[57]]:={{(-3Sqrt[3])/2,-1/2},{-Sqrt[3],-1},{-Sqrt[3],0},{-Sqrt[3],1},{(-1-Sqrt[3])/2,(-1-Sqrt[3])/2},{(-1-Sqrt[3])/2,(1+Sqrt[3])/2},{-Sqrt[3]/2,-1/2},{-Sqrt[3]/2,1/2},{-1/2,(-2-Sqrt[3])/2},{-1/2,(2+Sqrt[3])/2},{0,-2},{0,-1},{0,1},{0,2},{Sqrt[3]/2,-3/2},{Sqrt[3]/2,-1/2},{Sqrt[3]/2,1/2},{Sqrt[3]/2,3/2},{Sqrt[3]/2,5/2},{Sqrt[3],-2},{Sqrt[3],-1},{Sqrt[3],1},{(2+Sqrt[3])/2,-1/2},{(2+Sqrt[3])/2,1/2},{1+Sqrt[3],-1},{1+Sqrt[3],1},{(2+3Sqrt[3])/2,-1/2},{(2+3Sqrt[3])/2,1/2}}
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NetVertices[JohnsonSolid[91]]:={{(-12Sqrt[2]-4Sqrt[6]-2Sqrt[5-Sqrt[5]]-2Sqrt[3(5-Sqrt[5])]-3Sqrt[5+Sqrt[5]]-3Sqrt[3(5+Sqrt[5])]-Sqrt[5(5+Sqrt[5])]-Sqrt[15(5+Sqrt[5])])/32,(-4Sqrt[2]-4Sqrt[6]-2Sqrt[5-Sqrt[5]]+2Sqrt[3(5-Sqrt[5])]-3Sqrt[5+Sqrt[5]]+3Sqrt[3(5+Sqrt[5])]-Sqrt[5(5+Sqrt[5])]+Sqrt[15(5+Sqrt[5])])/32},{(-10Sqrt[2]-6Sqrt[6]+2Sqrt[10]-2Sqrt[30]-Sqrt[5-Sqrt[5]]-Sqrt[3(5-Sqrt[5])]-Sqrt[5(5-Sqrt[5])]-Sqrt[15(5-Sqrt[5])]-2Sqrt[5+Sqrt[5]]-2Sqrt[3(5+Sqrt[5])])/32,(-6Sqrt[2]-6Sqrt[6]-2Sqrt[10]-2Sqrt[30]-Sqrt[5-Sqrt[5]]+Sqrt[3(5-Sqrt[5])]-Sqrt[5(5-Sqrt[5])]+Sqrt[15(5-Sqrt[5])]-2Sqrt[5+Sqrt[5]]+2Sqrt[3(5+Sqrt[5])])/32},{(-14Sqrt[2]-2Sqrt[6]-2Sqrt[10]+2Sqrt[30]+2Sqrt[5-Sqrt[5]]+2Sqrt[3(5-Sqrt[5])]-3Sqrt[5+Sqrt[5]]-3Sqrt[3(5+Sqrt[5])]-Sqrt[5(5+Sqrt[5])]-Sqrt[15(5+Sqrt[5])])/32,(-2Sqrt[2]-2Sqrt[6]+2Sqrt[10]+2Sqrt[30]+2Sqrt[5-Sqrt[5]]-2Sqrt[3(5-Sqrt[5])]-3Sqrt[5+Sqrt[5]]+3Sqrt[3(5+Sqrt[5])]-Sqrt[5(5+Sqrt[5])]+Sqrt[15(5+Sqrt[5])])/32},{(-6Sqrt[2]-6Sqrt[6]-2Sqrt[10]-2Sqrt[30]+2Sqrt[5-Sqrt[5]]-2Sqrt[3(5-Sqrt[5])]-3Sqrt[5+Sqrt[5]]+3Sqrt[3(5+Sqrt[5])]-Sqrt[5(5+Sqrt[5])]+Sqrt[15(5+Sqrt[5])])/32,(-10Sqrt[2]-6Sqrt[6]+2Sqrt[10]-2Sqrt[30]+2Sqrt[5-Sqrt[5]]+2Sqrt[3(5-Sqrt[5])]-3Sqrt[5+Sqrt[5]]-3Sqrt[3(5+Sqrt[5])]-Sqrt[5(5+Sqrt[5])]-Sqrt[15(5+Sqrt[5])])/32},{(-Sqrt[2]-Sqrt[6])/4,(-Sqrt[2]-Sqrt[6])/4},{(-Sqrt[2]-Sqrt[6])/4,(Sqrt[2]+Sqrt[6])/4},{(4Sqrt[6]-4Sqrt[10]-3Sqrt[5-Sqrt[5]]-5Sqrt[3(5-Sqrt[5])]+Sqrt[5(5-Sqrt[5])]-Sqrt[15(5-Sqrt[5])]+2Sqrt[5+Sqrt[5]]-2Sqrt[3(5+Sqrt[5])])/32,(16Sqrt[2]+4Sqrt[6]+4Sqrt[10]-3Sqrt[5-Sqrt[5]]+5Sqrt[3(5-Sqrt[5])]+Sqrt[5(5-Sqrt[5])]+Sqrt[15(5-Sqrt[5])]+2Sqrt[5+Sqrt[5]]+2Sqrt[3(5+Sqrt[5])])/32},{(-2Sqrt[2]-6Sqrt[6]+2Sqrt[10]-2Sqrt[30]-6Sqrt[5-Sqrt[5]]-2Sqrt[3(5-Sqrt[5])]-3Sqrt[5+Sqrt[5]]+3Sqrt[3(5+Sqrt[5])]-Sqrt[5(5+Sqrt[5])]+Sqrt[15(5+Sqrt[5])])/32,(-14Sqrt[2]-6Sqrt[6]-2Sqrt[10]-2Sqrt[30]-6Sqrt[5-Sqrt[5]]+2Sqrt[3(5-Sqrt[5])]-3Sqrt[5+Sqrt[5]]-3Sqrt[3(5+Sqrt[5])]-Sqrt[5(5+Sqrt[5])]-Sqrt[15(5+Sqrt[5])])/32},{(4Sqrt[2]+4Sqrt[6]-11Sqrt[5-Sqrt[5]]-5Sqrt[3(5-Sqrt[5])]+Sqrt[5(5-Sqrt[5])]-Sqrt[15(5-Sqrt[5])]+2Sqrt[5+Sqrt[5]]-2Sqrt[3(5+Sqrt[5])])/32,(20Sqrt[2]+4Sqrt[6]+8Sqrt[10]+5Sqrt[5-Sqrt[5]]+5Sqrt[3(5-Sqrt[5])]+Sqrt[5(5-Sqrt[5])]+Sqrt[15(5-Sqrt[5])]+2Sqrt[5+Sqrt[5]]+2Sqrt[3(5+Sqrt[5])])/32},{-(1/Sqrt[2]),0},{(2Sqrt[2]+2Sqrt[6]-2Sqrt[10]-2Sqrt[30]+Sqrt[5-Sqrt[5]]-Sqrt[3(5-Sqrt[5])]+Sqrt[5(5-Sqrt[5])]-Sqrt[15(5-Sqrt[5])]+2Sqrt[5+Sqrt[5]]-2Sqrt[3(5+Sqrt[5])])/32,(14Sqrt[2]+2Sqrt[6]+2Sqrt[10]-2Sqrt[30]+Sqrt[5-Sqrt[5]]+Sqrt[3(5-Sqrt[5])]+Sqrt[5(5-Sqrt[5])]+Sqrt[15(5-Sqrt[5])]+2Sqrt[5+Sqrt[5]]+2Sqrt[3(5+Sqrt[5])])/32},{(-4Sqrt[2]-4Sqrt[6]-2Sqrt[5-Sqrt[5]]+2Sqrt[3(5-Sqrt[5])]-3Sqrt[5+Sqrt[5]]+3Sqrt[3(5+Sqrt[5])]-Sqrt[5(5+Sqrt[5])]+Sqrt[15(5+Sqrt[5])])/32,(-12Sqrt[2]-4Sqrt[6]-2Sqrt[5-Sqrt[5]]-2Sqrt[3(5-Sqrt[5])]-3Sqrt[5+Sqrt[5]]-3Sqrt[3(5+Sqrt[5])]-Sqrt[5(5+Sqrt[5])]-Sqrt[15(5+Sqrt[5])])/32},{0,-(1/Sqrt[2])},{0,1/Sqrt[2]},{(4Sqrt[2]+4Sqrt[6]+2Sqrt[5-Sqrt[5]]-2Sqrt[3(5-Sqrt[5])]+3Sqrt[5+Sqrt[5]]-3Sqrt[3(5+Sqrt[5])]+Sqrt[5(5+Sqrt[5])]-Sqrt[15(5+Sqrt[5])])/32,(12Sqrt[2]+4Sqrt[6]+2Sqrt[5-Sqrt[5]]+2Sqrt[3(5-Sqrt[5])]+3Sqrt[5+Sqrt[5]]+3Sqrt[3(5+Sqrt[5])]+Sqrt[5(5+Sqrt[5])]+Sqrt[15(5+Sqrt[5])])/32},{(8Sqrt[2]+4Sqrt[6]+4Sqrt[10]-3Sqrt[5-Sqrt[5]]-5Sqrt[3(5-Sqrt[5])]+Sqrt[5(5-Sqrt[5])]-Sqrt[15(5-Sqrt[5])]+2Sqrt[5+Sqrt[5]]-2Sqrt[3(5+Sqrt[5])])/32,(16Sqrt[2]+4Sqrt[6]+4Sqrt[10]+13Sqrt[5-Sqrt[5]]+5Sqrt[3(5-Sqrt[5])]+Sqrt[5(5-Sqrt[5])]+Sqrt[15(5-Sqrt[5])]+2Sqrt[5+Sqrt[5]]+2Sqrt[3(5+Sqrt[5])])/32},{(-2Sqrt[2]-2Sqrt[6]+2Sqrt[10]+2Sqrt[30]-Sqrt[5-Sqrt[5]]+Sqrt[3(5-Sqrt[5])]-Sqrt[5(5-Sqrt[5])]+Sqrt[15(5-Sqrt[5])]-2Sqrt[5+Sqrt[5]]+2Sqrt[3(5+Sqrt[5])])/32,(-14Sqrt[2]-2Sqrt[6]-2Sqrt[10]+2Sqrt[30]-Sqrt[5-Sqrt[5]]-Sqrt[3(5-Sqrt[5])]-Sqrt[5(5-Sqrt[5])]-Sqrt[15(5-Sqrt[5])]-2Sqrt[5+Sqrt[5]]-2Sqrt[3(5+Sqrt[5])])/32},{1/Sqrt[2],0},{(2Sqrt[2]+6Sqrt[6]-2Sqrt[10]+2Sqrt[30]+6Sqrt[5-Sqrt[5]]+2Sqrt[3(5-Sqrt[5])]+3Sqrt[5+Sqrt[5]]-3Sqrt[3(5+Sqrt[5])]+Sqrt[5(5+Sqrt[5])]-Sqrt[15(5+Sqrt[5])])/32,(14Sqrt[2]+6Sqrt[6]+2Sqrt[10]+2Sqrt[30]+6Sqrt[5-Sqrt[5]]-2Sqrt[3(5-Sqrt[5])]+3Sqrt[5+Sqrt[5]]+3Sqrt[3(5+Sqrt[5])]+Sqrt[5(5+Sqrt[5])]+Sqrt[15(5+Sqrt[5])])/32},{(-4Sqrt[6]+4Sqrt[10]+3Sqrt[5-Sqrt[5]]+5Sqrt[3(5-Sqrt[5])]-Sqrt[5(5-Sqrt[5])]+Sqrt[15(5-Sqrt[5])]-2Sqrt[5+Sqrt[5]]+2Sqrt[3(5+Sqrt[5])])/32,(-16Sqrt[2]-4Sqrt[6]-4Sqrt[10]+3Sqrt[5-Sqrt[5]]-5Sqrt[3(5-Sqrt[5])]-Sqrt[5(5-Sqrt[5])]-Sqrt[15(5-Sqrt[5])]-2Sqrt[5+Sqrt[5]]-2Sqrt[3(5+Sqrt[5])])/32},{(Sqrt[2]+Sqrt[6])/4,(-Sqrt[2]-Sqrt[6])/4},{(Sqrt[2]+Sqrt[6])/4,(Sqrt[2]+Sqrt[6])/4},{(6Sqrt[2]+6Sqrt[6]+2Sqrt[10]+2Sqrt[30]-2Sqrt[5-Sqrt[5]]+2Sqrt[3(5-Sqrt[5])]+3Sqrt[5+Sqrt[5]]-3Sqrt[3(5+Sqrt[5])]+Sqrt[5(5+Sqrt[5])]-Sqrt[15(5+Sqrt[5])])/32,(10Sqrt[2]+6Sqrt[6]-2Sqrt[10]+2Sqrt[30]-2Sqrt[5-Sqrt[5]]-2Sqrt[3(5-Sqrt[5])]+3Sqrt[5+Sqrt[5]]+3Sqrt[3(5+Sqrt[5])]+Sqrt[5(5+Sqrt[5])]+Sqrt[15(5+Sqrt[5])])/32},{(14Sqrt[2]+2Sqrt[6]+2Sqrt[10]-2Sqrt[30]-2Sqrt[5-Sqrt[5]]-2Sqrt[3(5-Sqrt[5])]+3Sqrt[5+Sqrt[5]]+3Sqrt[3(5+Sqrt[5])]+Sqrt[5(5+Sqrt[5])]+Sqrt[15(5+Sqrt[5])])/32,(2Sqrt[2]+2Sqrt[6]-2Sqrt[10]-2Sqrt[30]-2Sqrt[5-Sqrt[5]]+2Sqrt[3(5-Sqrt[5])]+3Sqrt[5+Sqrt[5]]-3Sqrt[3(5+Sqrt[5])]+Sqrt[5(5+Sqrt[5])]-Sqrt[15(5+Sqrt[5])])/32},{(10Sqrt[2]+6Sqrt[6]-2Sqrt[10]+2Sqrt[30]+Sqrt[5-Sqrt[5]]+Sqrt[3(5-Sqrt[5])]+Sqrt[5(5-Sqrt[5])]+Sqrt[15(5-Sqrt[5])]+2Sqrt[5+Sqrt[5]]+2Sqrt[3(5+Sqrt[5])])/32,(6Sqrt[2]+6Sqrt[6]+2Sqrt[10]+2Sqrt[30]+Sqrt[5-Sqrt[5]]-Sqrt[3(5-Sqrt[5])]+Sqrt[5(5-Sqrt[5])]-Sqrt[15(5-Sqrt[5])]+2Sqrt[5+Sqrt[5]]-2Sqrt[3(5+Sqrt[5])])/32},{(12Sqrt[2]+4Sqrt[6]+2Sqrt[5-Sqrt[5]]+2Sqrt[3(5-Sqrt[5])]+3Sqrt[5+Sqrt[5]]+3Sqrt[3(5+Sqrt[5])]+Sqrt[5(5+Sqrt[5])]+Sqrt[15(5+Sqrt[5])])/32,(4Sqrt[2]+4Sqrt[6]+2Sqrt[5-Sqrt[5]]-2Sqrt[3(5-Sqrt[5])]+3Sqrt[5+Sqrt[5]]-3Sqrt[3(5+Sqrt[5])]+Sqrt[5(5+Sqrt[5])]-Sqrt[15(5+Sqrt[5])])/32}}
NetVertices[JohnsonSolid[92]]:={{(-50Sqrt[3]-30Sqrt[15]-50Sqrt[2(5-Sqrt[5])]-15Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/80,(2+6Sqrt[5]-6Sqrt[6(5-Sqrt[5])]+Sqrt[6(5+Sqrt[5])]+Sqrt[30(5+Sqrt[5])])/16},{(-40Sqrt[3]-40Sqrt[15]-50Sqrt[2(5-Sqrt[5])]-5Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/80,(4+4Sqrt[5]-6Sqrt[6(5-Sqrt[5])]-Sqrt[6(5+Sqrt[5])]+Sqrt[30(5+Sqrt[5])])/16},{(-10-20Sqrt[3]-10Sqrt[5]-20Sqrt[15]-10Sqrt[2(5-Sqrt[5])]-5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,-((1+2Sqrt[3])Sqrt[(5-Sqrt[5])/2])/2},{(-20Sqrt[3]-20Sqrt[15]-20Sqrt[2(5-Sqrt[5])]-5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,(1+Sqrt[5]-2Sqrt[6(5-Sqrt[5])])/4},{(-10Sqrt[3]-30Sqrt[15]-50Sqrt[2(5-Sqrt[5])]-15Sqrt[2(5+Sqrt[5])]-3Sqrt[10(5+Sqrt[5])])/80,(10+6Sqrt[5]-6Sqrt[6(5-Sqrt[5])]+Sqrt[6(5+Sqrt[5])]+Sqrt[30(5+Sqrt[5])])/16},{(-10-15Sqrt[3]-10Sqrt[5]-15Sqrt[15]-15Sqrt[2(5-Sqrt[5])]-5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,(1+Sqrt[5]-2Sqrt[2(5-Sqrt[5])]-3Sqrt[6(5-Sqrt[5])])/8},{(-15Sqrt[3]-15Sqrt[15]-25Sqrt[2(5-Sqrt[5])]-5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,(3(1+Sqrt[5]-Sqrt[6(5-Sqrt[5])]))/8},{(-10-15Sqrt[3]-10Sqrt[5]-15Sqrt[15]-5Sqrt[2(5-Sqrt[5])]-5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,(-1-Sqrt[5]-2Sqrt[2(5-Sqrt[5])]-3Sqrt[6(5-Sqrt[5])])/8},{(-10Sqrt[3]-10Sqrt[15]-30Sqrt[2(5-Sqrt[5])]-5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,(2+2Sqrt[5]-Sqrt[6(5-Sqrt[5])])/4},{(-100Sqrt[3]-60Sqrt[15]+10Sqrt[2(5-Sqrt[5])]-35Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/160,(-36-12Sqrt[5]-14Sqrt[6(5-Sqrt[5])]-3Sqrt[6(5+Sqrt[5])]-Sqrt[30(5+Sqrt[5])])/32},{(-100Sqrt[3]-60Sqrt[15]+10Sqrt[2(5-Sqrt[5])]-35Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/160,(-4-12Sqrt[5]-14Sqrt[6(5-Sqrt[5])]-3Sqrt[6(5+Sqrt[5])]-Sqrt[30(5+Sqrt[5])])/32},{(-15Sqrt[3]-15Sqrt[15]-15Sqrt[2(5-Sqrt[5])]-5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,(1+Sqrt[5]-3Sqrt[6(5-Sqrt[5])])/8},{(-20Sqrt[3]-20Sqrt[15]-5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,(-1-Sqrt[5]-2Sqrt[6(5-Sqrt[5])])/4},{(5-10Sqrt[3]+5Sqrt[5]-10Sqrt[15]-20Sqrt[2(5-Sqrt[5])]-5Sqrt[6(5-Sqrt[5])]-5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,(2+Sqrt[3]+2Sqrt[5]+Sqrt[15]+Sqrt[2(5-Sqrt[5])]-2Sqrt[6(5-Sqrt[5])])/8},{(-10Sqrt[3]-10Sqrt[15]-20Sqrt[2(5-Sqrt[5])]-5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,(1+Sqrt[5]-Sqrt[6(5-Sqrt[5])])/4},{(-15Sqrt[3]-15Sqrt[15]-5Sqrt[2(5-Sqrt[5])]-5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,(-1-Sqrt[5]-3Sqrt[6(5-Sqrt[5])])/8},{(5-5Sqrt[3]+5Sqrt[5]-5Sqrt[15]-25Sqrt[2(5-Sqrt[5])]-5Sqrt[6(5-Sqrt[5])]-5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,(3+Sqrt[3]+3Sqrt[5]+Sqrt[15]+Sqrt[2(5-Sqrt[5])]-Sqrt[6(5-Sqrt[5])])/8},{(-20Sqrt[3]-60Sqrt[15]+10Sqrt[2(5-Sqrt[5])]-35Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/160,(-20-12Sqrt[5]-14Sqrt[6(5-Sqrt[5])]-3Sqrt[6(5+Sqrt[5])]-Sqrt[30(5+Sqrt[5])])/32},{(-15Sqrt[3]-15Sqrt[15]+5Sqrt[2(5-Sqrt[5])]-5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,(-3(1+Sqrt[5]+Sqrt[6(5-Sqrt[5])]))/8},{(5-5Sqrt[3]+5Sqrt[5]-5Sqrt[15]-15Sqrt[2(5-Sqrt[5])]-5Sqrt[6(5-Sqrt[5])]-5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,(1+Sqrt[3]+Sqrt[5]+Sqrt[15]+Sqrt[2(5-Sqrt[5])]-Sqrt[6(5-Sqrt[5])])/8},{(-5Sqrt[3]-5Sqrt[15]-15Sqrt[2(5-Sqrt[5])]-5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,(1+Sqrt[5]-Sqrt[6(5-Sqrt[5])])/8},{(-40Sqrt[3]-40Sqrt[15]+15Sqrt[2(5-Sqrt[5])]+5Sqrt[10(5-Sqrt[5])]-25Sqrt[2(5+Sqrt[5])]-Sqrt[10(5+Sqrt[5])])/160,(-16-16Sqrt[5]-13Sqrt[6(5-Sqrt[5])]+Sqrt[30(5-Sqrt[5])]-Sqrt[6(5+Sqrt[5])]-Sqrt[30(5+Sqrt[5])])/32},{(-10Sqrt[3]-10Sqrt[15]-5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,(-1-Sqrt[5]-Sqrt[6(5-Sqrt[5])])/4},{(-20Sqrt[2(5-Sqrt[5])]-5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,(1+Sqrt[5])/4},{(-5Sqrt[3]-5Sqrt[15]-5Sqrt[2(5-Sqrt[5])]-5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,(-1-Sqrt[5]-Sqrt[6(5-Sqrt[5])])/8},{(-10Sqrt[3]-10Sqrt[15]+10Sqrt[2(5-Sqrt[5])]-5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,(-2-2Sqrt[5]-Sqrt[6(5-Sqrt[5])])/4},{-Sqrt[(5+Sqrt[5])/10],0},{(5-10Sqrt[3]+5Sqrt[5]-10Sqrt[15]+5Sqrt[6(5-Sqrt[5])]-5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,(-2-Sqrt[3]-2Sqrt[5]-Sqrt[15]+Sqrt[2(5-Sqrt[5])]-2Sqrt[6(5-Sqrt[5])])/8},{(5-5Sqrt[3]+5Sqrt[5]-5Sqrt[15]-5Sqrt[2(5-Sqrt[5])]+5Sqrt[6(5-Sqrt[5])]-5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,(-1-Sqrt[3]-Sqrt[5]-Sqrt[15]+Sqrt[2(5-Sqrt[5])]-Sqrt[6(5-Sqrt[5])])/8},{((-5+Sqrt[5])Sqrt[(5+Sqrt[5])/2])/20,(-1-Sqrt[5])/4},{((-5+Sqrt[5])Sqrt[(5+Sqrt[5])/2])/20,(1+Sqrt[5])/4},{(5-5Sqrt[3]+5Sqrt[5]-5Sqrt[15]+5Sqrt[2(5-Sqrt[5])]+5Sqrt[6(5-Sqrt[5])]-5Sqrt[2(5+Sqrt[5])]+Sqrt[10(5+Sqrt[5])])/40,(-3-Sqrt[3]-3Sqrt[5]-Sqrt[15]+Sqrt[2(5-Sqrt[5])]-Sqrt[6(5-Sqrt[5])])/8},{(5+Sqrt[5])^(3/2)/(20Sqrt[2]),-1/2},{(5+Sqrt[5])^(3/2)/(20Sqrt[2]),1/2}}

(* Solid Faces *)

PolyhedronFaces[JohnsonSolid[ 1]]:={{3,4,1},{4,2,1},{2,5,1},{5,3,1},{5,2,4,3}}
PolyhedronFaces[JohnsonSolid[ 2]]:={{4,6,1},{6,5,1},{5,3,1},{3,2,1},{2,4,1},{2,3,5,6,4}}
PolyhedronFaces[JohnsonSolid[ 3]]:={{4,3,5},{9,8,1,6,7,2},{3,4,7,6},{5,3,1,8},{4,5,9,2},{2,7,4},{6,1,3},{8,9,5}}
PolyhedronFaces[JohnsonSolid[ 4]]:={{2,3,1,4},{12,11,10,9,8,7,6,5},{3,2,6,7},{1,3,8,9},{4,1,10,11},{2,4,12,5},{5,6,2},{7,8,3},{9,10,1},{11,12,4}}
PolyhedronFaces[JohnsonSolid[ 5]]:={{5,11,10,4,3},{15,14,8,1,6,12,13,7,2,9},{11,5,2,7},{10,11,13,12},{4,10,6,1},{3,4,8,14},{5,3,15,9},{9,2,5},{7,13,11},{12,6,10},{1,8,4},{14,15,3}}
PolyhedronFaces[JohnsonSolid[ 6]]:={{18,16,17,19,20},{15,16,18},{14,17,16},{12,19,17},{11,20,19},{13,18,20},{6,2,13},{4,8,15},{7,3,14},{1,5,12},{9,10,11},{2,4,15,18,13},{8,7,14,16,15},{3,1,12,17,14},{5,9,11,19,12},{10,6,13,20,11},{10,9,5,1,3,7,8,4,2,6}}
PolyhedronFaces[JohnsonSolid[ 7]]:={{5,3,1},{3,7,1},{7,5,1},{4,2,3,5},{2,6,7,3},{6,4,5,7},{6,2,4}}
PolyhedronFaces[JohnsonSolid[ 8]]:={{5,7,1},{7,3,1},{3,9,1},{9,5,1},{4,6,7,5},{6,2,3,7},{2,8,9,3},{8,4,5,9},{8,2,6,4}}
PolyhedronFaces[JohnsonSolid[ 9]]:={{7,11,1},{11,9,1},{9,5,1},{5,3,1},{3,7,1},{6,10,11,7},{10,8,9,11},{8,4,5,9},{4,2,3,5},{2,6,7,3},{2,4,8,10,6}}
PolyhedronFaces[JohnsonSolid[10]]:={{6,1,2,7},{7,5,9},{2,8,5},{1,4,8},{6,9,4},{7,2,5},{2,1,8},{1,6,4},{6,7,9},{5,8,3},{8,4,3},{4,9,3},{9,5,3}}
PolyhedronFaces[JohnsonSolid[11]]:={{10,4,2,5,11},{11,7,3},{5,9,7},{2,8,9},{4,6,8},{10,3,6},{11,5,7},{5,2,9},{2,4,8},{4,10,6},{10,11,3},{7,9,1},{9,8,1},{8,6,1},{6,3,1},{3,7,1}}
PolyhedronFaces[JohnsonSolid[12]]:={{4,3,2},{3,5,2},{5,4,2},{1,3,4},{1,5,3},{1,4,5}}
PolyhedronFaces[JohnsonSolid[13]]:={{5,7,2},{7,6,2},{6,4,2},{4,3,2},{3,5,2},{1,7,5},{1,6,7},{1,4,6},{1,3,4},{1,5,3}}
PolyhedronFaces[JohnsonSolid[14]]:={{6,4,2},{4,8,2},{8,6,2},{1,3,5},{1,7,3},{1,5,7},{5,3,4,6},{3,7,8,4},{7,5,6,8}}
PolyhedronFaces[JohnsonSolid[15]]:={{6,8,2},{8,4,2},{4,10,2},{10,6,2},{1,7,5},{1,3,7},{1,9,3},{1,5,9},{5,7,8,6},{7,3,4,8},{3,9,10,4},{9,5,6,10}}
PolyhedronFaces[JohnsonSolid[16]]:={{8,12,2},{12,10,2},{10,6,2},{6,4,2},{4,8,2},{1,11,7},{1,9,11},{1,5,9},{1,3,5},{1,7,3},{7,11,12,8},{11,9,10,12},{9,5,6,10},{5,3,4,6},{3,7,8,4}}
PolyhedronFaces[JohnsonSolid[17]]:={{10,6,4},{6,9,4},{9,5,4},{5,10,4},{7,1,3},{1,2,3},{2,8,3},{8,7,3},{8,6,10},{2,9,6},{1,5,9},{7,10,5},{8,2,6},{2,1,9},{1,7,5},{7,8,10}}
PolyhedronFaces[JohnsonSolid[18]]:={{5,6,11,9},{7,5,2,13},{6,7,15,4},{4,11,6},{9,2,5},{13,15,7},{6,5,7},{14,12,1,8,10,3},{3,10,11,4},{10,8,9,11},{8,1,2,9},{1,12,13,2},{12,14,15,13},{14,3,4,15}}
PolyhedronFaces[JohnsonSolid[19]]:={{3,2,8,10},{1,3,12,14},{4,1,16,18},{2,4,20,6},{6,8,2},{10,12,3},{14,16,1},{18,20,4},{2,3,1,4},{19,17,15,13,11,9,7,5},{5,7,8,6},{7,9,10,8},{9,11,12,10},{11,13,14,12},{13,15,16,14},{15,17,18,16},{17,19,20,18},{19,5,6,20}}
PolyhedronFaces[JohnsonSolid[20]]:={{17,7,4,11},{16,17,21,19},{6,16,9,2},{5,6,13,23},{7,5,25,15},{15,4,7},{11,21,17},{19,9,16},{2,13,6},{23,25,5},{7,17,16,6,5},{24,22,12,1,8,18,20,10,3,14},{14,3,4,15},{3,10,11,4},{10,20,21,11},{20,18,19,21},{18,8,9,19},{8,1,2,9},{1,12,13,2},{12,22,23,13},{22,24,25,23},{24,14,15,25}}
PolyhedronFaces[JohnsonSolid[21]]:={{28,26,27,29,30},{25,26,28},{24,27,26},{22,29,27},{21,30,29},{23,28,30},{12,4,23},{8,16,25},{14,6,24},{2,10,22},{18,20,21},{4,8,25,28,23},{16,14,24,26,25},{6,2,22,27,24},{10,18,21,29,22},{20,12,23,30,21},{19,17,9,1,5,13,15,7,3,11},{11,3,4,12},{3,7,8,4},{7,15,16,8},{15,13,14,16},{13,5,6,14},{5,1,2,6},{1,9,10,2},{9,17,18,10},{17,19,20,18},{19,11,12,20}}
PolyhedronFaces[JohnsonSolid[22]]:={{3,7,8,6,2,1},{1,2,12},{2,6,4},{6,8,14},{8,7,15},{7,3,5},{3,1,13},{2,4,12},{6,14,4},{8,15,14},{7,5,15},{3,13,5},{1,12,13},{10,9,11},{9,10,13,12},{11,9,4,14},{10,11,15,5},{5,13,10},{12,4,9},{14,15,11}}
PolyhedronFaces[JohnsonSolid[23]]:={{11,4,13,9,12,3,10,8},{8,10,7},{10,3,14},{3,12,16},{12,9,18},{9,13,20},{13,4,22},{4,11,24},{11,8,26},{10,14,7},{3,16,14},{12,18,16},{9,20,18},{13,22,20},{4,24,22},{11,26,24},{8,28,26},{2,5,1,6},{5,2,17,19},{1,5,21,23},{6,1,25,27},{2,6,29,15},{15,17,2},{19,21,5},{23,25,1},{27,29,6}}
PolyhedronFaces[JohnsonSolid[24]]:={{19,15,17,21,7,20,16,14,18,3},{3,18,22},{18,14,8},{14,16,1},{16,20,10},{20,7,24},{7,21,25},{21,17,11},{17,15,2},{15,19,9},{19,3,23},{18,8,22},{14,1,8},{16,10,1},{20,24,10},{7,25,24},{21,11,25},{17,2,11},{15,9,2},{19,23,9},{3,22,23},{6,13,12,5,4},{13,6,2,9},{12,13,23,22},{5,12,8,1},{4,5,10,24},{6,4,25,11},{11,2,6},{9,23,13},{22,8,12},{1,10,5},{24,25,4}}
PolyhedronFaces[JohnsonSolid[25]]:={{14,10,12,16,4,15,11,9,13,3},{3,13,17},{13,9,5},{9,11,1},{11,15,7},{15,4,19},{4,16,20},{16,12,8},{12,10,2},{10,14,6},{14,3,18},{13,5,17},{9,1,5},{11,7,1},{15,19,7},{4,20,19},{16,8,20},{12,2,8},{10,6,2},{14,18,6},{3,17,18},{28,26,27,29,30},{25,26,28},{24,27,26},{22,29,27},{21,30,29},{23,28,30},{8,2,23},{6,18,25},{17,5,24},{1,7,22},{19,20,21},{2,6,25,28,23},{18,17,24,26,25},{5,1,22,27,24},{7,19,21,29,22},{20,8,23,30,21}}
PolyhedronFaces[JohnsonSolid[26]]:={{6,8,7},{2,3,1},{7,2,1,6},{2,7,8,3},{4,6,1},{3,8,5},{1,3,5,4},{8,6,4,5}}
PolyhedronFaces[JohnsonSolid[27]]:={{6,4,8},{4,6,10,9},{8,4,1,11},{6,8,12,2},{2,10,6},{9,1,4},{11,12,8},{7,3,5},{9,10,5,3},{11,1,3,7},{2,12,7,5},{5,10,2},{3,1,9},{7,12,11}}
PolyhedronFaces[JohnsonSolid[28]]:={{4,6,2,8},{6,4,10,11},{2,6,12,13},{8,2,14,15},{4,8,16,9},{9,10,4},{11,12,6},{13,14,2},{15,16,8},{7,1,5,3},{11,10,3,5},{13,12,5,1},{15,14,1,7},{9,16,7,3},{3,10,9},{5,12,11},{1,14,13},{7,16,15}}
PolyhedronFaces[JohnsonSolid[29]]:={{4,7,3,8},{7,4,10,11},{3,7,12,13},{8,3,14,15},{4,8,16,9},{9,10,4},{11,12,7},{13,14,3},{15,16,8},{5,1,2,6},{19,18,6,2},{21,20,2,1},{23,22,1,5},{17,24,5,6},{6,18,17},{2,20,19},{1,22,21},{5,24,23}}
PolyhedronFaces[JohnsonSolid[30]]:={{8,16,14,6,4},{16,8,2,10},{14,16,18,17},{6,14,9,1},{4,6,11,19},{8,4,20,12},{12,2,8},{10,18,16},{17,9,14},{1,11,6},{19,20,4},{3,5,13,15,7},{10,2,7,15},{17,18,15,13},{1,9,13,5},{19,11,5,3},{12,20,3,7},{7,2,12},{15,18,10},{13,9,17},{5,11,1},{3,20,19}}
PolyhedronFaces[JohnsonSolid[31]]:={{8,14,13,7,5},{14,8,2,10},{13,14,19,18},{7,13,9,1},{5,7,11,20},{8,5,21,12},{12,2,8},{10,19,14},{18,9,13},{1,11,7},{20,21,5},{15,24,22,23,25},{4,12,25,23},{26,27,23,22},{6,16,22,24},{28,3,24,15},{17,29,15,25},{25,12,17},{23,27,4},{22,16,26},{24,3,6},{15,29,28}}
PolyhedronFaces[JohnsonSolid[32]]:={{24,22,23,25,26},{21,22,24},{20,23,22},{18,25,23},{17,26,25},{19,24,26},{9,2,19},{7,14,21},{13,6,20},{1,8,18},{15,16,17},{2,7,21,24,19},{14,13,20,22,21},{6,1,18,23,20},{8,15,17,25,18},{16,9,19,26,17},{10,29,27,28,30},{4,9,30,28},{31,32,28,27},{5,11,27,29},{33,3,29,10},{12,34,10,30},{30,9,12},{28,32,4},{27,11,31},{29,3,5},{10,34,33}}
PolyhedronFaces[JohnsonSolid[33]]:={{23,21,22,24,25},{20,21,23},{19,22,21},{17,24,22},{16,25,24},{18,23,25},{9,2,18},{7,13,20},{12,6,19},{1,8,17},{14,15,16},{2,7,20,23,18},{13,12,19,21,20},{6,1,17,22,19},{8,14,16,24,17},{15,9,18,25,16},{3,4,10,11,5},{7,2,5,11},{12,13,11,10},{1,6,10,4},{14,8,4,3},{9,15,3,5},{5,2,9},{11,13,7},{10,6,12},{4,8,1},{3,15,14}}
PolyhedronFaces[JohnsonSolid[34]]:={{26,22,24,28,30},{20,22,26},{18,24,22},{14,28,24},{12,30,28},{16,26,30},{6,2,16},{4,8,20},{7,3,18},{1,5,14},{9,10,12},{2,4,20,26,16},{8,7,18,22,20},{3,1,14,24,18},{5,9,12,28,14},{10,6,16,30,12},{29,27,23,21,25},{25,21,19},{21,23,17},{23,27,13},{27,29,11},{29,25,15},{15,2,6},{19,8,4},{17,3,7},{13,5,1},{11,10,9},{15,25,19,4,2},{19,21,17,7,8},{17,23,13,1,3},{13,27,11,9,5},{11,29,15,6,10}}
PolyhedronFaces[JohnsonSolid[35]]:={{6,8,14,12},{10,6,2,16},{8,10,18,4},{4,14,8},{12,2,6},{16,18,10},{8,6,10},{11,13,7,5},{15,1,5,9},{3,17,9,7},{7,13,3},{5,1,11},{9,17,15},{9,5,7},{3,13,14,4},{13,11,12,14},{11,1,2,12},{1,15,16,2},{15,17,18,16},{17,3,4,18}}
PolyhedronFaces[JohnsonSolid[36]]:={{6,5,8},{5,6,12,10},{8,5,2,14},{6,8,16,4},{4,12,6},{10,2,5},{14,16,8},{7,17,18},{11,3,18,17},{1,9,17,7},{15,13,7,18},{18,3,15},{17,9,11},{7,13,1},{3,11,12,4},{11,9,10,12},{9,1,2,10},{1,13,14,2},{13,15,16,14},{15,3,4,16}}
PolyhedronFaces[JohnsonSolid[37]]:={{4,7,3,8},{7,4,12,14},{3,7,16,18},{8,3,20,22},{4,8,24,10},{10,12,4},{14,16,7},{18,20,3},{22,24,8},{5,1,2,6},{27,26,6,2},{29,28,2,1},{31,30,1,5},{25,32,5,6},{6,26,25},{2,28,27},{1,30,29},{5,32,31},{9,11,12,10},{11,13,14,12},{13,15,16,14},{15,17,18,16},{17,19,20,18},{19,21,22,20},{21,23,24,22},{23,9,10,24}}
PolyhedronFaces[JohnsonSolid[38]]:={{22,10,4,14},{20,22,26,24},{8,20,12,2},{6,8,16,28},{10,6,30,18},{18,4,10},{14,26,22},{24,12,20},{2,16,8},{28,30,6},{10,22,20,8,6},{13,3,9,21},{23,25,21,19},{1,11,19,7},{27,15,7,5},{17,29,5,9},{9,3,17},{21,25,13},{19,11,23},{7,15,1},{5,29,27},{5,7,19,21,9},{17,3,4,18},{3,13,14,4},{13,25,26,14},{25,23,24,26},{23,11,12,24},{11,1,2,12},{1,15,16,2},{15,27,28,16},{27,29,30,28},{29,17,18,30}}
PolyhedronFaces[JohnsonSolid[39]]:={{10,20,19,9,7},{20,10,4,14},{19,20,27,25},{9,19,12,2},{7,9,16,29},{10,7,31,18},{18,4,10},{14,27,20},{25,12,19},{2,16,9},{29,31,7},{21,34,32,33,35},{6,17,35,33},{36,37,33,32},{8,22,32,34},{38,5,34,21},{23,39,21,35},{35,17,23},{33,37,6},{32,22,36},{34,5,8},{21,39,38},{17,3,4,18},{3,13,14,4},{13,26,27,14},{26,24,25,27},{24,11,12,25},{11,1,2,12},{1,15,16,2},{15,28,29,16},{28,30,31,29},{30,17,18,31}}
PolyhedronFaces[JohnsonSolid[40]]:={{34,32,33,35,36},{31,32,34},{30,33,32},{28,35,33},{27,36,35},{29,34,36},{15,4,29},{11,22,31},{20,9,30},{2,13,28},{24,26,27},{4,11,31,34,29},{22,20,30,32,31},{9,2,28,33,30},{13,24,27,35,28},{26,15,29,36,27},{16,39,37,38,40},{6,14,40,38},{41,42,38,37},{7,17,37,39},{43,5,39,16},{18,44,16,40},{40,14,18},{38,42,6},{37,17,41},{39,5,7},{16,44,43},{14,3,4,15},{3,10,11,4},{10,21,22,11},{21,19,20,22},{19,8,9,20},{8,1,2,9},{1,12,13,2},{12,23,24,13},{23,25,26,24},{25,14,15,26}}
PolyhedronFaces[JohnsonSolid[41]]:={{33,31,32,34,35},{30,31,33},{29,32,31},{27,34,32},{26,35,34},{28,33,35},{15,4,28},{11,21,30},{19,9,29},{2,13,27},{23,25,26},{4,11,30,33,28},{21,19,29,31,30},{9,2,27,32,29},{13,23,26,34,27},{25,15,28,35,26},{5,6,16,17,7},{10,3,7,17},{18,20,17,16},{1,8,16,6},{22,12,6,5},{14,24,5,7},{7,3,14},{17,20,10},{16,8,18},{6,12,1},{5,24,22},{14,3,4,15},{3,10,11,4},{10,20,21,11},{20,18,19,21},{18,8,9,19},{8,1,2,9},{1,12,13,2},{12,22,23,13},{22,24,25,23},{24,14,15,25}}
PolyhedronFaces[JohnsonSolid[42]]:={{36,32,34,38,40},{30,32,36},{28,34,32},{24,38,34},{22,40,38},{26,36,40},{12,4,26},{8,16,30},{14,6,28},{2,10,24},{18,20,22},{4,8,30,36,26},{16,14,28,32,30},{6,2,24,34,28},{10,18,22,38,24},{20,12,26,40,22},{39,37,33,31,35},{35,31,29},{31,33,27},{33,37,23},{37,39,21},{39,35,25},{25,3,11},{29,15,7},{27,5,13},{23,9,1},{21,19,17},{25,35,29,7,3},{29,31,27,13,15},{27,33,23,1,5},{23,37,21,17,9},{21,39,25,11,19},{11,3,4,12},{3,7,8,4},{7,15,16,8},{15,13,14,16},{13,5,6,14},{5,1,2,6},{1,9,10,2},{9,17,18,10},{17,19,20,18},{19,11,12,20}}
PolyhedronFaces[JohnsonSolid[43]]:={{34,32,33,36,37},{30,32,34},{29,33,32},{27,36,33},{26,37,36},{28,34,37},{15,4,28},{11,21,30},{19,9,29},{2,13,27},{23,25,26},{4,11,30,34,28},{21,19,29,32,30},{9,2,27,33,29},{13,23,26,36,27},{25,15,28,37,26},{47,46,48,35,49},{49,35,43},{35,48,42},{48,46,44},{46,47,31},{47,49,45},{45,14,17},{43,39,6},{42,16,38},{44,5,7},{31,41,40},{45,49,43,6,14},{43,35,42,38,39},{42,48,44,7,16},{44,46,31,40,5},{31,47,45,17,41},{14,3,4,15},{3,10,11,4},{10,20,21,11},{20,18,19,21},{18,8,9,19},{8,1,2,9},{1,12,13,2},{12,22,23,13},{22,24,25,23},{24,14,15,25}}
PolyhedronFaces[JohnsonSolid[44]]:={{30,17,19},{11,6,17},{26,2,6},{24,4,2},{9,15,4},{28,19,15},{30,11,17},{11,26,6},{26,24,2},{24,9,4},{9,28,15},{28,30,19},{12,7,13},{7,12,5,1},{13,7,3,14},{12,13,18,16},{16,5,12},{1,3,7},{14,18,13},{20,22,21},{29,27,21,22},{25,10,22,20},{8,23,20,21},{21,27,8},{22,10,29},{20,23,25}}
PolyhedronFaces[JohnsonSolid[45]]:={{9,15,11},{17,4,15},{19,13,4},{21,10,13},{23,12,10},{25,3,12},{27,14,3},{29,11,14},{9,17,15},{17,19,4},{19,21,13},{21,23,10},{23,25,12},{25,27,3},{27,29,14},{29,31,11},{33,7,32,8},{7,33,35,36},{32,7,37,38},{8,32,39,40},{33,8,41,34},{34,35,33},{36,37,7},{38,39,32},{40,41,8},{5,2,6,1},{20,18,1,6},{24,22,6,2},{28,26,2,5},{16,30,5,1},{1,18,16},{6,22,20},{2,26,24},{5,30,28}}
PolyhedronFaces[JohnsonSolid[46]]:={{38,30,11},{19,26,30},{4,23,26},{15,28,23},{34,6,28},{32,27,6},{13,22,27},{2,24,22},{17,29,24},{36,11,29},{38,19,30},{19,4,26},{4,15,23},{15,34,28},{34,32,6},{32,13,27},{13,2,22},{2,17,24},{17,36,29},{36,38,11},{40,46,45,5,39},{46,40,25,48},{45,46,42,41},{5,45,47,8},{39,5,49,43},{40,39,44,50},{50,25,40},{48,42,46},{41,47,45},{8,49,5},{43,44,39},{7,10,21,20,9},{16,1,9,20},{37,35,20,21},{3,18,21,10},{33,14,10,7},{12,31,7,9},{9,1,12},{20,35,16},{21,18,37},{10,14,3},{7,31,33}}
PolyhedronFaces[JohnsonSolid[47]]:={{39,31,12},{20,27,31},{4,24,27},{16,29,24},{35,7,29},{33,28,7},{14,23,28},{2,25,23},{18,30,25},{37,12,30},{39,20,31},{20,4,27},{4,16,24},{16,35,29},{35,33,7},{33,14,28},{14,2,23},{2,18,25},{18,37,30},{37,39,12},{8,10,21,22,11},{15,3,11,22},{32,34,22,21},{1,13,21,10},{36,17,10,8},{19,38,8,11},{11,3,19},{22,34,15},{21,13,32},{10,17,1},{8,38,36},{6,42,43,54,55},{53,42,6},{52,43,42},{5,54,43},{40,55,54},{41,6,55},{51,26,41},{49,45,53},{44,48,52},{9,50,5},{46,47,40},{26,49,53,6,41},{45,44,52,42,53},{48,9,5,43,52},{50,46,40,54,5},{47,51,41,55,40}}
PolyhedronFaces[JohnsonSolid[48]]:={{34,26,9},{17,22,26},{4,19,22},{13,24,19},{30,7,24},{28,23,7},{11,18,23},{2,20,18},{15,25,20},{32,9,25},{34,17,26},{17,4,22},{4,13,19},{13,30,24},{30,28,7},{28,11,23},{11,2,18},{2,15,20},{15,32,25},{32,34,9},{6,47,48,59,60},{58,47,6},{57,48,47},{5,59,48},{40,60,59},{41,6,60},{56,21,41},{54,50,58},{49,53,57},{8,55,5},{51,52,40},{21,54,58,6,41},{50,49,57,47,58},{53,8,5,48,57},{55,51,40,59,5},{52,56,41,60,40},{46,45,43,42,44},{44,42,39},{42,43,38},{43,45,36},{45,46,35},{46,44,37},{37,3,16},{39,29,12},{38,10,27},{36,14,1},{35,33,31},{37,44,39,12,3},{39,42,38,27,29},{38,43,36,1,10},{36,45,35,31,14},{35,46,37,16,33}}
PolyhedronFaces[JohnsonSolid[49]]:={{1,3,5},{6,4,2},{3,1,2,4},{1,5,6,2},{7,5,3},{7,3,4},{7,4,6},{7,6,5}}
PolyhedronFaces[JohnsonSolid[50]]:={{1,3,5},{6,4,2},{1,5,6,2},{8,5,3},{8,3,4},{8,4,6},{8,6,5},{7,3,1},{7,1,2},{7,2,4},{7,4,3}}
PolyhedronFaces[JohnsonSolid[51]]:={{1,3,5},{6,4,2},{9,5,3},{9,3,4},{9,4,6},{9,6,5},{7,3,1},{7,1,2},{7,2,4},{7,4,3},{8,1,5},{8,5,6},{8,6,2},{8,2,1}}
PolyhedronFaces[JohnsonSolid[52]]:={{3,7,9,5,1},{2,6,10,8,4},{5,9,10,6},{9,7,8,10},{7,3,4,8},{3,1,2,4},{11,1,5},{11,5,6},{11,6,2},{11,2,1}}
PolyhedronFaces[JohnsonSolid[53]]:={{3,7,9,5,1},{2,6,10,8,4},{5,9,10,6},{7,3,4,8},{3,1,2,4},{12,1,5},{12,5,6},{12,6,2},{12,2,1},{11,9,7},{11,7,8},{11,8,10},{11,10,9}}
PolyhedronFaces[JohnsonSolid[54]]:={{7,3,1,5,9,11},{12,10,6,2,4,8},{9,5,6,10},{5,1,2,6},{1,3,4,2},{3,7,8,4},{7,11,12,8},{13,11,9},{13,9,10},{13,10,12},{13,12,11}}
PolyhedronFaces[JohnsonSolid[55]]:={{7,3,1,5,9,11},{12,10,6,2,4,8},{9,5,6,10},{5,1,2,6},{3,7,8,4},{7,11,12,8},{14,11,9},{14,9,10},{14,10,12},{14,12,11},{13,1,3},{13,3,4},{13,4,2},{13,2,1}}
PolyhedronFaces[JohnsonSolid[56]]:={{7,3,1,5,9,11},{12,10,6,2,4,8},{9,5,6,10},{1,3,4,2},{3,7,8,4},{7,11,12,8},{14,11,9},{14,9,10},{14,10,12},{14,12,11},{13,5,1},{13,1,2},{13,2,6},{13,6,5}}
PolyhedronFaces[JohnsonSolid[57]]:={{8,3,1,5,10,12},{13,11,6,2,4,9},{10,5,6,11},{1,3,4,2},{8,12,13,9},{15,12,10},{15,10,11},{15,11,13},{15,13,12},{14,5,1},{14,1,2},{14,2,6},{14,6,5},{7,3,8},{7,8,9},{7,9,4},{7,4,3}}
PolyhedronFaces[JohnsonSolid[58]]:={{2,6,12,11,5},{5,11,7,3,20},{11,12,8,16,7},{12,6,21,4,8},{6,2,13,18,21},{2,5,20,17,13},{4,21,18,10,15},{18,13,17,9,10},{17,20,3,14,9},{3,7,16,1,14},{16,8,4,15,1},{19,15,10},{19,10,9},{19,9,14},{19,14,1},{19,1,15}}
PolyhedronFaces[JohnsonSolid[59]]:={{5,11,7,3,21},{11,12,8,16,7},{12,6,22,4,8},{6,2,13,18,22},{2,5,21,17,13},{4,22,18,10,15},{18,13,17,9,10},{17,21,3,14,9},{3,7,16,1,14},{16,8,4,15,1},{20,15,10},{20,10,9},{20,9,14},{20,14,1},{20,1,15},{19,2,6},{19,6,12},{19,12,11},{19,11,5},{19,5,2}}
PolyhedronFaces[JohnsonSolid[60]]:={{2,6,12,11,5},{11,12,8,16,7},{12,6,21,4,8},{6,2,13,18,21},{2,5,20,17,13},{4,21,18,10,15},{18,13,17,9,10},{17,20,3,14,9},{3,7,16,1,14},{16,8,4,15,1},{19,15,10},{19,10,9},{19,9,14},{19,14,1},{19,1,15},{22,5,11},{22,11,7},{22,7,3},{22,3,20},{22,20,5}}
PolyhedronFaces[JohnsonSolid[61]]:={{2,6,12,11,5},{11,12,8,16,7},{6,2,13,18,21},{2,5,20,17,13},{4,21,18,10,15},{18,13,17,9,10},{17,20,3,14,9},{3,7,16,1,14},{16,8,4,15,1},{19,15,10},{19,10,9},{19,9,14},{19,14,1},{19,1,15},{23,5,11},{23,11,7},{23,7,3},{23,3,20},{23,20,5},{22,12,6},{22,6,21},{22,21,4},{22,4,8},{22,8,12}}
PolyhedronFaces[JohnsonSolid[62]]:={{7,8,1},{7,1,3},{7,2,8},{8,4,1},{1,4,5},{1,5,3},{3,5,9},{4,10,5},{5,10,9},{6,9,10},{9,6,2,7,3},{4,8,2,6,10}}
PolyhedronFaces[JohnsonSolid[63]]:={{9,2,3},{1,5,4},{3,2,6},{3,6,8},{2,7,6},{2,9,1,4,7},{5,1,9,3,8},{8,6,7,4,5}}
PolyhedronFaces[JohnsonSolid[64]]:={{8,3,4},{4,3,7},{4,7,10},{3,9,7},{3,8,2,5,9},{6,2,8,4,10},{10,7,9,5,6},{1,2,6},{1,6,5},{1,5,2}}
PolyhedronFaces[JohnsonSolid[65]]:={{12,13,9},{3,10,1},{2,11,4},{6,5,8},{12,9,8,5,3,1},{2,4,6,8,9,13},{10,3,5,6,4,11},{7,14,15},{11,2,15,14},{1,10,14,7},{13,12,7,15},{15,2,13},{14,10,11},{7,12,1}}
PolyhedronFaces[JohnsonSolid[66]]:={{7,15,13,9,3,21,23,1},{9,19,20,10,4,28,27,3},{1,25,26,2,8,18,17,7},{15,17,18,16,14,20,19,13},{21,27,28,22,24,26,25,23},{9,13,19},{17,15,7},{27,21,3},{1,23,25},{20,14,10},{8,16,18},{4,22,28},{26,24,2},{6,11,5,12},{11,6,30,31},{5,11,32,33},{12,5,34,35},{6,12,36,29},{29,30,6},{31,32,11},{33,34,5},{35,36,12}}
PolyhedronFaces[JohnsonSolid[67]]:={{3,32,31,4,12,23,24,11},{9,22,21,10,2,29,30,1},{17,24,23,18,20,21,22,19},{27,30,29,28,26,31,32,25},{24,17,11},{9,19,22},{3,25,32},{30,27,1},{12,18,23},{21,20,10},{31,26,4},{2,28,29},{16,6,14,8},{38,36,8,14},{42,40,14,6},{46,44,6,16},{34,48,16,8},{8,36,34},{14,40,38},{6,44,42},{16,48,46},{7,13,5,15},{13,7,35,37},{5,13,39,41},{15,5,43,45},{7,15,47,33},{33,35,7},{37,39,13},{41,43,5},{45,47,15}}

PolyhedronFaces[JohnsonSolid[84]]:={{2,3,7},{3,2,8},{4,2,7},{2,4,8},{6,8,4},{4,1,6},{7,1,4},{3,5,7},{8,5,3},{5,8,6},{5,6,1},{5,1,7}}


PolyhedronFaces[JohnsonSolid[62]]:={{2,4,8,10,6},{3,6,10,9,5},{7,4,1},{1,4,2},{2,6,3},{2,3,1},{3,5,1},{5,9,7},{5,7,1},{10,8,9},{8,7,9},{8,4,7}}
PolyhedronFaces[JohnsonSolid[63]]:={{7,9,8,6,3},{1,3,6,4,2},{1,5,7},{5,9,7},{7,3,1},{1,2,5},{8,9,5,2,4},{6,8,4}}
PolyhedronFaces[JohnsonSolid[64]]:={{9,5,2,4,8},{7,8,4,1,3},{7,6,9},{6,5,9},{9,8,7},{7,3,6},{2,5,6,3,1},{10,2,1},{10,1,4},{10,4,2}}
PolyhedronFaces[JohnsonSolid[65]]:={{13,9,5,8},{13,8,12},{12,8,3,1,6,11},{11,14,12},{14,15,13,12},{15,14,10},{14,11,6,2,4,10},{10,4,7},{1,2,6},{4,2,1,3,5,7},{5,3,8},{15,10,7,9},{7,5,9},{15,9,13}}
PolyhedronFaces[JohnsonSolid[66]]:={{26,25,27,28},{25,26,19,17},{25,17,18},{27,25,18,20},{27,20,22},{28,27,22,24},{28,24,23},{26,28,23,21},{26,21,19},{24,16,12,8,7,11,15,23},{21,15,11,5,3,9,13,19},{17,13,9,1,2,10,14,18},{20,14,10,4,6,12,16,22},{5,7,8,6,4,2,1,3},{22,16,24},{6,8,12},{18,14,20},{2,4,10},{19,13,17},{3,1,9},{23,15,21},{7,5,11}}
PolyhedronFaces[JohnsonSolid[67]]:={{29,23,16,20},{23,29,32,26},{23,26,18},{16,23,18,11},{16,11,8},{20,16,8,14},{20,14,21},{29,20,21,30},{29,30,32},{17,10,4,13},{10,17,22,15},{10,15,7},{4,10,7,1},{4,1,3},{13,4,3,12},{13,12,19},{17,13,19,25},{17,25,22},{14,5,2,7,15,24,27,21},{30,27,24,22,25,28,31,32},{26,31,28,19,12,6,9,18},{11,9,6,3,1,2,5,8},{8,5,14},{1,7,2},{18,9,11},{12,3,6},{32,31,26},{25,19,28},{21,27,30},{15,22,24}}
PolyhedronFaces[JohnsonSolid[68]]:={{32,27,20,10,6,4,8,18,25,31},{35,31,25},{60,54,45,37,32,31,35,42,52,58},{46,52,42},{37,27,32},{38,28,22,17,20,27,37,45,50,48},{54,50,45},{17,10,20},{9,5,1,2,6,10,17,22,21,15},{28,21,22},{2,4,6},{11,13,14,12,8,4,2,1,3,7},{5,3,1},{12,18,8},{40,46,42,35,25,18,12,14,23,33},{13,23,14},{51,41,34,29,30,36,43,53,59,57},{63,57,59},{46,40,39,44,51,57,63,64,58,52},{60,58,64},{44,41,51},{13,11,16,24,34,41,44,39,33,23},{40,33,39},{24,29,34},{5,9,19,26,30,29,24,16,7,3},{11,7,16},{26,36,30},{28,38,47,49,43,36,26,19,15,21},{9,15,19},{49,53,43},{48,47,38},{61,55,56,62,65},{55,61,53,49},{55,49,47},{56,55,47,48},{56,48,50},{62,56,50,54},{62,54,60},{65,62,60,64},{63,65,64},{61,65,63,59},{61,59,53}}
PolyhedronFaces[JohnsonSolid[69]]:={{16,29,38,46,51,44,35,24,13,9},{5,9,13},{6,10,17,19,16,9,5,2,1,3},{4,1,2},{19,29,16},{43,53,56,50,38,29,19,17,22,34},{10,22,17},{50,46,38},{70,68,64,57,51,46,50,56,63,67},{53,63,56},{57,44,51},{68,70,69,66,62,55,52,54,61,65},{65,64,68},{30,24,35},{8,4,2,5,13,24,30,32,28,18},{37,28,32},{25,33,42,55,62,58,47,36,27,20},{14,20,27},{4,8,15,21,25,20,14,7,3,1},{6,3,7},{21,33,25},{37,49,54,52,42,33,21,15,18,28},{8,18,15},{52,55,42},{61,54,49},{66,58,62},{53,43,39,41,47,58,66,69,67,63},{70,67,69},{41,36,47},{34,39,43},{26,31,23,11,12},{31,26,36,41},{31,41,39},{23,31,39,34},{23,34,22},{11,23,22,10},{11,10,6},{12,11,6,7},{14,12,7},{26,12,14,27},{26,27,36},{45,59,60,48,40},{59,45,44,57},{59,57,64},{60,59,64,65},{60,65,61},{48,60,61,49},{48,49,37},{40,48,37,32},{30,40,32},{45,40,30,35},{45,35,44}}
PolyhedronFaces[JohnsonSolid[70]]:={{41,45,50,59,66,70,68,63,54,47},{42,47,54},{12,15,22,34,41,47,42,35,23,16},{27,23,35},{34,45,41},{21,32,44,49,50,45,34,22,19,18},{15,19,22},{49,59,50},{51,60,67,69,66,59,49,44,39,46},{32,39,44},{69,70,66},{57,58,62,65,68,70,69,67,64,61},{60,64,67},{65,63,68},{33,27,35,42,54,63,65,62,53,40},{58,53,62},{17,24,28,31,26,20,13,7,3,10},{4,10,3},{27,33,29,25,17,10,4,8,16,23},{12,16,8},{25,24,17},{58,57,52,38,28,24,25,29,40,53},{33,40,29},{38,31,28},{61,52,57},{30,20,26},{32,21,14,11,13,20,30,37,46,39},{51,46,37},{11,7,13},{18,14,21},{1,5,9,6,2},{5,1,7,11},{5,11,14},{9,5,14,18},{9,18,19},{6,9,19,15},{6,15,12},{2,6,12,8},{4,2,8},{1,2,4,3},{1,3,7},{36,48,56,55,43},{48,36,31,38},{48,38,52},{56,48,52,61},{56,61,64},{55,56,64,60},{55,60,51},{43,55,51,37},{30,43,37},{36,43,30,26},{36,26,31}}
PolyhedronFaces[JohnsonSolid[71]]:={{69,65,60,57,59,63,68,72,75,74},{71,74,75},{41,44,52,61,69,74,71,64,54,46},{55,54,64},{61,65,69},{55,64,71,75,72,66,56,51,48,49},{44,40,52},{50,57,60},{21,31,42,53,59,57,50,38,28,20},{29,28,38},{53,63,59},{37,47,56,66,68,63,53,42,36,33},{31,36,42},{66,72,68},{51,56,47},{3,7,12,16,17,15,10,5,2,1},{4,1,2},{29,26,18,9,3,1,4,11,20,28},{21,20,11},{9,7,3},{44,41,35,23,12,7,9,18,30,40},{26,30,18},{23,16,12},{46,35,41},{27,15,17},{47,37,24,13,10,15,27,39,48,51},{49,48,39},{13,5,10},{33,24,37},{6,14,22,19,8},{14,6,5,13},{14,13,24},{22,14,24,33},{22,33,36},{19,22,36,31},{19,31,21},{8,19,21,11},{4,8,11},{6,8,4,2},{6,2,5},{25,32,43,45,34},{32,25,16,23},{32,23,35},{43,32,35,46},{43,46,54},{45,43,54,55},{45,55,49},{34,45,49,39},{27,34,39},{25,34,27,17},{25,17,16},{73,67,58,62,70},{67,73,52,40},{67,40,30},{67,30,26,58},{58,26,29},{58,29,38,62},{62,38,50},{62,50,60,70},{70,60,65},{70,65,61,73},{73,61,52}}
PolyhedronFaces[JohnsonSolid[72]]:={{37,34,46,49},{49,46,56},{46,48,57,56},{34,28,36,48,46},{36,43,48},{48,43,51,57},{24,32,43,36},{24,19,32},{19,21,35,32},{32,35,47,51,43},{21,31,35},{35,31,42,47},{49,56,58,50},{50,58,52},{58,60,54,52},{56,57,59,60,58},{60,55,54},{54,55,45,44},{59,53,55,60},{57,51,59},{51,47,53,59},{53,42,39,45,55},{47,42,53},{42,31,25,39},{50,52,41,40},{40,41,29},{41,38,23,29},{52,54,44,38,41},{38,26,23},{23,26,14,12},{44,33,26,38},{44,45,33},{45,39,27,33},{33,27,15,14,26},{39,25,27},{27,25,13,15},{40,29,20,30},{30,20,18},{20,10,8,18},{29,23,12,10,20},{10,2,8},{8,2,1,6},{12,4,2,10},{12,14,4},{14,15,5,4},{4,5,3,1,2},{15,13,5},{5,13,11,3},{30,18,22,37},{37,22,34},{22,16,28,34},{18,8,6,16,22},{16,17,28},{28,17,24,36},{6,7,17,16},{6,1,7},{1,3,9,7},{7,9,19,24,17},{3,11,9},{9,11,21,19},{37,49,50,40,30},{11,13,25,31,21}}
PolyhedronFaces[JohnsonSolid[73]]:={{25,22,34,37},{37,34,45},{34,33,44,45},{22,12,20,33,34},{20,29,33},{33,29,42,44},{10,23,29,20},{10,11,23},{11,21,30,23},{23,30,43,42,29},{21,35,30},{30,35,46,43},{37,45,54,48},{48,54,56},{54,58,60,56},{45,44,52,58,54},{58,59,60},{60,59,55,57},{52,53,59,58},{44,42,52},{42,43,53,52},{53,46,47,55,59},{43,46,53},{46,35,36,47},{48,56,50,40},{50,51,38},{51,41,32,38},{56,60,57,51,50},{41,28,32},{32,28,17,19},{57,49,41,51},{57,55,49},{55,47,39,49},{49,39,27,28,41},{47,36,39},{39,36,24,27},{50,38,31,40},{40,31,26},{31,18,15,26},{38,32,19,18,31},{18,8,15},{8,9,3,2},{19,9,8,18},{19,17,9},{28,27,16,17},{17,16,7,3,9},{27,24,16},{16,24,13,7},{26,15,14,25},{25,14,22},{14,6,12,22},{15,8,2,6,14},{6,4,12},{12,4,10,20},{2,1,4,6},{2,3,1},{3,7,5,1},{1,5,11,10,4},{7,13,5},{5,13,21,11},{25,37,48,40,26},{13,24,36,35,21}}
PolyhedronFaces[JohnsonSolid[74]]:={{20,15,26,32},{32,26,38},{26,25,37,38},{15,6,14,25,26},{14,24,25},{25,24,36,37},{8,18,24,14},{8,11,18},{11,21,30,18},{18,30,42,36,24},{21,33,30},{30,33,45,42},{32,38,49,44},{49,55,57},{55,59,60,57},{38,37,48,55,49},{59,58,60},{60,58,54,56},{48,53,59,55},{37,36,48},{36,42,53,48},{53,45,52,58,59},{42,45,53},{45,33,41,52},{49,57,51,44},{44,51,40},{51,50,39,40},{57,60,56,50,51},{50,43,39},{43,47,35,31},{56,47,43,50},{56,54,47},{58,52,46,54},{54,46,34,35,47},{52,41,46},{46,41,29,34},{40,39,27,28},{28,27,16},{27,19,9,16},{39,43,31,19,27},{19,13,9},{9,13,5,3},{31,23,13,19},{31,35,23},{35,34,22,23},{23,22,12,5,13},{34,29,22},{22,29,17,12},{28,16,10,20},{20,10,15},{10,4,6,15},{16,9,3,4,10},{4,2,6},{6,2,8,14},{3,1,2,4},{3,5,1},{5,12,7,1},{1,7,11,8,2},{12,17,7},{7,17,21,11},{20,32,44,40,28},{17,29,41,33,21}}
PolyhedronFaces[JohnsonSolid[75]]:={{19,13,25,31},{31,25,38},{25,27,40,38},{13,8,15,27,25},{15,29,27},{27,29,42,40},{12,23,29,15},{12,14,23},{14,26,32,23},{23,32,44,42,29},{26,39,32},{32,39,50,44},{31,38,49,41},{49,57,56},{57,60,59,56},{38,40,51,57,49},{60,58,59},{59,58,53,54},{51,55,60,57},{40,42,51},{42,44,55,51},{55,50,52,58,60},{44,50,55},{50,39,43,52},{49,56,47,41},{41,47,35},{47,46,33,35},{56,59,54,46,47},{46,37,33},{37,45,34,28},{54,45,37,46},{54,53,45},{58,52,48,53},{53,48,36,34,45},{52,43,48},{48,43,30,36},{35,33,20,22},{22,20,11},{20,17,6,11},{33,37,28,17,20},{28,16,17},{17,16,5,6},{34,24,16,28},{34,36,24},{36,30,18,24},{24,18,7,5,16},{30,21,18},{18,21,10,7},{22,11,9,19},{19,9,13},{9,2,8,13},{11,6,1,2,9},{2,4,8},{8,4,12,15},{1,3,4,2},{6,5,1},{5,7,3,1},{3,10,14,12,4},{7,10,3},{10,21,26,14},{19,31,41,35,22},{21,30,43,39,26}}
PolyhedronFaces[JohnsonSolid[76]]:={{32,21,18,27},{27,18,19},{18,7,10,19},{21,13,4,7,18},{7,2,10},{10,2,5,12},{4,1,2,7},{4,6,1},{6,9,3,1},{1,3,8,5,2},{9,11,3},{3,11,14,8},{27,19,29,36},{36,29,38},{29,24,35,38},{19,10,12,24,29},{24,25,35},{35,25,33,41},{12,16,25,24},{12,5,16},{5,8,17,16},{16,17,28,33,25},{8,14,17},{17,14,26,28},{36,38,47,45},{45,47,53},{47,50,55,53},{38,35,41,50,47},{50,52,55},{55,52,49,54},{41,43,52,50},{41,33,43},{33,28,37,43},{43,37,40,49,52},{28,26,37},{37,26,31,40},{45,53,48,42},{42,48,51,44,34,23,15,13,21,32},{53,55,54,51,48},{54,46,44,51},{54,49,46},{49,40,39,46},{46,39,30,34,44},{40,31,39},{39,31,22,30},{13,15,6,4},{34,30,20,23},{23,20,9,6,15},{30,22,20},{20,22,11,9},{32,27,36,45,42},{22,31,26,14,11}}
PolyhedronFaces[JohnsonSolid[77]]:={{33,23,17,28},{28,17,20},{17,8,10,20},{23,15,6,8,17},{8,2,10},{10,2,4,12},{6,1,2,8},{6,7,1},{7,9,3,1},{1,3,5,4,2},{9,11,3},{3,11,14,5},{28,20,27,37},{37,27,39},{27,25,35,39},{20,10,12,25,27},{12,19,25},{25,19,31,35},{4,13,19,12},{4,5,13},{5,14,21,13},{13,21,32,31,19},{14,26,21},{21,26,36,32},{37,39,47,45},{45,47,53},{47,49,55,53},{39,35,41,49,47},{49,50,55},{55,50,48,54},{41,42,50,49},{35,31,41},{31,32,42,41},{42,36,40,48,50},{32,36,42},{36,26,29,40},{45,53,51,43},{43,51,52,44,34,24,16,15,23,33},{53,55,54,52,51},{54,46,44,52},{54,48,46},{48,40,38,46},{46,38,30,34,44},{40,29,38},{38,29,22,30},{15,16,7,6},{34,30,18,24},{24,18,9,7,16},{30,22,18},{18,22,11,9},{33,28,37,45,43},{22,29,26,14,11}}
PolyhedronFaces[JohnsonSolid[78]]:={{32,22,20,30},{30,20,27},{20,11,17,27},{22,13,5,11,20},{11,7,17},{17,7,12,21},{5,1,7,11},{5,2,1},{2,4,3,1},{1,3,8,12,7},{4,6,3},{3,6,14,8},{30,27,36,41},{41,36,45},{36,31,42,45},{27,17,21,31,36},{31,33,42},{42,33,37,46},{21,23,33,31},{21,12,23},{12,8,18,23},{23,18,28,37,33},{8,14,18},{18,14,24,28},{41,45,52,50},{50,52,53},{52,54,55,53},{45,42,46,54,52},{46,49,54},{54,49,51,55},{37,38,49,46},{37,28,38},{28,24,34,38},{38,34,40,51,49},{24,26,34},{34,26,29,40},{50,53,47,43},{43,47,44,35,25,15,9,13,22,32},{53,55,48,44,47},{48,39,35,44},{55,51,48},{51,40,39,48},{39,29,19,25,35},{40,29,39},{29,26,16,19},{13,9,2,5},{25,19,10,15},{15,10,4,2,9},{19,16,10},{10,16,6,4},{32,30,41,50,43},{16,26,24,14,6}}
PolyhedronFaces[JohnsonSolid[79]]:={{1,3,8,5},{8,18,17},{18,31,29,17},{3,10,19,18,8},{31,42,29},{29,42,45,35},{19,32,31,18},{19,26,32},{26,38,43,32},{32,43,51,42,31},{38,48,43},{43,48,54,51},{8,17,12,5},{5,12,7},{12,24,16,7},{17,29,35,24,12},{24,28,16},{28,41,46,36},{35,41,28,24},{35,45,41},{42,51,52,45},{45,52,53,46,41},{51,54,52},{52,54,55,53},{7,16,13,6},{6,13,9},{13,25,20,9},{16,28,36,25,13},{36,37,25},{25,37,30,20},{46,47,37,36},{46,53,47},{53,55,50,47},{47,50,40,30,37},{55,49,50},{50,49,39,40},{6,9,4,2},{2,4,11,15,23,22,14,10,3,1},{9,20,21,11,4},{21,27,15,11},{20,30,21},{30,40,27,21},{27,39,34,23,15},{40,39,27},{39,49,44,34},{10,14,26,19},{23,34,33,22},{22,33,38,26,14},{34,44,33},{33,44,48,38},{1,5,7,6,2},{44,49,55,54,48}}
PolyhedronFaces[JohnsonSolid[80]]:={{23,20,11,13},{13,11,5},{11,9,3,5},{20,26,17,9,11},{9,7,3},{3,7,6,1},{17,15,7,9},{17,27,15},{27,32,18,15},{15,18,14,6,7},{32,30,18},{18,30,22,14},{13,5,4,12},{12,4,10},{4,2,8,10},{5,3,1,2,4},{25,16,8,2,1,6,14,22,28,31},{12,10,19,21},{21,19,33},{19,24,36,33},{10,8,16,24,19},{24,34,36},{36,34,42,44},{16,25,34,24},{25,31,40,42,34},{31,28,38,40},{21,33,37,29},{29,37,45,50,49,43,35,26,20,23},{33,36,44,45,37},{44,48,50,45},{44,42,48},{42,40,46,48},{48,46,47,49,50},{40,38,46},{46,38,39,47},{26,35,27,17},{49,47,41,43},{43,41,32,27,35},{47,39,41},{41,39,30,32},{23,13,12,21,29},{39,38,28,22,30}}
PolyhedronFaces[JohnsonSolid[81]]:={{45,40,30,36},{36,30,26},{30,22,16,26},{40,35,25,22,30},{22,12,16},{16,12,4,7},{25,15,12,22},{25,23,15},{23,18,9,15},{15,9,2,4,12},{18,13,9},{9,13,5,2},{36,26,24,34},{34,24,28},{24,14,17,28},{26,16,7,14,24},{14,8,17},{17,8,11,21},{7,3,8,14},{7,4,3},{4,2,1,3},{3,1,6,11,8},{2,5,1},{1,5,10,6},{34,28,37,43},{43,37,46},{37,32,41,46},{28,17,21,32,37},{39,44,41,32,21,11,6,10,19,29},{43,46,50,48},{48,50,49,47,42,38,33,35,40,45},{46,41,44,49,50},{44,39,47,49},{39,29,31,42,47},{29,19,20,31},{35,33,23,25},{42,31,27,38},{38,27,18,23,33},{31,20,27},{27,20,13,18},{45,36,34,43,48},{20,19,10,5,13}}
PolyhedronFaces[JohnsonSolid[82]]:={{37,34,24,26},{26,24,16},{24,20,10,16},{34,36,28,20,24},{28,17,20},{20,17,7,10},{31,21,17,28},{31,29,21},{29,22,13,21},{21,13,5,7,17},{22,14,13},{13,14,6,5},{26,16,12,23},{23,12,18},{12,4,9,18},{16,10,2,4,12},{4,3,9},{9,3,8,15},{2,1,3,4},{10,7,2},{7,5,1,2},{1,6,11,8,3},{5,6,1},{6,14,19,11},{23,18,27,33},{33,27,38},{27,25,35,38},{18,9,15,25,27},{45,44,35,25,15,8,11,19,30,40},{33,38,47,42},{42,47,49,50,48,46,41,36,34,37},{38,35,44,49,47},{44,45,50,49},{45,40,43,48,50},{40,30,32,43},{36,41,31,28},{48,43,39,46},{46,39,29,31,41},{43,32,39},{39,32,22,29},{37,26,23,33,42},{32,30,19,14,22}}
PolyhedronFaces[JohnsonSolid[83]]:={{36,38,28,26},{26,28,18},{28,22,14,18},{38,40,30,22,28},{2,8,14,22,30,32,24,16,10,4},{26,18,12,21},{21,12,13},{12,6,7,13},{18,14,8,6,12},{6,1,7},{7,1,3,9},{8,2,1,6},{2,4,5,3,1},{4,10,11,5},{21,13,19,27},{27,19,29},{19,15,23,29},{13,7,9,15,19},{33,31,23,15,9,3,5,11,17,25},{27,29,39,37},{37,39,41,43,45,44,42,40,38,36},{29,23,31,41,39},{31,33,43,41},{33,25,35,45,43},{25,17,20,35},{40,42,32,30},{45,35,34,44},{44,34,24,32,42},{35,20,34},{34,20,16,24},{36,26,21,27,37},{20,17,11,10,16}}
PolyhedronFaces[JohnsonSolid[85]]:={{14,12,5,7},{14,10,15},{10,14,7},{10,7,4},{12,16,13},{16,12,14},{16,14,15},{5,8,2},{8,5,12},{8,12,13},{7,1,4},{1,7,5},{1,5,2},{3,6,11,9},{3,4,1},{4,3,9},{4,9,10},{6,2,8},{2,6,3},{2,3,1},{11,13,16},{13,11,6},{13,6,8},{9,15,10},{15,9,11},{15,11,16}}
PolyhedronFaces[JohnsonSolid[86]]:={{2,1,5,6},{6,5,9,10},{8,4,3},{3,2,7},{3,4,1},{3,1,2},{2,6,7},{7,6,10},{4,5,1},{5,4,9},{10,8,7},{10,9,8},{8,9,4},{7,8,3}}
PolyhedronFaces[JohnsonSolid[87]]:={{6,1,2},{6,2,5,8},{10,4,9},{9,7,11},{9,4,3},{9,3,7},{7,6,11},{11,6,8},{4,2,3},{2,4,5},{8,10,11},{8,5,10},{10,5,4},{11,10,9},{7,1,6},{7,3,1},{1,3,2}}
PolyhedronFaces[JohnsonSolid[88]]:={{8,6,10,11},{7,8,11},{4,8,2},{4,6,8},{11,12,9},{11,10,12},{2,8,7},{7,11,9},{5,10,6,1},{3,5,1},{12,5,9},{12,10,5},{1,4,2},{1,6,4},{9,3,7},{9,5,3},{3,2,7},{3,1,2}}
PolyhedronFaces[JohnsonSolid[89]]:={{9,8,13,14},{10,9,14},{2,9,3},{2,8,9},{14,12,11},{14,13,12},{3,9,10},{10,14,11},{4,7,13,8},{13,7,12},{4,8,2},{6,7,4,1},{5,6,1},{12,6,11},{12,7,6},{1,2,3},{1,4,2},{11,5,10},{11,6,5},{5,3,10},{5,1,3}}
PolyhedronFaces[JohnsonSolid[90]]:={{14,7,8},{10,7,14,16},{10,16,12},{13,14,8},{16,14,13,15},{16,15,12},{13,11,15},{13,8,6},{13,6,11},{11,6,4},{6,3,1,4},{8,7,3},{6,8,3},{3,7,2},{3,2,1},{4,1,5,9},{11,4,9},{11,9,15},{9,12,15},{9,5,12},{12,5,10},{10,2,7},{5,2,10},{1,2,5}}
PolyhedronFaces[JohnsonSolid[91]]:={{14,12,8,11},{12,14,13},{8,12,5},{11,8,6},{14,11,10},{13,14,10,7,9},{12,13,9,4,5},{6,8,5,1,2},{11,6,2,3,10},{4,9,7},{5,4,1},{3,2,1},{10,3,7},{7,3,1,4}}
PolyhedronFaces[JohnsonSolid[92]]:={{12,13,17,18,16},{13,12,5},{12,7,5},{7,12,16},{7,16,14,6},{14,16,18},{9,14,18,15,10},{6,14,9},{3,6,9},{3,9,10},{8,10,15},{3,10,8,1},{17,15,18},{17,11,4,8,15},{4,1,8},{4,2,1},{2,4,11},{13,5,2,11},{13,11,17},{2,5,7,6,3,1}}


(* Solid Vertices *)

PolyhedronVertices[JohnsonSolid[ 1]]:={{0,0,1/Sqrt[2]},{0,-(1/Sqrt[2]),0},{0,1/Sqrt[2],0},{-(1/Sqrt[2]),0,0},{1/Sqrt[2],0,0}}
PolyhedronVertices[JohnsonSolid[ 2]]:={{0,0,Sqrt[(5-Sqrt[5])/10]},{Sqrt[1/2+1/(2Sqrt[5])],0,0},{Sqrt[(5-Sqrt[5])/10]/2,(-1-Sqrt[5])/4,0},{Sqrt[(5-Sqrt[5])/10]/2,(1+Sqrt[5])/4,0},{-Sqrt[1/4+1/(2Sqrt[5])],-1/2,0},{-Sqrt[1/4+1/(2Sqrt[5])],1/2,0}}
PolyhedronVertices[JohnsonSolid[ 3]]:={{0,-1,0},{0,1,0},{-1/(2Sqrt[3]),-1/2,Sqrt[2/3]},{-1/(2Sqrt[3]),1/2,Sqrt[2/3]},{1/Sqrt[3],0,Sqrt[2/3]},{-Sqrt[3]/2,-1/2,0},{-Sqrt[3]/2,1/2,0},{Sqrt[3]/2,-1/2,0},{Sqrt[3]/2,1/2,0}}
PolyhedronVertices[JohnsonSolid[ 4]]:={{0,-(1/Sqrt[2]),1/Sqrt[2]},{0,1/Sqrt[2],1/Sqrt[2]},{-(1/Sqrt[2]),0,1/Sqrt[2]},{1/Sqrt[2],0,1/Sqrt[2]},{1/2,(1+Sqrt[2])/2,0},{-1/2,(1+Sqrt[2])/2,0},{(-1-Sqrt[2])/2,1/2,0},{(-1-Sqrt[2])/2,-1/2,0},{-1/2,(-1-Sqrt[2])/2,0},{1/2,(-1-Sqrt[2])/2,0},{(1+Sqrt[2])/2,-1/2,0},{(1+Sqrt[2])/2,1/2,0}}
PolyhedronVertices[JohnsonSolid[ 5]]:={{0,(-1-Sqrt[5])/2,0},{0,(1+Sqrt[5])/2,0},{Sqrt[1/2+1/(2Sqrt[5])],0,Sqrt[(5-Sqrt[5])/10]},{Sqrt[(5-Sqrt[5])/10]/2,(-1-Sqrt[5])/4,Sqrt[(5-Sqrt[5])/10]},{Sqrt[(5-Sqrt[5])/10]/2,(1+Sqrt[5])/4,Sqrt[(5-Sqrt[5])/10]},{-Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,0},{-Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,0},{Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,0},{Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,0},{-Sqrt[1/4+1/(2Sqrt[5])],-1/2,Sqrt[(5-Sqrt[5])/10]},{-Sqrt[1/4+1/(2Sqrt[5])],1/2,Sqrt[(5-Sqrt[5])/10]},{-Sqrt[5/4+Sqrt[5]/2],-1/2,0},{-Sqrt[5/4+Sqrt[5]/2],1/2,0},{Sqrt[5/4+Sqrt[5]/2],-1/2,0},{Sqrt[5/4+Sqrt[5]/2],1/2,0}}
PolyhedronVertices[JohnsonSolid[ 6]]:={{0,(-1-Sqrt[5])/2,0},{0,(1+Sqrt[5])/2,0},{-Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,0},{-Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,0},{Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,0},{Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,0},{-Sqrt[5/4+Sqrt[5]/2],-1/2,0},{-Sqrt[5/4+Sqrt[5]/2],1/2,0},{Sqrt[5/4+Sqrt[5]/2],-1/2,0},{Sqrt[5/4+Sqrt[5]/2],1/2,0},{Sqrt[(5+2Sqrt[5])/5],0,Sqrt[1/2+1/(2Sqrt[5])]},{Sqrt[1/8+1/(8Sqrt[5])],(-3-Sqrt[5])/4,Sqrt[1/2+1/(2Sqrt[5])]},{Sqrt[1/8+1/(8Sqrt[5])],(3+Sqrt[5])/4,Sqrt[1/2+1/(2Sqrt[5])]},{-Sqrt[5/8+11/(8Sqrt[5])],(-1-Sqrt[5])/4,Sqrt[1/2+1/(2Sqrt[5])]},{-Sqrt[5/8+11/(8Sqrt[5])],(1+Sqrt[5])/4,Sqrt[1/2+1/(2Sqrt[5])]},{-Sqrt[1/2+1/(2Sqrt[5])],0,Sqrt[(5+2Sqrt[5])/5]},{-Sqrt[(5-Sqrt[5])/10]/2,(-1-Sqrt[5])/4,Sqrt[(5+2Sqrt[5])/5]},{-Sqrt[(5-Sqrt[5])/10]/2,(1+Sqrt[5])/4,Sqrt[(5+2Sqrt[5])/5]},{Sqrt[1/4+1/(2Sqrt[5])],-1/2,Sqrt[(5+2Sqrt[5])/5]},{Sqrt[1/4+1/(2Sqrt[5])],1/2,Sqrt[(5+2Sqrt[5])/5]}}
PolyhedronVertices[JohnsonSolid[ 7]]:={{0,0,(3+Sqrt[6])/3},{-1/(2Sqrt[3]),-1/2,0},{-1/(2Sqrt[3]),-1/2,1},{-1/(2Sqrt[3]),1/2,0},{-1/(2Sqrt[3]),1/2,1},{1/Sqrt[3],0,0},{1/Sqrt[3],0,1}}
PolyhedronVertices[JohnsonSolid[ 8]]:={{0,0,(2+Sqrt[2])/2},{0,-(1/Sqrt[2]),0},{0,-(1/Sqrt[2]),1},{0,1/Sqrt[2],0},{0,1/Sqrt[2],1},{-(1/Sqrt[2]),0,0},{-(1/Sqrt[2]),0,1},{1/Sqrt[2],0,0},{1/Sqrt[2],0,1}}
PolyhedronVertices[JohnsonSolid[ 9]]:={{0,0,(10+Sqrt[10(5-Sqrt[5])])/10},{Sqrt[1/2+1/(2Sqrt[5])],0,0},{Sqrt[1/2+1/(2Sqrt[5])],0,1},{Sqrt[(5-Sqrt[5])/10]/2,(-1-Sqrt[5])/4,0},{Sqrt[(5-Sqrt[5])/10]/2,(-1-Sqrt[5])/4,1},{Sqrt[(5-Sqrt[5])/10]/2,(1+Sqrt[5])/4,0},{Sqrt[(5-Sqrt[5])/10]/2,(1+Sqrt[5])/4,1},{-Sqrt[1/4+1/(2Sqrt[5])],-1/2,0},{-Sqrt[1/4+1/(2Sqrt[5])],-1/2,1},{-Sqrt[1/4+1/(2Sqrt[5])],1/2,0},{-Sqrt[1/4+1/(2Sqrt[5])],1/2,1}}
PolyhedronVertices[JohnsonSolid[10]]:={{-1/2,-1/2,-1/(2*2^(1/4))},{-1/2,1/2,-1/(2*2^(1/4))},{0,0,Root[7-16#1-32#1^2+32#1^4&,2,0]},{0,-(1/Sqrt[2]),1/(2*2^(1/4))},{0,1/Sqrt[2],1/(2*2^(1/4))},{1/2,-1/2,-1/(2*2^(1/4))},{1/2,1/2,-1/(2*2^(1/4))},{-(1/Sqrt[2]),0,1/(2*2^(1/4))},{1/Sqrt[2],0,1/(2*2^(1/4))}}
PolyhedronVertices[JohnsonSolid[11]]:={{0,0,Sqrt[5/8+Sqrt[5]/8]},{-Sqrt[1/2+1/(2Sqrt[5])],0,-Sqrt[1/8+1/(8Sqrt[5])]},{Sqrt[1/2+1/(2Sqrt[5])],0,Sqrt[1/8+1/(8Sqrt[5])]},{-Sqrt[(5-Sqrt[5])/10]/2,(-1-Sqrt[5])/4,-Sqrt[1/8+1/(8Sqrt[5])]},{-Sqrt[(5-Sqrt[5])/10]/2,(1+Sqrt[5])/4,-Sqrt[1/8+1/(8Sqrt[5])]},{Sqrt[(5-Sqrt[5])/10]/2,(-1-Sqrt[5])/4,Sqrt[1/8+1/(8Sqrt[5])]},{Sqrt[(5-Sqrt[5])/10]/2,(1+Sqrt[5])/4,Sqrt[1/8+1/(8Sqrt[5])]},{-Sqrt[1/4+1/(2Sqrt[5])],-1/2,Sqrt[1/8+1/(8Sqrt[5])]},{-Sqrt[1/4+1/(2Sqrt[5])],1/2,Sqrt[1/8+1/(8Sqrt[5])]},{Sqrt[1/4+1/(2Sqrt[5])],-1/2,-Sqrt[1/8+1/(8Sqrt[5])]},{Sqrt[1/4+1/(2Sqrt[5])],1/2,-Sqrt[1/8+1/(8Sqrt[5])]}}
PolyhedronVertices[JohnsonSolid[12]]:={{0,0,-Sqrt[2/3]},{0,0,Sqrt[2/3]},{-1/(2Sqrt[3]),-1/2,0},{-1/(2Sqrt[3]),1/2,0},{1/Sqrt[3],0,0}}
PolyhedronVertices[JohnsonSolid[13]]:={{0,0,-Sqrt[(5-Sqrt[5])/10]},{0,0,Sqrt[(5-Sqrt[5])/10]},{Sqrt[1/2+1/(2Sqrt[5])],0,0},{Sqrt[(5-Sqrt[5])/10]/2,(-1-Sqrt[5])/4,0},{Sqrt[(5-Sqrt[5])/10]/2,(1+Sqrt[5])/4,0},{-Sqrt[1/4+1/(2Sqrt[5])],-1/2,0},{-Sqrt[1/4+1/(2Sqrt[5])],1/2,0}}
PolyhedronVertices[JohnsonSolid[14]]:={{0,0,(-3-2Sqrt[6])/6},{0,0,(3+2Sqrt[6])/6},{-1/(2Sqrt[3]),-1/2,-1/2},{-1/(2Sqrt[3]),-1/2,1/2},{-1/(2Sqrt[3]),1/2,-1/2},{-1/(2Sqrt[3]),1/2,1/2},{1/Sqrt[3],0,-1/2},{1/Sqrt[3],0,1/2}}
PolyhedronVertices[JohnsonSolid[15]]:={{0,0,(-1-Sqrt[2])/2},{0,0,(1+Sqrt[2])/2},{0,-(1/Sqrt[2]),-1/2},{0,-(1/Sqrt[2]),1/2},{0,1/Sqrt[2],-1/2},{0,1/Sqrt[2],1/2},{-(1/Sqrt[2]),0,-1/2},{-(1/Sqrt[2]),0,1/2},{1/Sqrt[2],0,-1/2},{1/Sqrt[2],0,1/2}}
PolyhedronVertices[JohnsonSolid[16]]:={{0,0,(-5-Sqrt[10*(5-Sqrt[5])])/10},{0,0,(5+Sqrt[10*(5-Sqrt[5])])/10},{Sqrt[1/2+1/(2Sqrt[5])],0,-1/2},{Sqrt[1/2+1/(2Sqrt[5])],0,1/2},{Sqrt[(5-Sqrt[5])/10]/2,(-1-Sqrt[5])/4,-1/2},{Sqrt[(5-Sqrt[5])/10]/2,(-1-Sqrt[5])/4,1/2},{Sqrt[(5-Sqrt[5])/10]/2,(1+Sqrt[5])/4,-1/2},{Sqrt[(5-Sqrt[5])/10]/2,(1+Sqrt[5])/4,1/2},{-Sqrt[1/4+1/(2Sqrt[5])],-1/2,-1/2},{-Sqrt[1/4+1/(2Sqrt[5])],-1/2,1/2},{-Sqrt[1/4+1/(2Sqrt[5])],1/2,-1/2},{-Sqrt[1/4+1/(2Sqrt[5])],1/2,1/2}}
PolyhedronVertices[JohnsonSolid[17]]:={{-1/2,-1/2,-1/(2*2^(1/4))},{-1/2,1/2,-1/(2*2^(1/4))},{0,0,Root[7+16#1-32#1^2+32#1^4&,1,0]},{0,0,Root[7-16#1-32#1^2+32#1^4&,2,0]},{0,-(1/Sqrt[2]),1/(2*2^(1/4))},{0,1/Sqrt[2],1/(2*2^(1/4))},{1/2,-1/2,-1/(2*2^(1/4))},{1/2,1/2,-1/(2*2^(1/4))},{-(1/Sqrt[2]),0,1/(2*2^(1/4))},{1/Sqrt[2],0,1/(2*2^(1/4))}}
PolyhedronVertices[JohnsonSolid[18]]:={{0,-1,-1},{0,-1,0},{0,1,-1},{0,1,0},{-1/(2Sqrt[3]),-1/2,Sqrt[2/3]},{-1/(2Sqrt[3]),1/2,Sqrt[2/3]},{1/Sqrt[3],0,Sqrt[2/3]},{-Sqrt[3]/2,-1/2,-1},{-Sqrt[3]/2,-1/2,0},{-Sqrt[3]/2,1/2,-1},{-Sqrt[3]/2,1/2,0},{Sqrt[3]/2,-1/2,-1},{Sqrt[3]/2,-1/2,0},{Sqrt[3]/2,1/2,-1},{Sqrt[3]/2,1/2,0}}
PolyhedronVertices[JohnsonSolid[19]]:={{0,-(1/Sqrt[2]),1/Sqrt[2]},{0,1/Sqrt[2],1/Sqrt[2]},{-(1/Sqrt[2]),0,1/Sqrt[2]},{1/Sqrt[2],0,1/Sqrt[2]},{1/2,(1+Sqrt[2])/2,-1},{1/2,(1+Sqrt[2])/2,0},{-1/2,(1+Sqrt[2])/2,-1},{-1/2,(1+Sqrt[2])/2,0},{(-1-Sqrt[2])/2,1/2,-1},{(-1-Sqrt[2])/2,1/2,0},{(-1-Sqrt[2])/2,-1/2,-1},{(-1-Sqrt[2])/2,-1/2,0},{-1/2,(-1-Sqrt[2])/2,-1},{-1/2,(-1-Sqrt[2])/2,0},{1/2,(-1-Sqrt[2])/2,-1},{1/2,(-1-Sqrt[2])/2,0},{(1+Sqrt[2])/2,-1/2,-1},{(1+Sqrt[2])/2,-1/2,0},{(1+Sqrt[2])/2,1/2,-1},{(1+Sqrt[2])/2,1/2,0}}
PolyhedronVertices[JohnsonSolid[20]]:={{0,(-1-Sqrt[5])/2,-1},{0,(-1-Sqrt[5])/2,0},{0,(1+Sqrt[5])/2,-1},{0,(1+Sqrt[5])/2,0},{Sqrt[1/2+1/(2Sqrt[5])],0,Sqrt[(5-Sqrt[5])/10]},{Sqrt[(5-Sqrt[5])/10]/2,(-1-Sqrt[5])/4,Sqrt[(5-Sqrt[5])/10]},{Sqrt[(5-Sqrt[5])/10]/2,(1+Sqrt[5])/4,Sqrt[(5-Sqrt[5])/10]},{-Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,-1},{-Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,0},{-Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,-1},{-Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,0},{Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,-1},{Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,0},{Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,-1},{Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,0},{-Sqrt[1/4+1/(2Sqrt[5])],-1/2,Sqrt[(5-Sqrt[5])/10]},{-Sqrt[1/4+1/(2Sqrt[5])],1/2,Sqrt[(5-Sqrt[5])/10]},{-Sqrt[5/4+Sqrt[5]/2],-1/2,-1},{-Sqrt[5/4+Sqrt[5]/2],-1/2,0},{-Sqrt[5/4+Sqrt[5]/2],1/2,-1},{-Sqrt[5/4+Sqrt[5]/2],1/2,0},{Sqrt[5/4+Sqrt[5]/2],-1/2,-1},{Sqrt[5/4+Sqrt[5]/2],-1/2,0},{Sqrt[5/4+Sqrt[5]/2],1/2,-1},{Sqrt[5/4+Sqrt[5]/2],1/2,0}}
PolyhedronVertices[JohnsonSolid[21]]:={{0,(-1-Sqrt[5])/2,-1},{0,(-1-Sqrt[5])/2,0},{0,(1+Sqrt[5])/2,-1},{0,(1+Sqrt[5])/2,0},{-Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,-1},{-Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,0},{-Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,-1},{-Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,0},{Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,-1},{Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,0},{Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,-1},{Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,0},{-Sqrt[5/4+Sqrt[5]/2],-1/2,-1},{-Sqrt[5/4+Sqrt[5]/2],-1/2,0},{-Sqrt[5/4+Sqrt[5]/2],1/2,-1},{-Sqrt[5/4+Sqrt[5]/2],1/2,0},{Sqrt[5/4+Sqrt[5]/2],-1/2,-1},{Sqrt[5/4+Sqrt[5]/2],-1/2,0},{Sqrt[5/4+Sqrt[5]/2],1/2,-1},{Sqrt[5/4+Sqrt[5]/2],1/2,0},{Sqrt[(5+2Sqrt[5])/5],0,Sqrt[1/2+1/(2Sqrt[5])]},{Sqrt[1/8+1/(8Sqrt[5])],(-3-Sqrt[5])/4,Sqrt[1/2+1/(2Sqrt[5])]},{Sqrt[1/8+1/(8Sqrt[5])],(3+Sqrt[5])/4,Sqrt[1/2+1/(2Sqrt[5])]},{-Sqrt[5/8+11/(8Sqrt[5])],(-1-Sqrt[5])/4,Sqrt[1/2+1/(2Sqrt[5])]},{-Sqrt[5/8+11/(8Sqrt[5])],(1+Sqrt[5])/4,Sqrt[1/2+1/(2Sqrt[5])]},{-Sqrt[1/2+1/(2Sqrt[5])],0,Sqrt[(5+2Sqrt[5])/5]},{-Sqrt[(5-Sqrt[5])/10]/2,(-1-Sqrt[5])/4,Sqrt[(5+2Sqrt[5])/5]},{-Sqrt[(5-Sqrt[5])/10]/2,(1+Sqrt[5])/4,Sqrt[(5+2Sqrt[5])/5]},{Sqrt[1/4+1/(2Sqrt[5])],-1/2,Sqrt[(5+2Sqrt[5])/5]},{Sqrt[1/4+1/(2Sqrt[5])],1/2,Sqrt[(5+2Sqrt[5])/5]}}
PolyhedronVertices[JohnsonSolid[22]]:={{-1,0,-Sqrt[-1+Sqrt[3]]},{-1/2,-Sqrt[3]/2,-Sqrt[-1+Sqrt[3]]},{-1/2,Sqrt[3]/2,-Sqrt[-1+Sqrt[3]]},{0,-1,0},{0,1,0},{1/2,-Sqrt[3]/2,-Sqrt[-1+Sqrt[3]]},{1/2,Sqrt[3]/2,-Sqrt[-1+Sqrt[3]]},{1,0,-Sqrt[-1+Sqrt[3]]},{-1/(2Sqrt[3]),-1/2,Sqrt[2/3]},{-1/(2Sqrt[3]),1/2,Sqrt[2/3]},{1/Sqrt[3],0,Sqrt[2/3]},{-Sqrt[3]/2,-1/2,0},{-Sqrt[3]/2,1/2,0},{Sqrt[3]/2,-1/2,0},{Sqrt[3]/2,1/2,0}}
PolyhedronVertices[JohnsonSolid[23]]:={{0,-(1/Sqrt[2]),1/Sqrt[2]},{0,1/Sqrt[2],1/Sqrt[2]},{0,Root[1-4#1^2+2#1^4&,1,0],Root[-1+8#1^2-16#1^4+8#1^6+2#1^8&,1,0]},{0,Root[1-4#1^2+2#1^4&,4,0],Root[-1+8#1^2-16#1^4+8#1^6+2#1^8&,1,0]},{-(1/Sqrt[2]),0,1/Sqrt[2]},{1/Sqrt[2],0,1/Sqrt[2]},{(-1-Sqrt[2])/2,-1/2,0},{Root[1-4#1^2+2#1^4&,1,0],0,Root[-1+8#1^2-16#1^4+8#1^6+2#1^8&,1,0]},{Root[1-4#1^2+2#1^4&,4,0],0,Root[-1+8#1^2-16#1^4+8#1^6+2#1^8&,1,0]},{Root[1-8#1^2+8#1^4&,1,0],Root[1-8#1^2+8#1^4&,1,0],Root[-1+8#1^2-16#1^4+8#1^6+2#1^8&,1,0]},{Root[1-8#1^2+8#1^4&,1,0],Root[1-8#1^2+8#1^4&,4,0],Root[-1+8#1^2-16#1^4+8#1^6+2#1^8&,1,0]},{Root[1-8#1^2+8#1^4&,4,0],Root[1-8#1^2+8#1^4&,1,0],Root[-1+8#1^2-16#1^4+8#1^6+2#1^8&,1,0]},{Root[1-8#1^2+8#1^4&,4,0],Root[1-8#1^2+8#1^4&,4,0],Root[-1+8#1^2-16#1^4+8#1^6+2#1^8&,1,0]},{-1/2,(-1-Sqrt[2])/2,0},{1/2,(1+Sqrt[2])/2,0},{1/2,(-1-Sqrt[2])/2,0},{-1/2,(1+Sqrt[2])/2,0},{(1+Sqrt[2])/2,-1/2,0},{(-1-Sqrt[2])/2,1/2,0},{(1+Sqrt[2])/2,1/2,0},{(-1-Sqrt[2])/2,-1/2,0},{1/2,(1+Sqrt[2])/2,0},{-1/2,(-1-Sqrt[2])/2,0},{-1/2,(1+Sqrt[2])/2,0},{1/2,(-1-Sqrt[2])/2,0},{(-1-Sqrt[2])/2,1/2,0},{(1+Sqrt[2])/2,-1/2,0},{(-1-Sqrt[2])/2,-1/2,0},{(1+Sqrt[2])/2,1/2,0}}
PolyhedronVertices[JohnsonSolid[24]]:={{0,(-1-Sqrt[5])/2,0},{0,(1+Sqrt[5])/2,0},{(-1-Sqrt[5])/2,0,Root[1+2#1^2-11#1^4+8#1^6+#1^8&,1,0]},{Root[1-5#1^2+5#1^4&,4,0],0,Root[1-5#1^2+5#1^4&,3,0]},{Root[1-20#1^2+80#1^4&,3,0],(-1-Sqrt[5])/4,Root[1-5#1^2+5#1^4&,3,0]},{Root[1-20#1^2+80#1^4&,3,0],(1+Sqrt[5])/4,Root[1-5#1^2+5#1^4&,3,0]},{(1+Sqrt[5])/2,0,Root[1+2#1^2-11#1^4+8#1^6+#1^8&,1,0]},{Root[5-20#1^2+16#1^4&,1,0],(-3-Sqrt[5])/4,0},{Root[5-20#1^2+16#1^4&,1,0],(3+Sqrt[5])/4,0},{Root[5-20#1^2+16#1^4&,4,0],(-3-Sqrt[5])/4,0},{Root[5-20#1^2+16#1^4&,4,0],(3+Sqrt[5])/4,0},{Root[1-40#1^2+80#1^4&,1,0],-1/2,Root[1-5#1^2+5#1^4&,3,0]},{Root[1-40#1^2+80#1^4&,1,0],1/2,Root[1-5#1^2+5#1^4&,3,0]},{-1/2,Root[5-40#1^2+16#1^4&,1,0],Root[1+2#1^2-11#1^4+8#1^6+#1^8&,1,0]},{-1/2,Root[5-40#1^2+16#1^4&,4,0],Root[1+2#1^2-11#1^4+8#1^6+#1^8&,1,0]},{1/2,Root[5-40#1^2+16#1^4&,1,0],Root[1+2#1^2-11#1^4+8#1^6+#1^8&,1,0]},{1/2,Root[5-40#1^2+16#1^4&,4,0],Root[1+2#1^2-11#1^4+8#1^6+#1^8&,1,0]},{(-3-Sqrt[5])/4,Root[5-20#1^2+16#1^4&,1,0],Root[1+2#1^2-11#1^4+8#1^6+#1^8&,1,0]},{(-3-Sqrt[5])/4,Root[5-20#1^2+16#1^4&,4,0],Root[1+2#1^2-11#1^4+8#1^6+#1^8&,1,0]},{(3+Sqrt[5])/4,Root[5-20#1^2+16#1^4&,1,0],Root[1+2#1^2-11#1^4+8#1^6+#1^8&,1,0]},{(3+Sqrt[5])/4,Root[5-20#1^2+16#1^4&,4,0],Root[1+2#1^2-11#1^4+8#1^6+#1^8&,1,0]},{Root[5-40#1^2+16#1^4&,1,0],-1/2,0},{Root[5-40#1^2+16#1^4&,1,0],1/2,0},{Root[5-40#1^2+16#1^4&,4,0],-1/2,0},{Root[5-40#1^2+16#1^4&,4,0],1/2,0}}
PolyhedronVertices[JohnsonSolid[25]]:={{0,(-1-Sqrt[5])/2,0},{0,(1+Sqrt[5])/2,0},{(-1-Sqrt[5])/2,0,Root[1+2#1^2-11#1^4+8#1^6+#1^8&,1,0]},{(1+Sqrt[5])/2,0,Root[1+2#1^2-11#1^4+8#1^6+#1^8&,1,0]},{Root[5-20#1^2+16#1^4&,1,0],(-3-Sqrt[5])/4,0},{Root[5-20#1^2+16#1^4&,1,0],(3+Sqrt[5])/4,0},{Root[5-20#1^2+16#1^4&,4,0],(-3-Sqrt[5])/4,0},{Root[5-20#1^2+16#1^4&,4,0],(3+Sqrt[5])/4,0},{-1/2,Root[5-40#1^2+16#1^4&,1,0],Root[1+2#1^2-11#1^4+8#1^6+#1^8&,1,0]},{-1/2,Root[5-40#1^2+16#1^4&,4,0],Root[1+2#1^2-11#1^4+8#1^6+#1^8&,1,0]},{1/2,Root[5-40#1^2+16#1^4&,1,0],Root[1+2#1^2-11#1^4+8#1^6+#1^8&,1,0]},{1/2,Root[5-40#1^2+16#1^4&,4,0],Root[1+2#1^2-11#1^4+8#1^6+#1^8&,1,0]},{(-3-Sqrt[5])/4,Root[5-20#1^2+16#1^4&,1,0],Root[1+2#1^2-11#1^4+8#1^6+#1^8&,1,0]},{(-3-Sqrt[5])/4,Root[5-20#1^2+16#1^4&,4,0],Root[1+2#1^2-11#1^4+8#1^6+#1^8&,1,0]},{(3+Sqrt[5])/4,Root[5-20#1^2+16#1^4&,1,0],Root[1+2#1^2-11#1^4+8#1^6+#1^8&,1,0]},{(3+Sqrt[5])/4,Root[5-20#1^2+16#1^4&,4,0],Root[1+2#1^2-11#1^4+8#1^6+#1^8&,1,0]},{Root[5-40#1^2+16#1^4&,1,0],-1/2,0},{Root[5-40#1^2+16#1^4&,1,0],1/2,0},{Root[5-40#1^2+16#1^4&,4,0],-1/2,0},{Root[5-40#1^2+16#1^4&,4,0],1/2,0},{Root[1-10#1^2+5#1^4&,4,0],0,Root[1-5#1^2+5#1^4&,4,0]},{Root[1-20#1^2+80#1^4&,4,0],(-3-Sqrt[5])/4,Root[1-5#1^2+5#1^4&,4,0]},{Root[1-20#1^2+80#1^4&,4,0],(3+Sqrt[5])/4,Root[1-5#1^2+5#1^4&,4,0]},{Root[1-100#1^2+80#1^4&,1,0],(-1-Sqrt[5])/4,Root[1-5#1^2+5#1^4&,4,0]},{Root[1-100#1^2+80#1^4&,1,0],(1+Sqrt[5])/4,Root[1-5#1^2+5#1^4&,4,0]},{Root[1-5#1^2+5#1^4&,1,0],0,Root[1-10#1^2+5#1^4&,4,0]},{Root[1-20#1^2+80#1^4&,2,0],(-1-Sqrt[5])/4,Root[1-10#1^2+5#1^4&,4,0]},{Root[1-20#1^2+80#1^4&,2,0],(1+Sqrt[5])/4,Root[1-10#1^2+5#1^4&,4,0]},{Root[1-40#1^2+80#1^4&,4,0],-1/2,Root[1-10#1^2+5#1^4&,4,0]},{Root[1-40#1^2+80#1^4&,4,0],1/2,Root[1-10#1^2+5#1^4&,4,0]}}
PolyhedronVertices[JohnsonSolid[26]]:={{-1/2,-1/2,0},{-1/2,0,Sqrt[3]/2},{-1/2,1/2,0},{0,-1/2,-Sqrt[3]/2},{0,1/2,-Sqrt[3]/2},{1/2,-1/2,0},{1/2,0,Sqrt[3]/2},{1/2,1/2,0}}
PolyhedronVertices[JohnsonSolid[27]]:={{0,-1,0},{0,1,0},{-1/(2Sqrt[3]),-1/2,-Sqrt[2/3]},{-1/(2Sqrt[3]),-1/2,Sqrt[2/3]},{-1/(2Sqrt[3]),1/2,-Sqrt[2/3]},{-1/(2Sqrt[3]),1/2,Sqrt[2/3]},{1/Sqrt[3],0,-Sqrt[2/3]},{1/Sqrt[3],0,Sqrt[2/3]},{-Sqrt[3]/2,-1/2,0},{-Sqrt[3]/2,1/2,0},{Sqrt[3]/2,-1/2,0},{Sqrt[3]/2,1/2,0}}
PolyhedronVertices[JohnsonSolid[28]]:={{0,-(1/Sqrt[2]),-(1/Sqrt[2])},{0,-(1/Sqrt[2]),1/Sqrt[2]},{0,1/Sqrt[2],-(1/Sqrt[2])},{0,1/Sqrt[2],1/Sqrt[2]},{-(1/Sqrt[2]),0,-(1/Sqrt[2])},{-(1/Sqrt[2]),0,1/Sqrt[2]},{1/Sqrt[2],0,-(1/Sqrt[2])},{1/Sqrt[2],0,1/Sqrt[2]},{1/2,(1+Sqrt[2])/2,0},{-1/2,(1+Sqrt[2])/2,0},{(-1-Sqrt[2])/2,1/2,0},{(-1-Sqrt[2])/2,-1/2,0},{-1/2,(-1-Sqrt[2])/2,0},{1/2,(-1-Sqrt[2])/2,0},{(1+Sqrt[2])/2,-1/2,0},{(1+Sqrt[2])/2,1/2,0}}
PolyhedronVertices[JohnsonSolid[29]]:={{-1/2,-1/2,-(1/Sqrt[2])},{-1/2,1/2,-(1/Sqrt[2])},{0,-(1/Sqrt[2]),1/Sqrt[2]},{0,1/Sqrt[2],1/Sqrt[2]},{1/2,-1/2,-(1/Sqrt[2])},{1/2,1/2,-(1/Sqrt[2])},{-(1/Sqrt[2]),0,1/Sqrt[2]},{1/Sqrt[2],0,1/Sqrt[2]},{1/2,(1+Sqrt[2])/2,0},{-1/2,(1+Sqrt[2])/2,0},{(-1-Sqrt[2])/2,1/2,0},{(-1-Sqrt[2])/2,-1/2,0},{-1/2,(-1-Sqrt[2])/2,0},{1/2,(-1-Sqrt[2])/2,0},{(1+Sqrt[2])/2,-1/2,0},{(1+Sqrt[2])/2,1/2,0},{(1+Sqrt[2])/2,1/2,0},{1/2,(1+Sqrt[2])/2,0},{-1/2,(1+Sqrt[2])/2,0},{(-1-Sqrt[2])/2,1/2,0},{(-1-Sqrt[2])/2,-1/2,0},{-1/2,(-1-Sqrt[2])/2,0},{1/2,(-1-Sqrt[2])/2,0},{(1+Sqrt[2])/2,-1/2,0}}
PolyhedronVertices[JohnsonSolid[30]]:={{0,(-1-Sqrt[5])/2,0},{0,(1+Sqrt[5])/2,0},{Sqrt[1/2+1/(2Sqrt[5])],0,-Sqrt[(5-Sqrt[5])/10]},{Sqrt[1/2+1/(2Sqrt[5])],0,Sqrt[(5-Sqrt[5])/10]},{Sqrt[(5-Sqrt[5])/10]/2,(-1-Sqrt[5])/4,-Sqrt[(5-Sqrt[5])/10]},{Sqrt[(5-Sqrt[5])/10]/2,(-1-Sqrt[5])/4,Sqrt[(5-Sqrt[5])/10]},{Sqrt[(5-Sqrt[5])/10]/2,(1+Sqrt[5])/4,-Sqrt[(5-Sqrt[5])/10]},{Sqrt[(5-Sqrt[5])/10]/2,(1+Sqrt[5])/4,Sqrt[(5-Sqrt[5])/10]},{-Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,0},{-Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,0},{Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,0},{Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,0},{-Sqrt[1/4+1/(2Sqrt[5])],-1/2,-Sqrt[(5-Sqrt[5])/10]},{-Sqrt[1/4+1/(2Sqrt[5])],-1/2,Sqrt[(5-Sqrt[5])/10]},{-Sqrt[1/4+1/(2Sqrt[5])],1/2,-Sqrt[(5-Sqrt[5])/10]},{-Sqrt[1/4+1/(2Sqrt[5])],1/2,Sqrt[(5-Sqrt[5])/10]},{-Sqrt[5/4+Sqrt[5]/2],-1/2,0},{-Sqrt[5/4+Sqrt[5]/2],1/2,0},{Sqrt[5/4+Sqrt[5]/2],-1/2,0},{Sqrt[5/4+Sqrt[5]/2],1/2,0}}
PolyhedronVertices[JohnsonSolid[31]]:={{0,(-1-Sqrt[5])/2,0},{0,(1+Sqrt[5])/2,0},{0,(-1-Sqrt[5])/2,0},{0,(1+Sqrt[5])/2,0},{Sqrt[1/2+1/(2Sqrt[5])],0,Sqrt[(5-Sqrt[5])/10]},{-Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,0},{Sqrt[(5-Sqrt[5])/10]/2,(-1-Sqrt[5])/4,Sqrt[(5-Sqrt[5])/10]},{Sqrt[(5-Sqrt[5])/10]/2,(1+Sqrt[5])/4,Sqrt[(5-Sqrt[5])/10]},{-Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,0},{-Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,0},{Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,0},{Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,0},{-Sqrt[1/4+1/(2Sqrt[5])],-1/2,Sqrt[(5-Sqrt[5])/10]},{-Sqrt[1/4+1/(2Sqrt[5])],1/2,Sqrt[(5-Sqrt[5])/10]},{Sqrt[1/4+1/(2Sqrt[5])],-1/2,-Sqrt[(5-Sqrt[5])/10]},{-Sqrt[5/4+Sqrt[5]/2],-1/2,0},{Sqrt[5/4+Sqrt[5]/2],1/2,0},{-Sqrt[5/4+Sqrt[5]/2],-1/2,0},{-Sqrt[5/4+Sqrt[5]/2],1/2,0},{Sqrt[5/4+Sqrt[5]/2],-1/2,0},{Sqrt[5/4+Sqrt[5]/2],1/2,0},{-Sqrt[1/2+1/(2Sqrt[5])],0,-Sqrt[(5-Sqrt[5])/10]},{-Sqrt[(5-Sqrt[5])/10]/2,(1+Sqrt[5])/4,-Sqrt[(5-Sqrt[5])/10]},{-Sqrt[(5-Sqrt[5])/10]/2,(-1-Sqrt[5])/4,-Sqrt[(5-Sqrt[5])/10]},{Sqrt[1/4+1/(2Sqrt[5])],1/2,-Sqrt[(5-Sqrt[5])/10]},{-Sqrt[5/4+Sqrt[5]/2],1/2,0},{-Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,0},{Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,0},{Sqrt[5/4+Sqrt[5]/2],-1/2,0}}
PolyhedronVertices[JohnsonSolid[32]]:={{0,(-1-Sqrt[5])/2,0},{0,(1+Sqrt[5])/2,0},{0,(-1-Sqrt[5])/2,0},{0,(1+Sqrt[5])/2,0},{-Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,0},{-Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,0},{-Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,0},{Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,0},{Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,0},{Sqrt[1/4+1/(2Sqrt[5])],-1/2,-Sqrt[(5-Sqrt[5])/10]},{-Sqrt[5/4+Sqrt[5]/2],-1/2,0},{Sqrt[5/4+Sqrt[5]/2],1/2,0},{-Sqrt[5/4+Sqrt[5]/2],-1/2,0},{-Sqrt[5/4+Sqrt[5]/2],1/2,0},{Sqrt[5/4+Sqrt[5]/2],-1/2,0},{Sqrt[5/4+Sqrt[5]/2],1/2,0},{Sqrt[(5+2Sqrt[5])/5],0,Sqrt[1/2+1/(2Sqrt[5])]},{Sqrt[1/8+1/(8Sqrt[5])],(-3-Sqrt[5])/4,Sqrt[1/2+1/(2Sqrt[5])]},{Sqrt[1/8+1/(8Sqrt[5])],(3+Sqrt[5])/4,Sqrt[1/2+1/(2Sqrt[5])]},{-Sqrt[5/8+11/(8Sqrt[5])],(-1-Sqrt[5])/4,Sqrt[1/2+1/(2Sqrt[5])]},{-Sqrt[5/8+11/(8Sqrt[5])],(1+Sqrt[5])/4,Sqrt[1/2+1/(2Sqrt[5])]},{-Sqrt[1/2+1/(2Sqrt[5])],0,Sqrt[(5+2Sqrt[5])/5]},{-Sqrt[(5-Sqrt[5])/10]/2,(-1-Sqrt[5])/4,Sqrt[(5+2Sqrt[5])/5]},{-Sqrt[(5-Sqrt[5])/10]/2,(1+Sqrt[5])/4,Sqrt[(5+2Sqrt[5])/5]},{Sqrt[1/4+1/(2Sqrt[5])],-1/2,Sqrt[(5+2Sqrt[5])/5]},{Sqrt[1/4+1/(2Sqrt[5])],1/2,Sqrt[(5+2Sqrt[5])/5]},{-Sqrt[1/2+1/(2Sqrt[5])],0,-Sqrt[(5-Sqrt[5])/10]},{-Sqrt[(5-Sqrt[5])/10]/2,(1+Sqrt[5])/4,-Sqrt[(5-Sqrt[5])/10]},{-Sqrt[(5-Sqrt[5])/10]/2,(-1-Sqrt[5])/4,-Sqrt[(5-Sqrt[5])/10]},{Sqrt[1/4+1/(2Sqrt[5])],1/2,-Sqrt[(5-Sqrt[5])/10]},{-Sqrt[5/4+Sqrt[5]/2],1/2,0},{-Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,0},{Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,0},{Sqrt[5/4+Sqrt[5]/2],-1/2,0}}
PolyhedronVertices[JohnsonSolid[33]]:={{0,(-1-Sqrt[5])/2,0},{0,(1+Sqrt[5])/2,0},{Sqrt[1/2+1/(2Sqrt[5])],0,-Sqrt[(5-Sqrt[5])/10]},{Sqrt[(5-Sqrt[5])/10]/2,(-1-Sqrt[5])/4,-Sqrt[(5-Sqrt[5])/10]},{Sqrt[(5-Sqrt[5])/10]/2,(1+Sqrt[5])/4,-Sqrt[(5-Sqrt[5])/10]},{-Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,0},{-Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,0},{Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,0},{Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,0},{-Sqrt[1/4+1/(2Sqrt[5])],-1/2,-Sqrt[(5-Sqrt[5])/10]},{-Sqrt[1/4+1/(2Sqrt[5])],1/2,-Sqrt[(5-Sqrt[5])/10]},{-Sqrt[5/4+Sqrt[5]/2],-1/2,0},{-Sqrt[5/4+Sqrt[5]/2],1/2,0},{Sqrt[5/4+Sqrt[5]/2],-1/2,0},{Sqrt[5/4+Sqrt[5]/2],1/2,0},{Sqrt[(5+2Sqrt[5])/5],0,Sqrt[1/2+1/(2Sqrt[5])]},{Sqrt[1/8+1/(8Sqrt[5])],(-3-Sqrt[5])/4,Sqrt[1/2+1/(2Sqrt[5])]},{Sqrt[1/8+1/(8Sqrt[5])],(3+Sqrt[5])/4,Sqrt[1/2+1/(2Sqrt[5])]},{-Sqrt[5/8+11/(8Sqrt[5])],(-1-Sqrt[5])/4,Sqrt[1/2+1/(2Sqrt[5])]},{-Sqrt[5/8+11/(8Sqrt[5])],(1+Sqrt[5])/4,Sqrt[1/2+1/(2Sqrt[5])]},{-Sqrt[1/2+1/(2Sqrt[5])],0,Sqrt[(5+2Sqrt[5])/5]},{-Sqrt[(5-Sqrt[5])/10]/2,(-1-Sqrt[5])/4,Sqrt[(5+2Sqrt[5])/5]},{-Sqrt[(5-Sqrt[5])/10]/2,(1+Sqrt[5])/4,Sqrt[(5+2Sqrt[5])/5]},{Sqrt[1/4+1/(2Sqrt[5])],-1/2,Sqrt[(5+2Sqrt[5])/5]},{Sqrt[1/4+1/(2Sqrt[5])],1/2,Sqrt[(5+2Sqrt[5])/5]}}
PolyhedronVertices[JohnsonSolid[34]]:={{0,(-1-Sqrt[5])/2,0},{0,(1+Sqrt[5])/2,0},{-Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,0},{-Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,0},{Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,0},{Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,0},{-Sqrt[5/4+Sqrt[5]/2],-1/2,0},{-Sqrt[5/4+Sqrt[5]/2],1/2,0},{Sqrt[5/4+Sqrt[5]/2],-1/2,0},{Sqrt[5/4+Sqrt[5]/2],1/2,0},{Sqrt[(5+2Sqrt[5])/5],0,-Sqrt[1/2+1/(2Sqrt[5])]},{Sqrt[(5+2Sqrt[5])/5],0,Sqrt[1/2+1/(2Sqrt[5])]},{Sqrt[1/8+1/(8Sqrt[5])],(-3-Sqrt[5])/4,-Sqrt[1/2+1/(2Sqrt[5])]},{Sqrt[1/8+1/(8Sqrt[5])],(-3-Sqrt[5])/4,Sqrt[1/2+1/(2Sqrt[5])]},{Sqrt[1/8+1/(8Sqrt[5])],(3+Sqrt[5])/4,-Sqrt[1/2+1/(2Sqrt[5])]},{Sqrt[1/8+1/(8Sqrt[5])],(3+Sqrt[5])/4,Sqrt[1/2+1/(2Sqrt[5])]},{-Sqrt[5/8+11/(8Sqrt[5])],(-1-Sqrt[5])/4,-Sqrt[1/2+1/(2Sqrt[5])]},{-Sqrt[5/8+11/(8Sqrt[5])],(-1-Sqrt[5])/4,Sqrt[1/2+1/(2Sqrt[5])]},{-Sqrt[5/8+11/(8Sqrt[5])],(1+Sqrt[5])/4,-Sqrt[1/2+1/(2Sqrt[5])]},{-Sqrt[5/8+11/(8Sqrt[5])],(1+Sqrt[5])/4,Sqrt[1/2+1/(2Sqrt[5])]},{-Sqrt[1/2+1/(2Sqrt[5])],0,-Sqrt[(5+2Sqrt[5])/5]},{-Sqrt[1/2+1/(2Sqrt[5])],0,Sqrt[(5+2Sqrt[5])/5]},{-Sqrt[(5-Sqrt[5])/10]/2,(-1-Sqrt[5])/4,-Sqrt[(5+2Sqrt[5])/5]},{-Sqrt[(5-Sqrt[5])/10]/2,(-1-Sqrt[5])/4,Sqrt[(5+2Sqrt[5])/5]},{-Sqrt[(5-Sqrt[5])/10]/2,(1+Sqrt[5])/4,-Sqrt[(5+2Sqrt[5])/5]},{-Sqrt[(5-Sqrt[5])/10]/2,(1+Sqrt[5])/4,Sqrt[(5+2Sqrt[5])/5]},{Sqrt[1/4+1/(2Sqrt[5])],-1/2,-Sqrt[(5+2Sqrt[5])/5]},{Sqrt[1/4+1/(2Sqrt[5])],-1/2,Sqrt[(5+2Sqrt[5])/5]},{Sqrt[1/4+1/(2Sqrt[5])],1/2,-Sqrt[(5+2Sqrt[5])/5]},{Sqrt[1/4+1/(2Sqrt[5])],1/2,Sqrt[(5+2Sqrt[5])/5]}}
PolyhedronVertices[JohnsonSolid[35]]:={{0,-1,-1/2},{0,-1,1/2},{0,1,-1/2},{0,1,1/2},{-1/(2Sqrt[3]),-1/2,(-3-2Sqrt[6])/6},{-1/(2Sqrt[3]),-1/2,(3+2Sqrt[6])/6},{-1/(2Sqrt[3]),1/2,(-3-2Sqrt[6])/6},{-1/(2Sqrt[3]),1/2,(3+2Sqrt[6])/6},{1/Sqrt[3],0,(-3-2Sqrt[6])/6},{1/Sqrt[3],0,(3+2Sqrt[6])/6},{-Sqrt[3]/2,-1/2,-1/2},{-Sqrt[3]/2,-1/2,1/2},{-Sqrt[3]/2,1/2,-1/2},{-Sqrt[3]/2,1/2,1/2},{Sqrt[3]/2,-1/2,-1/2},{Sqrt[3]/2,-1/2,1/2},{Sqrt[3]/2,1/2,-1/2},{Sqrt[3]/2,1/2,1/2}}
PolyhedronVertices[JohnsonSolid[36]]:={{0,-1,-1/2},{0,-1,1/2},{0,1,-1/2},{0,1,1/2},{-1/(2Sqrt[3]),-1/2,(3+2Sqrt[6])/6},{-1/(2Sqrt[3]),1/2,(3+2Sqrt[6])/6},{1/(2Sqrt[3]),-1/2,(-3-2Sqrt[6])/6},{1/Sqrt[3],0,(3+2Sqrt[6])/6},{-Sqrt[3]/2,-1/2,-1/2},{-Sqrt[3]/2,-1/2,1/2},{-Sqrt[3]/2,1/2,-1/2},{-Sqrt[3]/2,1/2,1/2},{Sqrt[3]/2,-1/2,-1/2},{Sqrt[3]/2,-1/2,1/2},{Sqrt[3]/2,1/2,-1/2},{Sqrt[3]/2,1/2,1/2},{-(1/Sqrt[3]),0,(-3-2Sqrt[6])/6},{1/(2Sqrt[3]),1/2,(-3-2Sqrt[6])/6}}
PolyhedronVertices[JohnsonSolid[37]]:={{-1/2,-1/2,(-1-Sqrt[2])/2},{-1/2,1/2,(-1-Sqrt[2])/2},{0,-(1/Sqrt[2]),(1+Sqrt[2])/2},{0,1/Sqrt[2],(1+Sqrt[2])/2},{1/2,-1/2,(-1-Sqrt[2])/2},{1/2,1/2,(-1-Sqrt[2])/2},{-(1/Sqrt[2]),0,(1+Sqrt[2])/2},{1/Sqrt[2],0,(1+Sqrt[2])/2},{1/2,(1+Sqrt[2])/2,-1/2},{1/2,(1+Sqrt[2])/2,1/2},{-1/2,(1+Sqrt[2])/2,-1/2},{-1/2,(1+Sqrt[2])/2,1/2},{(-1-Sqrt[2])/2,1/2,-1/2},{(-1-Sqrt[2])/2,1/2,1/2},{(-1-Sqrt[2])/2,-1/2,-1/2},{(-1-Sqrt[2])/2,-1/2,1/2},{-1/2,(-1-Sqrt[2])/2,-1/2},{-1/2,(-1-Sqrt[2])/2,1/2},{1/2,(-1-Sqrt[2])/2,-1/2},{1/2,(-1-Sqrt[2])/2,1/2},{(1+Sqrt[2])/2,-1/2,-1/2},{(1+Sqrt[2])/2,-1/2,1/2},{(1+Sqrt[2])/2,1/2,-1/2},{(1+Sqrt[2])/2,1/2,1/2},{(1+Sqrt[2])/2,1/2,-1/2},{1/2,(1+Sqrt[2])/2,-1/2},{-1/2,(1+Sqrt[2])/2,-1/2},{(-1-Sqrt[2])/2,1/2,-1/2},{(-1-Sqrt[2])/2,-1/2,-1/2},{-1/2,(-1-Sqrt[2])/2,-1/2},{1/2,(-1-Sqrt[2])/2,-1/2},{(1+Sqrt[2])/2,-1/2,-1/2}}
PolyhedronVertices[JohnsonSolid[38]]:={{0,(-1-Sqrt[5])/2,-1/2},{0,(-1-Sqrt[5])/2,1/2},{0,(1+Sqrt[5])/2,-1/2},{0,(1+Sqrt[5])/2,1/2},{Sqrt[1/2+1/(2Sqrt[5])],0,(-5-Sqrt[10*(5-Sqrt[5])])/10},{Sqrt[1/2+1/(2Sqrt[5])],0,(5+Sqrt[10*(5-Sqrt[5])])/10},{Sqrt[(5-Sqrt[5])/10]/2,(-1-Sqrt[5])/4,(-5-Sqrt[10*(5-Sqrt[5])])/10},{Sqrt[(5-Sqrt[5])/10]/2,(-1-Sqrt[5])/4,(5+Sqrt[10*(5-Sqrt[5])])/10},{Sqrt[(5-Sqrt[5])/10]/2,(1+Sqrt[5])/4,(-5-Sqrt[10*(5-Sqrt[5])])/10},{Sqrt[(5-Sqrt[5])/10]/2,(1+Sqrt[5])/4,(5+Sqrt[10*(5-Sqrt[5])])/10},{-Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,-1/2},{-Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,1/2},{-Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,-1/2},{-Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,1/2},{Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,-1/2},{Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,1/2},{Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,-1/2},{Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,1/2},{-Sqrt[1/4+1/(2Sqrt[5])],-1/2,(-5-Sqrt[10*(5-Sqrt[5])])/10},{-Sqrt[1/4+1/(2Sqrt[5])],-1/2,(5+Sqrt[10*(5-Sqrt[5])])/10},{-Sqrt[1/4+1/(2Sqrt[5])],1/2,(-5-Sqrt[10*(5-Sqrt[5])])/10},{-Sqrt[1/4+1/(2Sqrt[5])],1/2,(5+Sqrt[10*(5-Sqrt[5])])/10},{-Sqrt[5/4+Sqrt[5]/2],-1/2,-1/2},{-Sqrt[5/4+Sqrt[5]/2],-1/2,1/2},{-Sqrt[5/4+Sqrt[5]/2],1/2,-1/2},{-Sqrt[5/4+Sqrt[5]/2],1/2,1/2},{Sqrt[5/4+Sqrt[5]/2],-1/2,-1/2},{Sqrt[5/4+Sqrt[5]/2],-1/2,1/2},{Sqrt[5/4+Sqrt[5]/2],1/2,-1/2},{Sqrt[5/4+Sqrt[5]/2],1/2,1/2}}
PolyhedronVertices[JohnsonSolid[39]]:={{0,(-1-Sqrt[5])/2,-1/2},{0,(-1-Sqrt[5])/2,1/2},{0,(1+Sqrt[5])/2,-1/2},{0,(1+Sqrt[5])/2,1/2},{0,(-1-Sqrt[5])/2,-1/2},{0,(1+Sqrt[5])/2,-1/2},{Sqrt[1/2+1/(2Sqrt[5])],0,(5+Sqrt[10*(5-Sqrt[5])])/10},{-Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,-1/2},{Sqrt[(5-Sqrt[5])/10]/2,(-1-Sqrt[5])/4,(5+Sqrt[10*(5-Sqrt[5])])/10},{Sqrt[(5-Sqrt[5])/10]/2,(1+Sqrt[5])/4,(5+Sqrt[10*(5-Sqrt[5])])/10},{-Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,-1/2},{-Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,1/2},{-Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,-1/2},{-Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,1/2},{Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,-1/2},{Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,1/2},{Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,-1/2},{Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,1/2},{-Sqrt[1/4+1/(2Sqrt[5])],-1/2,(5+Sqrt[10*(5-Sqrt[5])])/10},{-Sqrt[1/4+1/(2Sqrt[5])],1/2,(5+Sqrt[10*(5-Sqrt[5])])/10},{Sqrt[1/4+1/(2Sqrt[5])],-1/2,(-5-Sqrt[10*(5-Sqrt[5])])/10},{-Sqrt[5/4+Sqrt[5]/2],-1/2,-1/2},{Sqrt[5/4+Sqrt[5]/2],1/2,-1/2},{-Sqrt[5/4+Sqrt[5]/2],-1/2,-1/2},{-Sqrt[5/4+Sqrt[5]/2],-1/2,1/2},{-Sqrt[5/4+Sqrt[5]/2],1/2,-1/2},{-Sqrt[5/4+Sqrt[5]/2],1/2,1/2},{Sqrt[5/4+Sqrt[5]/2],-1/2,-1/2},{Sqrt[5/4+Sqrt[5]/2],-1/2,1/2},{Sqrt[5/4+Sqrt[5]/2],1/2,-1/2},{Sqrt[5/4+Sqrt[5]/2],1/2,1/2},{-Sqrt[1/2+1/(2Sqrt[5])],0,(-5-Sqrt[10*(5-Sqrt[5])])/10},{-Sqrt[(5-Sqrt[5])/10]/2,(1+Sqrt[5])/4,(-5-Sqrt[10*(5-Sqrt[5])])/10},{-Sqrt[(5-Sqrt[5])/10]/2,(-1-Sqrt[5])/4,(-5-Sqrt[10*(5-Sqrt[5])])/10},{Sqrt[1/4+1/(2Sqrt[5])],1/2,(-5-Sqrt[10*(5-Sqrt[5])])/10},{-Sqrt[5/4+Sqrt[5]/2],1/2,-1/2},{-Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,-1/2},{Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,-1/2},{Sqrt[5/4+Sqrt[5]/2],-1/2,-1/2}}
PolyhedronVertices[JohnsonSolid[40]]:={{0,(-1-Sqrt[5])/2,-1/2},{0,(-1-Sqrt[5])/2,1/2},{0,(1+Sqrt[5])/2,-1/2},{0,(1+Sqrt[5])/2,1/2},{0,(-1-Sqrt[5])/2,-1/2},{0,(1+Sqrt[5])/2,-1/2},{-Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,-1/2},{-Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,-1/2},{-Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,1/2},{-Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,-1/2},{-Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,1/2},{Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,-1/2},{Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,1/2},{Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,-1/2},{Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,1/2},{Sqrt[1/4+1/(2Sqrt[5])],-1/2,(-5-Sqrt[10*(5-Sqrt[5])])/10},{-Sqrt[5/4+Sqrt[5]/2],-1/2,-1/2},{Sqrt[5/4+Sqrt[5]/2],1/2,-1/2},{-Sqrt[5/4+Sqrt[5]/2],-1/2,-1/2},{-Sqrt[5/4+Sqrt[5]/2],-1/2,1/2},{-Sqrt[5/4+Sqrt[5]/2],1/2,-1/2},{-Sqrt[5/4+Sqrt[5]/2],1/2,1/2},{Sqrt[5/4+Sqrt[5]/2],-1/2,-1/2},{Sqrt[5/4+Sqrt[5]/2],-1/2,1/2},{Sqrt[5/4+Sqrt[5]/2],1/2,-1/2},{Sqrt[5/4+Sqrt[5]/2],1/2,1/2},{Sqrt[(5+2Sqrt[5])/5],0,(5+Sqrt[10*(5+Sqrt[5])])/10},{Sqrt[1/8+1/(8Sqrt[5])],(-3-Sqrt[5])/4,(5+Sqrt[10*(5+Sqrt[5])])/10},{Sqrt[1/8+1/(8Sqrt[5])],(3+Sqrt[5])/4,(5+Sqrt[10*(5+Sqrt[5])])/10},{-Sqrt[5/8+11/(8Sqrt[5])],(-1-Sqrt[5])/4,(5+Sqrt[10*(5+Sqrt[5])])/10},{-Sqrt[5/8+11/(8Sqrt[5])],(1+Sqrt[5])/4,(5+Sqrt[10*(5+Sqrt[5])])/10},{-Sqrt[1/2+1/(2Sqrt[5])],0,(5+2Sqrt[5*(5+2Sqrt[5])])/10},{-Sqrt[(5-Sqrt[5])/10]/2,(-1-Sqrt[5])/4,(5+2Sqrt[5*(5+2Sqrt[5])])/10},{-Sqrt[(5-Sqrt[5])/10]/2,(1+Sqrt[5])/4,(5+2Sqrt[5*(5+2Sqrt[5])])/10},{Sqrt[1/4+1/(2Sqrt[5])],-1/2,(5+2Sqrt[5*(5+2Sqrt[5])])/10},{Sqrt[1/4+1/(2Sqrt[5])],1/2,(5+2Sqrt[5*(5+2Sqrt[5])])/10},{-Sqrt[1/2+1/(2Sqrt[5])],0,(-5-Sqrt[10*(5-Sqrt[5])])/10},{-Sqrt[(5-Sqrt[5])/10]/2,(1+Sqrt[5])/4,(-5-Sqrt[10*(5-Sqrt[5])])/10},{-Sqrt[(5-Sqrt[5])/10]/2,(-1-Sqrt[5])/4,(-5-Sqrt[10*(5-Sqrt[5])])/10},{Sqrt[1/4+1/(2Sqrt[5])],1/2,(-5-Sqrt[10*(5-Sqrt[5])])/10},{-Sqrt[5/4+Sqrt[5]/2],1/2,-1/2},{-Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,-1/2},{Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,-1/2},{Sqrt[5/4+Sqrt[5]/2],-1/2,-1/2}}
PolyhedronVertices[JohnsonSolid[41]]:={{0,(-1-Sqrt[5])/2,-1/2},{0,(-1-Sqrt[5])/2,1/2},{0,(1+Sqrt[5])/2,-1/2},{0,(1+Sqrt[5])/2,1/2},{Sqrt[1/2+1/(2Sqrt[5])],0,(-5-Sqrt[10*(5-Sqrt[5])])/10},{Sqrt[(5-Sqrt[5])/10]/2,(-1-Sqrt[5])/4,(-5-Sqrt[10*(5-Sqrt[5])])/10},{Sqrt[(5-Sqrt[5])/10]/2,(1+Sqrt[5])/4,(-5-Sqrt[10*(5-Sqrt[5])])/10},{-Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,-1/2},{-Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,1/2},{-Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,-1/2},{-Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,1/2},{Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,-1/2},{Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,1/2},{Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,-1/2},{Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,1/2},{-Sqrt[1/4+1/(2Sqrt[5])],-1/2,(-5-Sqrt[10*(5-Sqrt[5])])/10},{-Sqrt[1/4+1/(2Sqrt[5])],1/2,(-5-Sqrt[10*(5-Sqrt[5])])/10},{-Sqrt[5/4+Sqrt[5]/2],-1/2,-1/2},{-Sqrt[5/4+Sqrt[5]/2],-1/2,1/2},{-Sqrt[5/4+Sqrt[5]/2],1/2,-1/2},{-Sqrt[5/4+Sqrt[5]/2],1/2,1/2},{Sqrt[5/4+Sqrt[5]/2],-1/2,-1/2},{Sqrt[5/4+Sqrt[5]/2],-1/2,1/2},{Sqrt[5/4+Sqrt[5]/2],1/2,-1/2},{Sqrt[5/4+Sqrt[5]/2],1/2,1/2},{Sqrt[(5+2Sqrt[5])/5],0,(5+Sqrt[10*(5+Sqrt[5])])/10},{Sqrt[1/8+1/(8Sqrt[5])],(-3-Sqrt[5])/4,(5+Sqrt[10*(5+Sqrt[5])])/10},{Sqrt[1/8+1/(8Sqrt[5])],(3+Sqrt[5])/4,(5+Sqrt[10*(5+Sqrt[5])])/10},{-Sqrt[5/8+11/(8Sqrt[5])],(-1-Sqrt[5])/4,(5+Sqrt[10*(5+Sqrt[5])])/10},{-Sqrt[5/8+11/(8Sqrt[5])],(1+Sqrt[5])/4,(5+Sqrt[10*(5+Sqrt[5])])/10},{-Sqrt[1/2+1/(2Sqrt[5])],0,(5+2Sqrt[5*(5+2Sqrt[5])])/10},{-Sqrt[(5-Sqrt[5])/10]/2,(-1-Sqrt[5])/4,(5+2Sqrt[5*(5+2Sqrt[5])])/10},{-Sqrt[(5-Sqrt[5])/10]/2,(1+Sqrt[5])/4,(5+2Sqrt[5*(5+2Sqrt[5])])/10},{Sqrt[1/4+1/(2Sqrt[5])],-1/2,(5+2Sqrt[5*(5+2Sqrt[5])])/10},{Sqrt[1/4+1/(2Sqrt[5])],1/2,(5+2Sqrt[5*(5+2Sqrt[5])])/10}}
PolyhedronVertices[JohnsonSolid[42]]:={{0,(-1-Sqrt[5])/2,-1/2},{0,(-1-Sqrt[5])/2,1/2},{0,(1+Sqrt[5])/2,-1/2},{0,(1+Sqrt[5])/2,1/2},{-Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,-1/2},{-Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,1/2},{-Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,-1/2},{-Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,1/2},{Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,-1/2},{Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,1/2},{Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,-1/2},{Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,1/2},{-Sqrt[5/4+Sqrt[5]/2],-1/2,-1/2},{-Sqrt[5/4+Sqrt[5]/2],-1/2,1/2},{-Sqrt[5/4+Sqrt[5]/2],1/2,-1/2},{-Sqrt[5/4+Sqrt[5]/2],1/2,1/2},{Sqrt[5/4+Sqrt[5]/2],-1/2,-1/2},{Sqrt[5/4+Sqrt[5]/2],-1/2,1/2},{Sqrt[5/4+Sqrt[5]/2],1/2,-1/2},{Sqrt[5/4+Sqrt[5]/2],1/2,1/2},{Sqrt[(5+2Sqrt[5])/5],0,(-5-Sqrt[10*(5+Sqrt[5])])/10},{Sqrt[(5+2Sqrt[5])/5],0,(5+Sqrt[10*(5+Sqrt[5])])/10},{Sqrt[1/8+1/(8Sqrt[5])],(-3-Sqrt[5])/4,(-5-Sqrt[10*(5+Sqrt[5])])/10},{Sqrt[1/8+1/(8Sqrt[5])],(-3-Sqrt[5])/4,(5+Sqrt[10*(5+Sqrt[5])])/10},{Sqrt[1/8+1/(8Sqrt[5])],(3+Sqrt[5])/4,(-5-Sqrt[10*(5+Sqrt[5])])/10},{Sqrt[1/8+1/(8Sqrt[5])],(3+Sqrt[5])/4,(5+Sqrt[10*(5+Sqrt[5])])/10},{-Sqrt[5/8+11/(8Sqrt[5])],(-1-Sqrt[5])/4,(-5-Sqrt[10*(5+Sqrt[5])])/10},{-Sqrt[5/8+11/(8Sqrt[5])],(-1-Sqrt[5])/4,(5+Sqrt[10*(5+Sqrt[5])])/10},{-Sqrt[5/8+11/(8Sqrt[5])],(1+Sqrt[5])/4,(-5-Sqrt[10*(5+Sqrt[5])])/10},{-Sqrt[5/8+11/(8Sqrt[5])],(1+Sqrt[5])/4,(5+Sqrt[10*(5+Sqrt[5])])/10},{-Sqrt[1/2+1/(2Sqrt[5])],0,(-5-2Sqrt[5*(5+2Sqrt[5])])/10},{-Sqrt[1/2+1/(2Sqrt[5])],0,(5+2Sqrt[5*(5+2Sqrt[5])])/10},{-Sqrt[(5-Sqrt[5])/10]/2,(-1-Sqrt[5])/4,(-5-2Sqrt[5*(5+2Sqrt[5])])/10},{-Sqrt[(5-Sqrt[5])/10]/2,(-1-Sqrt[5])/4,(5+2Sqrt[5*(5+2Sqrt[5])])/10},{-Sqrt[(5-Sqrt[5])/10]/2,(1+Sqrt[5])/4,(-5-2Sqrt[5*(5+2Sqrt[5])])/10},{-Sqrt[(5-Sqrt[5])/10]/2,(1+Sqrt[5])/4,(5+2Sqrt[5*(5+2Sqrt[5])])/10},{Sqrt[1/4+1/(2Sqrt[5])],-1/2,(-5-2Sqrt[5*(5+2Sqrt[5])])/10},{Sqrt[1/4+1/(2Sqrt[5])],-1/2,(5+2Sqrt[5*(5+2Sqrt[5])])/10},{Sqrt[1/4+1/(2Sqrt[5])],1/2,(-5-2Sqrt[5*(5+2Sqrt[5])])/10},{Sqrt[1/4+1/(2Sqrt[5])],1/2,(5+2Sqrt[5*(5+2Sqrt[5])])/10}}
PolyhedronVertices[JohnsonSolid[43]]:={{0,(-1-Sqrt[5])/2,-1/2},{0,(-1-Sqrt[5])/2,1/2},{0,(1+Sqrt[5])/2,-1/2},{0,(1+Sqrt[5])/2,1/2},{0,(-1-Sqrt[5])/2,-1/2},{0,(1+Sqrt[5])/2,-1/2},{-Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,-1/2},{-Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,-1/2},{-Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,1/2},{-Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,-1/2},{-Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,1/2},{Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,-1/2},{Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,1/2},{Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,-1/2},{Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,1/2},{-Sqrt[5/4+Sqrt[5]/2],-1/2,-1/2},{Sqrt[5/4+Sqrt[5]/2],1/2,-1/2},{-Sqrt[5/4+Sqrt[5]/2],-1/2,-1/2},{-Sqrt[5/4+Sqrt[5]/2],-1/2,1/2},{-Sqrt[5/4+Sqrt[5]/2],1/2,-1/2},{-Sqrt[5/4+Sqrt[5]/2],1/2,1/2},{Sqrt[5/4+Sqrt[5]/2],-1/2,-1/2},{Sqrt[5/4+Sqrt[5]/2],-1/2,1/2},{Sqrt[5/4+Sqrt[5]/2],1/2,-1/2},{Sqrt[5/4+Sqrt[5]/2],1/2,1/2},{Sqrt[(5+2Sqrt[5])/5],0,(5+Sqrt[10*(5+Sqrt[5])])/10},{Sqrt[1/8+1/(8Sqrt[5])],(-3-Sqrt[5])/4,(5+Sqrt[10*(5+Sqrt[5])])/10},{Sqrt[1/8+1/(8Sqrt[5])],(3+Sqrt[5])/4,(5+Sqrt[10*(5+Sqrt[5])])/10},{-Sqrt[5/8+11/(8Sqrt[5])],(-1-Sqrt[5])/4,(5+Sqrt[10*(5+Sqrt[5])])/10},{-Sqrt[5/8+11/(8Sqrt[5])],(1+Sqrt[5])/4,(5+Sqrt[10*(5+Sqrt[5])])/10},{Sqrt[5/8+11/(8Sqrt[5])],(-1-Sqrt[5])/4,(-5-Sqrt[10*(5+Sqrt[5])])/10},{-Sqrt[1/2+1/(2Sqrt[5])],0,(5+2Sqrt[5*(5+2Sqrt[5])])/10},{-Sqrt[(5-Sqrt[5])/10]/2,(-1-Sqrt[5])/4,(5+2Sqrt[5*(5+2Sqrt[5])])/10},{-Sqrt[(5-Sqrt[5])/10]/2,(1+Sqrt[5])/4,(5+2Sqrt[5*(5+2Sqrt[5])])/10},{-Sqrt[1/4+1/(2Sqrt[5])],1/2,(-5-2Sqrt[5*(5+2Sqrt[5])])/10},{Sqrt[1/4+1/(2Sqrt[5])],-1/2,(5+2Sqrt[5*(5+2Sqrt[5])])/10},{Sqrt[1/4+1/(2Sqrt[5])],1/2,(5+2Sqrt[5*(5+2Sqrt[5])])/10},{-Sqrt[5/4+Sqrt[5]/2],1/2,-1/2},{-Sqrt[5/8+Sqrt[5]/8],(3+Sqrt[5])/4,-1/2},{Sqrt[5/8+Sqrt[5]/8],(-3-Sqrt[5])/4,-1/2},{Sqrt[5/4+Sqrt[5]/2],-1/2,-1/2},{-Sqrt[(5+2Sqrt[5])/5],0,(-5-Sqrt[10*(5+Sqrt[5])])/10},{-Sqrt[1/8+1/(8Sqrt[5])],(3+Sqrt[5])/4,(-5-Sqrt[10*(5+Sqrt[5])])/10},{-Sqrt[1/8+1/(8Sqrt[5])],(-3-Sqrt[5])/4,(-5-Sqrt[10*(5+Sqrt[5])])/10},{Sqrt[5/8+11/(8Sqrt[5])],(1+Sqrt[5])/4,(-5-Sqrt[10*(5+Sqrt[5])])/10},{Sqrt[(5-Sqrt[5])/10]/2,(-1-Sqrt[5])/4,(-5-2Sqrt[5*(5+2Sqrt[5])])/10},{Sqrt[1/2+1/(2Sqrt[5])],0,(-5-2Sqrt[5*(5+2Sqrt[5])])/10},{-Sqrt[1/4+1/(2Sqrt[5])],-1/2,(-5-2Sqrt[5*(5+2Sqrt[5])])/10},{Sqrt[(5-Sqrt[5])/10]/2,(1+Sqrt[5])/4,(-5-2Sqrt[5*(5+2Sqrt[5])])/10}}
PolyhedronVertices[JohnsonSolid[44]]:={{-1,0,1/Sqrt[6]},{-1,0,Sqrt[-1/4+Sqrt[3]/4]},{-1/2,-Sqrt[3]/2,1/Sqrt[6]},{-1/2,-Sqrt[3]/2,Sqrt[-1/4+Sqrt[3]/4]},{-1/2,Sqrt[3]/2,1/Sqrt[6]},{-1/2,Sqrt[3]/2,Sqrt[-1/4+Sqrt[3]/4]},{-1/2,-1/(2Sqrt[3]),Sqrt[3/2]},{0,-1,-(1/Sqrt[6])},{0,-1,-Sqrt[-1/4+Sqrt[3]/4]},{0,1,-(1/Sqrt[6])},{0,1,-Sqrt[-1/4+Sqrt[3]/4]},{0,1/Sqrt[3],Sqrt[3/2]},{1/2,-1/(2Sqrt[3]),Sqrt[3/2]},{1/2,-Sqrt[3]/2,1/Sqrt[6]},{1/2,-Sqrt[3]/2,Sqrt[-1/4+Sqrt[3]/4]},{1/2,Sqrt[3]/2,1/Sqrt[6]},{1/2,Sqrt[3]/2,Sqrt[-1/4+Sqrt[3]/4]},{1,0,1/Sqrt[6]},{1,0,Sqrt[-1/4+Sqrt[3]/4]},{-(1/Sqrt[3]),0,-Sqrt[3/2]},{1/(2Sqrt[3]),-1/2,-Sqrt[3/2]},{1/(2Sqrt[3]),1/2,-Sqrt[3/2]},{-Sqrt[3]/2,-1/2,-(1/Sqrt[6])},{-Sqrt[3]/2,-1/2,-Sqrt[-1/4+Sqrt[3]/4]},{-Sqrt[3]/2,1/2,-(1/Sqrt[6])},{-Sqrt[3]/2,1/2,-Sqrt[-1/4+Sqrt[3]/4]},{Sqrt[3]/2,-1/2,-(1/Sqrt[6])},{Sqrt[3]/2,-1/2,-Sqrt[-1/4+Sqrt[3]/4]},{Sqrt[3]/2,1/2,-(1/Sqrt[6])},{Sqrt[3]/2,1/2,-Sqrt[-1/4+Sqrt[3]/4]}}
PolyhedronVertices[JohnsonSolid[45]]:={{0,-(1/Sqrt[2]),Root[7+16#1-32#1^2+32#1^4&,1,0]},{0,1/Sqrt[2],Root[7+16#1-32#1^2+32#1^4&,1,0]},{0,Root[1-4#1^2+2#1^4&,1,0],Root[-1+32#1^2-256#1^4+512#1^6+512#1^8&,6,0]},{0,Root[1-4#1^2+2#1^4&,4,0],Root[-1+32#1^2-256#1^4+512#1^6+512#1^8&,6,0]},{-(1/Sqrt[2]),0,Root[7+16#1-32#1^2+32#1^4&,1,0]},{1/Sqrt[2],0,Root[7+16#1-32#1^2+32#1^4&,1,0]},{Root[1-16#1^2+32#1^4&,1,0],Root[1-16#1^2+32#1^4&,3,0],Root[7-16#1-32#1^2+32#1^4&,2,0]},{Root[1-16#1^2+32#1^4&,4,0],Root[1-16#1^2+32#1^4&,2,0],Root[7-16#1-32#1^2+32#1^4&,2,0]},{(1+Sqrt[2])/2,1/2,Root[-1+32#1^2-256#1^4+512#1^6+512#1^8&,1,0]},{Root[1-4#1^2+2#1^4&,1,0],0,Root[-1+32#1^2-256#1^4+512#1^6+512#1^8&,6,0]},{Root[1-4#1^2+2#1^4&,4,0],0,Root[-1+32#1^2-256#1^4+512#1^6+512#1^8&,6,0]},{Root[1-8#1^2+8#1^4&,1,0],Root[1-8#1^2+8#1^4&,1,0],Root[-1+32#1^2-256#1^4+512#1^6+512#1^8&,6,0]},{Root[1-8#1^2+8#1^4&,1,0],Root[1-8#1^2+8#1^4&,4,0],Root[-1+32#1^2-256#1^4+512#1^6+512#1^8&,6,0]},{Root[1-8#1^2+8#1^4&,4,0],Root[1-8#1^2+8#1^4&,1,0],Root[-1+32#1^2-256#1^4+512#1^6+512#1^8&,6,0]},{Root[1-8#1^2+8#1^4&,4,0],Root[1-8#1^2+8#1^4&,4,0],Root[-1+32#1^2-256#1^4+512#1^6+512#1^8&,6,0]},{-1/2,(-1-Sqrt[2])/2,Root[-1+32#1^4&,1,0]},{1/2,(1+Sqrt[2])/2,Root[-1+32#1^2-256#1^4+512#1^6+512#1^8&,1,0]},{1/2,(-1-Sqrt[2])/2,Root[-1+32#1^4&,1,0]},{-1/2,(1+Sqrt[2])/2,Root[-1+32#1^2-256#1^4+512#1^6+512#1^8&,1,0]},{(1+Sqrt[2])/2,-1/2,Root[-1+32#1^4&,1,0]},{(-1-Sqrt[2])/2,1/2,Root[-1+32#1^2-256#1^4+512#1^6+512#1^8&,1,0]},{(1+Sqrt[2])/2,1/2,Root[-1+32#1^4&,1,0]},{(-1-Sqrt[2])/2,-1/2,Root[-1+32#1^2-256#1^4+512#1^6+512#1^8&,1,0]},{1/2,(1+Sqrt[2])/2,Root[-1+32#1^4&,1,0]},{-1/2,(-1-Sqrt[2])/2,Root[-1+32#1^2-256#1^4+512#1^6+512#1^8&,1,0]},{-1/2,(1+Sqrt[2])/2,Root[-1+32#1^4&,1,0]},{1/2,(-1-Sqrt[2])/2,Root[-1+32#1^2-256#1^4+512#1^6+512#1^8&,1,0]},{(-1-Sqrt[2])/2,1/2,Root[-1+32#1^4&,1,0]},{(1+Sqrt[2])/2,-1/2,Root[-1+32#1^2-256#1^4+512#1^6+512#1^8&,1,0]},{(-1-Sqrt[2])/2,-1/2,Root[-1+32#1^4&,1,0]},{(1+Sqrt[2])/2,1/2,Root[-1+32#1^2-256#1^4+512#1^6+512#1^8&,1,0]},{Root[1-16#1^2+32#1^4&,2,0],Root[1-16#1^2+32#1^4&,1,0],Root[7-16#1-32#1^2+32#1^4&,2,0]},{Root[1-16#1^2+32#1^4&,3,0],Root[1-16#1^2+32#1^4&,4,0],Root[7-16#1-32#1^2+32#1^4&,2,0]},{Root[1-8#1^2+8#1^4&,4,0],Root[1-8#1^2+8#1^4&,4,0],Root[-1+32#1^4&,2,0]},{0,Root[1-4#1^2+2#1^4&,4,0],Root[-1+32#1^4&,2,0]},{Root[1-8#1^2+8#1^4&,1,0],Root[1-8#1^2+8#1^4&,4,0],Root[-1+32#1^4&,2,0]},{Root[1-4#1^2+2#1^4&,1,0],0,Root[-1+32#1^4&,2,0]},{Root[1-8#1^2+8#1^4&,1,0],Root[1-8#1^2+8#1^4&,1,0],Root[-1+32#1^4&,2,0]},{0,Root[1-4#1^2+2#1^4&,1,0],Root[-1+32#1^4&,2,0]},{Root[1-8#1^2+8#1^4&,4,0],Root[1-8#1^2+8#1^4&,1,0],Root[-1+32#1^4&,2,0]},{Root[1-4#1^2+2#1^4&,4,0],0,Root[-1+32#1^4&,2,0]}}
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PolyhedronVertices[JohnsonSolid[49]]:={{-1/(2Sqrt[3]),-1/2,-1/2},{-1/(2Sqrt[3]),-1/2,1/2},{-1/(2Sqrt[3]),1/2,-1/2},{-1/(2Sqrt[3]),1/2,1/2},{1/Sqrt[3],0,-1/2},{1/Sqrt[3],0,1/2},{Root[25-672#1^2+2304#1^4&,4,0],(1+Sqrt[6])/4,0}}
PolyhedronVertices[JohnsonSolid[50]]:={{-1/(2Sqrt[3]),-1/2,-1/2},{-1/(2Sqrt[3]),-1/2,1/2},{-1/(2Sqrt[3]),1/2,-1/2},{-1/(2Sqrt[3]),1/2,1/2},{1/Sqrt[3],0,-1/2},{1/Sqrt[3],0,1/2},{Root[25-168#1^2+144#1^4&,1,0],0,0},{Root[25-672#1^2+2304#1^4&,4,0],(1+Sqrt[6])/4,0}}
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PolyhedronVertices[JohnsonSolid[52]]:={{Root[1-5#1^2+5#1^4&,4,0],0,-1/2},{Root[1-5#1^2+5#1^4&,4,0],0,1/2},{Root[1-20#1^2+80#1^4&,3,0],(-1-Sqrt[5])/4,-1/2},{Root[1-20#1^2+80#1^4&,3,0],(-1-Sqrt[5])/4,1/2},{Root[1-20#1^2+80#1^4&,3,0],(1+Sqrt[5])/4,-1/2},{Root[1-20#1^2+80#1^4&,3,0],(1+Sqrt[5])/4,1/2},{Root[1-40#1^2+80#1^4&,1,0],-1/2,-1/2},{Root[1-40#1^2+80#1^4&,1,0],-1/2,1/2},{Root[1-40#1^2+80#1^4&,1,0],1/2,-1/2},{Root[1-40#1^2+80#1^4&,1,0],1/2,1/2},{Root[1-4320#1^2+213760#1^4-2252800#1^6+1638400#1^8&,8,0],Root[1+88#1-176#1^2-128#1^3+256#1^4&,4,0],0}}
PolyhedronVertices[JohnsonSolid[53]]:={{Root[1-5#1^2+5#1^4&,4,0],0,-1/2},{Root[1-5#1^2+5#1^4&,4,0],0,1/2},{Root[1-20#1^2+80#1^4&,3,0],(-1-Sqrt[5])/4,-1/2},{Root[1-20#1^2+80#1^4&,3,0],(-1-Sqrt[5])/4,1/2},{Root[1-20#1^2+80#1^4&,3,0],(1+Sqrt[5])/4,-1/2},{Root[1-20#1^2+80#1^4&,3,0],(1+Sqrt[5])/4,1/2},{Root[1-40#1^2+80#1^4&,1,0],-1/2,-1/2},{Root[1-40#1^2+80#1^4&,1,0],-1/2,1/2},{Root[1-40#1^2+80#1^4&,1,0],1/2,-1/2},{Root[1-40#1^2+80#1^4&,1,0],1/2,1/2},{Root[1-2800#1^2+14560#1^4-19200#1^6+6400#1^8&,1,0],0,0},{Root[1-4320#1^2+213760#1^4-2252800#1^6+1638400#1^8&,8,0],Root[1+88#1-176#1^2-128#1^3+256#1^4&,4,0],0}}
PolyhedronVertices[JohnsonSolid[54]]:={{-1,0,-1/2},{-1,0,1/2},{-1/2,-Sqrt[3]/2,-1/2},{-1/2,-Sqrt[3]/2,1/2},{-1/2,Sqrt[3]/2,-1/2},{-1/2,Sqrt[3]/2,1/2},{1/2,-Sqrt[3]/2,-1/2},{1/2,-Sqrt[3]/2,1/2},{1/2,Sqrt[3]/2,-1/2},{1/2,Sqrt[3]/2,1/2},{1,0,-1/2},{1,0,1/2},{(3+Sqrt[6])/4,Root[1-160#1^2+256#1^4&,4,0],0}}
PolyhedronVertices[JohnsonSolid[55]]:={{-1,0,-1/2},{-1,0,1/2},{-1/2,-Sqrt[3]/2,-1/2},{-1/2,-Sqrt[3]/2,1/2},{-1/2,Sqrt[3]/2,-1/2},{-1/2,Sqrt[3]/2,1/2},{1/2,-Sqrt[3]/2,-1/2},{1/2,-Sqrt[3]/2,1/2},{1/2,Sqrt[3]/2,-1/2},{1/2,Sqrt[3]/2,1/2},{1,0,-1/2},{1,0,1/2},{(-3-Sqrt[6])/4,Root[1-160#1^2+256#1^4&,1,0],0},{(3+Sqrt[6])/4,Root[1-160#1^2+256#1^4&,4,0],0}}
PolyhedronVertices[JohnsonSolid[56]]:={{-1,0,-1/2},{-1,0,1/2},{-1/2,-Sqrt[3]/2,-1/2},{-1/2,-Sqrt[3]/2,1/2},{-1/2,Sqrt[3]/2,-1/2},{-1/2,Sqrt[3]/2,1/2},{1/2,-Sqrt[3]/2,-1/2},{1/2,-Sqrt[3]/2,1/2},{1/2,Sqrt[3]/2,-1/2},{1/2,Sqrt[3]/2,1/2},{1,0,-1/2},{1,0,1/2},{(-3-Sqrt[6])/4,Root[1-160#1^2+256#1^4&,4,0],0},{(3+Sqrt[6])/4,Root[1-160#1^2+256#1^4&,4,0],0}}
PolyhedronVertices[JohnsonSolid[57]]:={{-1,0,-1/2},{-1,0,1/2},{-1/2,-Sqrt[3]/2,-1/2},{-1/2,-Sqrt[3]/2,1/2},{-1/2,Sqrt[3]/2,-1/2},{-1/2,Sqrt[3]/2,1/2},{0,Root[1-40#1^2+16#1^4&,1,0],0},{1/2,-Sqrt[3]/2,-1/2},{1/2,-Sqrt[3]/2,1/2},{1/2,Sqrt[3]/2,-1/2},{1/2,Sqrt[3]/2,1/2},{1,0,-1/2},{1,0,1/2},{(-3-Sqrt[6])/4,Root[1-160#1^2+256#1^4&,4,0],0},{(3+Sqrt[6])/4,Root[1-160#1^2+256#1^4&,4,0],0}}
PolyhedronVertices[JohnsonSolid[58]]:={{0,0,Root[9-36#1^2+16#1^4&,4,0]},{0,0,Root[9-36#1^2+16#1^4&,1,0]},{Root[1-36#1^2+144#1^4&,3,0],(-3-Sqrt[5])/4,Root[1-36#1^2+144#1^4&,4,0]},{Root[1-36#1^2+144#1^4&,3,0],(3+Sqrt[5])/4,Root[1-36#1^2+144#1^4&,4,0]},{Root[1-36#1^2+144#1^4&,4,0],(-1-Sqrt[5])/4,Root[25-180#1^2+144#1^4&,1,0]},{Root[1-36#1^2+144#1^4&,4,0],(1+Sqrt[5])/4,Root[25-180#1^2+144#1^4&,1,0]},{Root[25-180#1^2+144#1^4&,4,0],(-1-Sqrt[5])/4,Root[1-36#1^2+144#1^4&,4,0]},{Root[25-180#1^2+144#1^4&,4,0],(1+Sqrt[5])/4,Root[1-36#1^2+144#1^4&,4,0]},{Root[1-216#1^2+144#1^4&,1,0],-1/2,Root[1-36#1^2+144#1^4&,4,0]},{Root[1-216#1^2+144#1^4&,1,0],1/2,Root[1-36#1^2+144#1^4&,4,0]},{Root[1-216#1^2+144#1^4&,4,0],-1/2,Root[1-36#1^2+144#1^4&,1,0]},{Root[1-216#1^2+144#1^4&,4,0],1/2,Root[1-36#1^2+144#1^4&,1,0]},{Root[1-9#1^2+9#1^4&,1,0],0,Root[25-180#1^2+144#1^4&,1,0]},{Root[1-36#1^2+144#1^4&,1,0],(-1-Sqrt[5])/4,Root[25-180#1^2+144#1^4&,4,0]},{Root[1-36#1^2+144#1^4&,1,0],(1+Sqrt[5])/4,Root[25-180#1^2+144#1^4&,4,0]},{Root[1-9#1^2+9#1^4&,4,0],0,Root[25-180#1^2+144#1^4&,4,0]},{Root[25-180#1^2+144#1^4&,1,0],(-1-Sqrt[5])/4,Root[1-36#1^2+144#1^4&,1,0]},{Root[25-180#1^2+144#1^4&,1,0],(1+Sqrt[5])/4,Root[1-36#1^2+144#1^4&,1,0]},{Root[121-345#1^2+225#1^4&,1,0],0,Root[121-6180#1^2+3600#1^4&,4,0]},{Root[1-36#1^2+144#1^4&,2,0],(-3-Sqrt[5])/4,Root[1-36#1^2+144#1^4&,1,0]},{Root[1-36#1^2+144#1^4&,2,0],(3+Sqrt[5])/4,Root[1-36#1^2+144#1^4&,1,0]}}
PolyhedronVertices[JohnsonSolid[59]]:={{0,0,Root[9-36#1^2+16#1^4&,4,0]},{0,0,Root[9-36#1^2+16#1^4&,1,0]},{Root[1-36#1^2+144#1^4&,3,0],(-3-Sqrt[5])/4,Root[1-36#1^2+144#1^4&,4,0]},{Root[1-36#1^2+144#1^4&,3,0],(3+Sqrt[5])/4,Root[1-36#1^2+144#1^4&,4,0]},{Root[1-36#1^2+144#1^4&,4,0],(-1-Sqrt[5])/4,Root[25-180#1^2+144#1^4&,1,0]},{Root[1-36#1^2+144#1^4&,4,0],(1+Sqrt[5])/4,Root[25-180#1^2+144#1^4&,1,0]},{Root[25-180#1^2+144#1^4&,4,0],(-1-Sqrt[5])/4,Root[1-36#1^2+144#1^4&,4,0]},{Root[25-180#1^2+144#1^4&,4,0],(1+Sqrt[5])/4,Root[1-36#1^2+144#1^4&,4,0]},{Root[1-216#1^2+144#1^4&,1,0],-1/2,Root[1-36#1^2+144#1^4&,4,0]},{Root[1-216#1^2+144#1^4&,1,0],1/2,Root[1-36#1^2+144#1^4&,4,0]},{Root[1-216#1^2+144#1^4&,4,0],-1/2,Root[1-36#1^2+144#1^4&,1,0]},{Root[1-216#1^2+144#1^4&,4,0],1/2,Root[1-36#1^2+144#1^4&,1,0]},{Root[1-9#1^2+9#1^4&,1,0],0,Root[25-180#1^2+144#1^4&,1,0]},{Root[1-36#1^2+144#1^4&,1,0],(-1-Sqrt[5])/4,Root[25-180#1^2+144#1^4&,4,0]},{Root[1-36#1^2+144#1^4&,1,0],(1+Sqrt[5])/4,Root[25-180#1^2+144#1^4&,4,0]},{Root[1-9#1^2+9#1^4&,4,0],0,Root[25-180#1^2+144#1^4&,4,0]},{Root[25-180#1^2+144#1^4&,1,0],(-1-Sqrt[5])/4,Root[1-36#1^2+144#1^4&,1,0]},{Root[25-180#1^2+144#1^4&,1,0],(1+Sqrt[5])/4,Root[1-36#1^2+144#1^4&,1,0]},{Root[121-345#1^2+225#1^4&,4,0],0,Root[121-6180#1^2+3600#1^4&,1,0]},{Root[121-345#1^2+225#1^4&,1,0],0,Root[121-6180#1^2+3600#1^4&,4,0]},{Root[1-36#1^2+144#1^4&,2,0],(-3-Sqrt[5])/4,Root[1-36#1^2+144#1^4&,1,0]},{Root[1-36#1^2+144#1^4&,2,0],(3+Sqrt[5])/4,Root[1-36#1^2+144#1^4&,1,0]}}
PolyhedronVertices[JohnsonSolid[60]]:={{0,0,Root[9-36#1^2+16#1^4&,4,0]},{0,0,Root[9-36#1^2+16#1^4&,1,0]},{Root[1-36#1^2+144#1^4&,3,0],(-3-Sqrt[5])/4,Root[1-36#1^2+144#1^4&,4,0]},{Root[1-36#1^2+144#1^4&,3,0],(3+Sqrt[5])/4,Root[1-36#1^2+144#1^4&,4,0]},{Root[1-36#1^2+144#1^4&,4,0],(-1-Sqrt[5])/4,Root[25-180#1^2+144#1^4&,1,0]},{Root[1-36#1^2+144#1^4&,4,0],(1+Sqrt[5])/4,Root[25-180#1^2+144#1^4&,1,0]},{Root[25-180#1^2+144#1^4&,4,0],(-1-Sqrt[5])/4,Root[1-36#1^2+144#1^4&,4,0]},{Root[25-180#1^2+144#1^4&,4,0],(1+Sqrt[5])/4,Root[1-36#1^2+144#1^4&,4,0]},{Root[1-216#1^2+144#1^4&,1,0],-1/2,Root[1-36#1^2+144#1^4&,4,0]},{Root[1-216#1^2+144#1^4&,1,0],1/2,Root[1-36#1^2+144#1^4&,4,0]},{Root[1-216#1^2+144#1^4&,4,0],-1/2,Root[1-36#1^2+144#1^4&,1,0]},{Root[1-216#1^2+144#1^4&,4,0],1/2,Root[1-36#1^2+144#1^4&,1,0]},{Root[1-9#1^2+9#1^4&,1,0],0,Root[25-180#1^2+144#1^4&,1,0]},{Root[1-36#1^2+144#1^4&,1,0],(-1-Sqrt[5])/4,Root[25-180#1^2+144#1^4&,4,0]},{Root[1-36#1^2+144#1^4&,1,0],(1+Sqrt[5])/4,Root[25-180#1^2+144#1^4&,4,0]},{Root[1-9#1^2+9#1^4&,4,0],0,Root[25-180#1^2+144#1^4&,4,0]},{Root[25-180#1^2+144#1^4&,1,0],(-1-Sqrt[5])/4,Root[1-36#1^2+144#1^4&,1,0]},{Root[25-180#1^2+144#1^4&,1,0],(1+Sqrt[5])/4,Root[1-36#1^2+144#1^4&,1,0]},{Root[121-345#1^2+225#1^4&,1,0],0,Root[121-6180#1^2+3600#1^4&,4,0]},{Root[1-36#1^2+144#1^4&,2,0],(-3-Sqrt[5])/4,Root[1-36#1^2+144#1^4&,1,0]},{Root[1-36#1^2+144#1^4&,2,0],(3+Sqrt[5])/4,Root[1-36#1^2+144#1^4&,1,0]},{Root[121-2520#1^2+3600#1^4&,4,0],(-5-4Sqrt[5])/10,Root[121-1620#1^2+3600#1^4&,2,0]}}
PolyhedronVertices[JohnsonSolid[61]]:={{0,0,Root[9-36#1^2+16#1^4&,4,0]},{0,0,Root[9-36#1^2+16#1^4&,1,0]},{Root[1-36#1^2+144#1^4&,3,0],(-3-Sqrt[5])/4,Root[1-36#1^2+144#1^4&,4,0]},{Root[1-36#1^2+144#1^4&,3,0],(3+Sqrt[5])/4,Root[1-36#1^2+144#1^4&,4,0]},{Root[1-36#1^2+144#1^4&,4,0],(-1-Sqrt[5])/4,Root[25-180#1^2+144#1^4&,1,0]},{Root[1-36#1^2+144#1^4&,4,0],(1+Sqrt[5])/4,Root[25-180#1^2+144#1^4&,1,0]},{Root[25-180#1^2+144#1^4&,4,0],(-1-Sqrt[5])/4,Root[1-36#1^2+144#1^4&,4,0]},{Root[25-180#1^2+144#1^4&,4,0],(1+Sqrt[5])/4,Root[1-36#1^2+144#1^4&,4,0]},{Root[1-216#1^2+144#1^4&,1,0],-1/2,Root[1-36#1^2+144#1^4&,4,0]},{Root[1-216#1^2+144#1^4&,1,0],1/2,Root[1-36#1^2+144#1^4&,4,0]},{Root[1-216#1^2+144#1^4&,4,0],-1/2,Root[1-36#1^2+144#1^4&,1,0]},{Root[1-216#1^2+144#1^4&,4,0],1/2,Root[1-36#1^2+144#1^4&,1,0]},{Root[1-9#1^2+9#1^4&,1,0],0,Root[25-180#1^2+144#1^4&,1,0]},{Root[1-36#1^2+144#1^4&,1,0],(-1-Sqrt[5])/4,Root[25-180#1^2+144#1^4&,4,0]},{Root[1-36#1^2+144#1^4&,1,0],(1+Sqrt[5])/4,Root[25-180#1^2+144#1^4&,4,0]},{Root[1-9#1^2+9#1^4&,4,0],0,Root[25-180#1^2+144#1^4&,4,0]},{Root[25-180#1^2+144#1^4&,1,0],(-1-Sqrt[5])/4,Root[1-36#1^2+144#1^4&,1,0]},{Root[25-180#1^2+144#1^4&,1,0],(1+Sqrt[5])/4,Root[1-36#1^2+144#1^4&,1,0]},{Root[121-345#1^2+225#1^4&,1,0],0,Root[121-6180#1^2+3600#1^4&,4,0]},{Root[1-36#1^2+144#1^4&,2,0],(-3-Sqrt[5])/4,Root[1-36#1^2+144#1^4&,1,0]},{Root[1-36#1^2+144#1^4&,2,0],(3+Sqrt[5])/4,Root[1-36#1^2+144#1^4&,1,0]},{Root[121-2520#1^2+3600#1^4&,4,0],(5+4Sqrt[5])/10,Root[121-1620#1^2+3600#1^4&,2,0]},{Root[121-2520#1^2+3600#1^4&,4,0],(-5-4Sqrt[5])/10,Root[121-1620#1^2+3600#1^4&,2,0]}}
PolyhedronVertices[JohnsonSolid[62]]:={{-1/2,0,(-1-Sqrt[5])/4},{-1/2,0,(1+Sqrt[5])/4},{0,(-1-Sqrt[5])/4,-1/2},{0,(1+Sqrt[5])/4,-1/2},{1/2,0,(-1-Sqrt[5])/4},{1/2,0,(1+Sqrt[5])/4},{(-1-Sqrt[5])/4,-1/2,0},{(-1-Sqrt[5])/4,1/2,0},{(1+Sqrt[5])/4,-1/2,0},{(1+Sqrt[5])/4,1/2,0}}
PolyhedronVertices[JohnsonSolid[63]]:={{-(1/Sqrt[3]),0,Root[1-84#1^2+144#1^4&,1,0]},{-1/(2Sqrt[3]),-1/2,Root[1-84#1^2+144#1^4&,4,0]},{-1/(2Sqrt[3]),1/2,Root[1-84#1^2+144#1^4&,4,0]},{1/(2Sqrt[3]),-1/2,Root[1-84#1^2+144#1^4&,1,0]},{1/(2Sqrt[3]),1/2,Root[1-84#1^2+144#1^4&,1,0]},{1/Sqrt[3],0,Root[1-84#1^2+144#1^4&,4,0]},{Root[1-36#1^2+144#1^4&,4,0],(-1-Sqrt[5])/4,Root[1-36#1^2+144#1^4&,3,0]},{Root[1-36#1^2+144#1^4&,4,0],(1+Sqrt[5])/4,Root[1-36#1^2+144#1^4&,3,0]},{Root[1-9#1^2+9#1^4&,1,0],0,Root[1-36#1^2+144#1^4&,3,0]}}
PolyhedronVertices[JohnsonSolid[64]]:={{0,0,Root[9-2472#1^2+8752#1^4-8832#1^6+2304#1^8&,1,0]},{-(1/Sqrt[3]),0,Root[1-84#1^2+144#1^4&,1,0]},{-1/(2Sqrt[3]),-1/2,Root[1-84#1^2+144#1^4&,4,0]},{-1/(2Sqrt[3]),1/2,Root[1-84#1^2+144#1^4&,4,0]},{1/(2Sqrt[3]),-1/2,Root[1-84#1^2+144#1^4&,1,0]},{1/(2Sqrt[3]),1/2,Root[1-84#1^2+144#1^4&,1,0]},{1/Sqrt[3],0,Root[1-84#1^2+144#1^4&,4,0]},{Root[1-9#1^2+9#1^4&,1,0],0,Root[1-36#1^2+144#1^4&,3,0]},{Root[1-36#1^2+144#1^4&,4,0],(-1-Sqrt[5])/4,Root[1-36#1^2+144#1^4&,3,0]},{Root[1-36#1^2+144#1^4&,4,0],(1+Sqrt[5])/4,Root[1-36#1^2+144#1^4&,3,0]}}
PolyhedronVertices[JohnsonSolid[65]]:={{0,-1,-Sqrt[3/2]/2},{0,1,-Sqrt[3/2]/2},{-(1/Sqrt[3]),-1,1/(2Sqrt[6])},{-(1/Sqrt[3]),1,1/(2Sqrt[6])},{-1/(2Sqrt[3]),-1/2,5/(2Sqrt[6])},{-1/(2Sqrt[3]),1/2,5/(2Sqrt[6])},{1/(2Sqrt[3]),-1/2,-7/(2Sqrt[6])},{1/Sqrt[3],0,5/(2Sqrt[6])},{2/Sqrt[3],0,1/(2Sqrt[6])},{-Sqrt[3]/2,-1/2,-Sqrt[3/2]/2},{-Sqrt[3]/2,1/2,-Sqrt[3/2]/2},{Sqrt[3]/2,-1/2,-Sqrt[3/2]/2},{Sqrt[3]/2,1/2,-Sqrt[3/2]/2},{-(1/Sqrt[3]),0,-7/(2Sqrt[6])},{1/(2Sqrt[3]),1/2,-7/(2Sqrt[6])}}
PolyhedronVertices[JohnsonSolid[66]]:={{-1/2,(1+Sqrt[2])/2,(-1-Sqrt[2])/2},{-1/2,(1+Sqrt[2])/2,(1+Sqrt[2])/2},{-1/2,(-1-Sqrt[2])/2,(-1-Sqrt[2])/2},{-1/2,(-1-Sqrt[2])/2,(1+Sqrt[2])/2},{0,-(1/Sqrt[2]),(1+2*Sqrt[2])/2},{0,1/Sqrt[2],(1+2*Sqrt[2])/2},{1/2,(1+Sqrt[2])/2,(-1-Sqrt[2])/2},{1/2,(1+Sqrt[2])/2,(1+Sqrt[2])/2},{1/2,(-1-Sqrt[2])/2,(-1-Sqrt[2])/2},{1/2,(-1-Sqrt[2])/2,(1+Sqrt[2])/2},{-(1/Sqrt[2]),0,(1+2*Sqrt[2])/2},{1/Sqrt[2],0,(1+2*Sqrt[2])/2},{(1+Sqrt[2])/2,-1/2,(-1-Sqrt[2])/2},{(1+Sqrt[2])/2,-1/2,(1+Sqrt[2])/2},{(1+Sqrt[2])/2,1/2,(-1-Sqrt[2])/2},{(1+Sqrt[2])/2,1/2,(1+Sqrt[2])/2},{(1+Sqrt[2])/2,(1+Sqrt[2])/2,-1/2},{(1+Sqrt[2])/2,(1+Sqrt[2])/2,1/2},{(1+Sqrt[2])/2,(-1-Sqrt[2])/2,-1/2},{(1+Sqrt[2])/2,(-1-Sqrt[2])/2,1/2},{(-1-Sqrt[2])/2,-1/2,(-1-Sqrt[2])/2},{(-1-Sqrt[2])/2,-1/2,(1+Sqrt[2])/2},{(-1-Sqrt[2])/2,1/2,(-1-Sqrt[2])/2},{(-1-Sqrt[2])/2,1/2,(1+Sqrt[2])/2},{(-1-Sqrt[2])/2,(1+Sqrt[2])/2,-1/2},{(-1-Sqrt[2])/2,(1+Sqrt[2])/2,1/2},{(-1-Sqrt[2])/2,(-1-Sqrt[2])/2,-1/2},{(-1-Sqrt[2])/2,(-1-Sqrt[2])/2,1/2},{1/2,(1+Sqrt[2])/2,(1+Sqrt[2])/2},{-1/2,(1+Sqrt[2])/2,(1+Sqrt[2])/2},{(-1-Sqrt[2])/2,1/2,(1+Sqrt[2])/2},{(-1-Sqrt[2])/2,-1/2,(1+Sqrt[2])/2},{-1/2,(-1-Sqrt[2])/2,(1+Sqrt[2])/2},{1/2,(-1-Sqrt[2])/2,(1+Sqrt[2])/2},{(1+Sqrt[2])/2,-1/2,(1+Sqrt[2])/2},{(1+Sqrt[2])/2,1/2,(1+Sqrt[2])/2}}
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PolyhedronVertices[JohnsonSolid[88]]:=Sqrt[2/5]{{-5.1393174775428929,-6.4356967096898658,-1.9798102036351451},{-5.0455221961774503,-6.133411136606041,-.43067346304621851},{-4.998510925243431,-4.9448397155615323,-1.472346965667925},{-4.987200696285929,-7.6257216533886811,-.94990726152031311},{-4.2812030101654498,-5.3967428988756334,-2.8069828215765951},{-4.0220885268230617,-7.5328780540256212,-2.1988822678401698},{-3.757197646615639,-5.2223186443843514,-.53312209825082306},{-3.6627656243198595,-6.7664241494868242,-.86342898750651935},{-3.5785887928366443,-4.3573356740054854,-1.8447122535767787},{-3.1639740594456185,-6.4939242432113889,-3.0260548857816199},{-2.8046511569424161,-5.7274703386725924,-1.6906016054479693},{-2.7851040190065284,-4.9595543509855794,-3.0726008446645943}}
PolyhedronVertices[JohnsonSolid[89]]:=Sqrt[2/5]{{-4.2813369860125861,-4.158584663717826,-9.2324279917721441},{-4.064004129378856,-5.7159902795237062,-9.3974914501135092},{-3.9497036951321801,-5.0488701026147272,-7.9685466334642676},{-3.8585827143091437,-4.7535491711769421,-10.635031307261034},{-3.6434350480078813,-3.4978543016274185,-7.9453706868700946},{-3.1896546100160458,-3.0249294192831716,-9.3842679624793113},{-2.7669003383126033,-3.619893926742288,-10.7868712779682},{-2.7294156219429735,-5.788302899981292,-10.242263693724961},{-2.6624683753743131,-5.6420943609337013,-8.669323412186452},{-2.5142756017623685,-4.5326011451306878,-7.5526066400990057},{-2.1251563847390059,-3.1541728237855979,-8.2223193573401523},{-1.6594738168469554,-3.2190108841061097,-9.7319328776267597},{-1.6377332459464324,-4.6546476555466378,-10.394103664432128},{-1.5707859993777726,-4.5084391164990477,-8.8211633828936189}}
PolyhedronVertices[JohnsonSolid[90]]:={{-1.0658431645866941,2.2918840422530855,-6.5391959450941427},{-.73346896560663036,1.54054675314574,-5.969090981207445},{-.70913815183803692,1.4606535748196595,-6.9655965125074483},{-.56128355834245895,2.8474301535959758,-7.2000954245299007},{-.35513905159035659,2.4592714630717195,-5.8559089518608637},{-.20457854559380171,2.0161996861625495,-7.6264959919432063},{-.01196565899586241,1.007041792358483,-6.4104542383217635},{.11578359713496122,1.0968290498803687,-7.3981867612802239},{.14942055465387838,3.0148175744146099,-6.5168084312966219},{.21876247129843253,1.6579191410975981,-5.6871828245674967},{.40328574228697522,2.7921723862228828,-7.458073414729167},{.63560639923831153,2.5632356893835294,-5.7686803539860885},{.76216165863513069,1.8593936369605565,-7.4244434233874521},{.90880007097631759,1.1012952879362935,-6.789002784213105},{.99288916388112487,2.5102713744296892,-6.7011721600595029},{1.1395267312315534,1.7521735030913253,-6.0657305275323528}}
PolyhedronVertices[JohnsonSolid[91]]:={{-2.4908877360426889,-2.6269398706441472,-6.7373538238346334},{-2.4423106009757433,-1.723572723315908,-6.3112528733371721},{-1.9760179382249007,-2.5011242494159426,-5.8893715095702832},{-1.9720612366592696,-3.4601132003551921,-6.9287481666965509},{-1.8384732297309525,-2.6467487471669611,-7.494954872599727},{-1.7598721355538361,-1.1850669844985983,-6.8055091936526758},{-1.4571892845175618,-3.3342958791814433,-6.0807674127021408},{-1.3866793877481302,-1.7556228169164642,-7.5370782027551546},{-1.084000674491266,-3.9048521010589228,-6.8123306504526101},{-1.0053946759488472,-2.4431699803004968,-6.1228885282757534},{-.87180856297529568,-1.6298034793133951,-6.68909381134051},{-.86784992116402571,-2.5887997191490422,-7.7284736943169402},{-.40155963109641405,-3.3663513517467601,-7.3065888474294439},{-.35297909639119118,-2.462980381545973,-6.8804893029022955}}
PolyhedronVertices[JohnsonSolid[92]]:={{-8.4234137123414135,.86122140608387621,1.7970352529065671},{-8.2820090961865886,.30729775520347504,.97656384967216516},{-8.2243952329259762,.47842626807070854,2.69917688655655},{-8.0112232180570357,1.2699175064431346,.98274630297987142},{-7.9415860006163294,-.62942103369008973,1.0582340800877317},{-7.8839721373557153,-.45829252082285976,2.7808471169721181},{-7.7425675212008919,-1.0122161717032608,1.9603757137377044},{-7.6448499874028328,1.4820550062343055,1.8887094975639214},{-7.4901859456277412,.26774657640986049,3.3445843469772589},{-7.4458314563019813,1.0992593526090058,2.7908514193017548},{-7.4160522623142019,.58578633587574526,.56115758420753304},{-7.1199837627619756,-1.1824447276164118,1.1965617949853747},{-7.0756288841024978,-.35093287599235748,.64282751011001626},{-6.8950145359835839,-.41638514232467971,2.9229971585309107},{-6.8232469955683897,.92903175396421505,2.027037632714718},{-6.753610092750165,-.97030884491695069,2.1025251905435213},{-6.6818421187457562,.37510683412913157,1.2065647988988588},{-6.4828238250398874,-.0076890129283274239,2.1087082937368414}}

End[] 

(* Protect[ ] *)

EndPackage[]

Classical  Polyhedra

Platonics

Table[
        Show[GraphicsArray[{Graphics3D[Polyhedron[Platonics[[i]]],  
                Boxed -> False], 
              UnfoldPolytope[
                Flatten[Flatten[Polyhedron[Platonics[[i]]]], 1, 
                  Polygon]]}],  
          DisplayFunction -> $DisplayFunction], {i, 1, 5}];

Archimedeans

Table[
        Show[GraphicsArray[{Graphics3D[Polyhedron[Archimedeans[[i]]],  
                Boxed -> False], 
              UnfoldPolytope[
                Flatten[Flatten[Polyhedron[Archimedeans[[i]]]], 1, 
                  Polygon]]}],  
          DisplayFunction -> $DisplayFunction], {i, 1, 
          13}];

Table[
        UnfoldPolytope[
          Flatten[Flatten[Polyhedron[Archimedeans[[i]]]], 1, 
            Polygon]], {i, 1, 13}];

Archimedean Duals

Table[
        Show[GraphicsArray[{Graphics3D[
                Polyhedron[ArchimedeanDuals[[i, 1]]],  
                Boxed -> False], 
              UnfoldPolytope[
                Flatten[Flatten[Polyhedron[ArchimedeanDuals[[i, 1]]]], 1, 
                  Polygon]]}],  
          DisplayFunction -> $DisplayFunction], {i, 1, 
          13}];

Table[
        UnfoldPolytope[
          Flatten[Flatten[
              Polyhedron[Flatten[ArchimedeanDuals][[i]]]], 1, 
            Polygon]], {i, 1, 13}];

Johnson Solids

Table[
        Show[GraphicsArray[{Graphics3D[Polyhedron[JohnsonSolid[i]],  
                Boxed -> False], 
              UnfoldPolytope[
                Flatten[Flatten[Polyhedron[JohnsonSolid[i]]], 1, 
                  Polygon]]}],  DisplayFunction -> $DisplayFunction,  
          PlotLabel -> i], {i, 1, 43}];

Table[
        Show[GraphicsArray[{Graphics3D[Polyhedron[JohnsonSolid[i]],  
                Boxed -> False], 
              UnfoldPolytope[
                Flatten[Flatten[Polyhedron[JohnsonSolid[i]]], 1, 
                  Polygon]]}],  DisplayFunction -> $DisplayFunction, 
          PlotLabel -> i], {i, 49, 92}];

Other

Show[
      UnfoldPolytope[
        Flatten[Flatten[Polyhedron[Antiprism[10]]], 1, Polygon]],  
      DisplayFunction -> $DisplayFunction]

Show[
      UnfoldPolytope[
        Flatten[Flatten[Polyhedron[DuerersSolid]], 1, Polygon]],  
      DisplayFunction -> $DisplayFunction]

Polyhedra Generated by  Linear Inequalities

UnfoldRandomLI[n_]:=Module[{m,d,b,vertices,zerosets,faces},
m = Table[Table[Random[Integer,{-5,5}],{3}],{n}];
d = Table[10,{3}];
b = (d.#+1)& /@ m;
{vertices,zerosets} = VertexEnumeration[m,b];
faces = MakeFacetsFromZerosets[vertices,zerosets];
faces = OrderFacets[faces];
Show[GraphicsArray[{Graphics3D[Polygon /@ faces,Boxed->False],
UnfoldPolytope[faces]}], DisplayFunction->$DisplayFunction]]

Do[UnfoldRandomLI[16], {5}]

Polyhedra Generated by Convex Hull of Points

randompts[n_] := 
        Table[{Random[Integer, {-50, 50}], 
            Random[Integer, {-50, 50}], 
            Random[Integer, {-50, 50}]}, {n}];

\(UnfoldConvexHull[pts_] := 
      Module[{datapoints, center, m, d, c, b, vlist, vlist1, facetlist, 
          faces, vindex, vertices, nonvertices}, 
datapoints\  = \ pts; 

        center\  = \ 
          Sum[datapoints[\([i]\)], {i, 1, Length[datapoints]}]/
			Length[
              datapoints]; 

        m\  = \ \(\((# - center)\) &\)\  /@ \ datapoints; 

        d\  = \ Table[1, {Length[datapoints]}]; 
c\  = \ Table[10, {3}]; 

        b\  = \ d\  + \ m . c; 

        vlist\  = \ VE[m, b]; \ 
 (*\ 
          VE\ is\ a\ abbreviated\ form\ of\ \(\(VertexEnumeration\)\(.\)\)\ *) 
vlist1 = \(\((# - c)\) &\)\  /@ \ vlist[\([1]\)]; 

        facetlist = Map[\((# - Table[3, {Length[#]}])\) &, vlist[\([2]\)]]; 

        faces = \((datapoints[\([#]\)] &)\)\  /@ \ facetlist; 

        vindex\  = \ Sort[Union[Flatten[facetlist]]]; 

        vertices\  = \ datapoints[\([vindex]\)]; 

        nonvertices\  = \ Complement[datapoints, vertices]; 

        faces\  = \ OrderFacets[faces]; 

        Show[GraphicsArray[{Graphics3D[Polygon /@ faces, 
                Boxed -> False], UnfoldPolytope[faces]}], 
          DisplayFunction -> $DisplayFunction]]\)

Do[UnfoldConvexHull[randompts[16]], {5}]

sphere[θ_, ϕ_] := {Cos[θ] Sin[ϕ],  
          Sin[θ] Sin[ϕ], Cos[ϕ]};
sphericalrandompts[n_] := 
        Table[sphere[Random[Real, {0, 2  Pi}], 
            Random[Real, {0, Pi}]], {n}];

UnfoldConvexHull[sphericalrandompts[100]]

UnfoldConvexHull[sphericalrandompts[150]]