Pseudo-Edge Unfoldings

Nicholas Barvinok and Mohammad Ghomi
January 2019

This notebook is an accompaniment to the paper 

              MouseOver, 

by the authors. Here we give the coordinates for the convex subdivision of the equilateral triangle discussed in Section 5 of our paper, and verify some basic properties of it.

The Coordinates

Vertices

p[1]={25.5,0};
p[2]={23.1,-0.1};
p[3]={23.1,-1.1};
p[4]={23.3,-2.1};
p[5]={23.9,-2.9};
p[6]={24.7,-3.5};
p[7]={25.6,-3.9};
p[8]={26.6,-4.1};
p[9]={27.6,-4.0};
p[10]={28.5,-3.7};
p[11]={29.4,-3.2};
p[12]={30.2,-2.6};
p[13]={30.8,-1.8};
p[14]={31.3,-0.9};
p[15]={31.9,0.8};
p[16]={32.4,6.7};
p[17]={28.9,11.9};
p[18]={23.2,14.7};
p[19]={16.9,14.8};
p[20]={11.0,12.6};
p[21]={6.1,8.7};
p[22]={2.5,3.5};
p[23]={2.4,-2.5};
p[24]={-2.3,-5.0};
p[25]={-2.9,-7.5};
p[26]={-4.0,-7.0};
p[27]={7.2,-12.4};
p[28]={-12.3,-23};
p[29]={-21.3,-37.6};
p[30]={-24.5,-41.7};
p[31]={-7.3,-40.4};
p[32]={4.6,-40.3};
p[33]={13.5,-40.7};
p[34]={21.0,-41.3};
p[35]={33.0,-42.8};
p[36]={44.9,-44.9};
p[37]={40.0,-2.5};
p[38]={40.1,7.6};
p[39]={40.6,15.8};
p[40]={41.6,26.3};
p[41]={44.9,44.9};
p[42]={27.5,42.1};
p[-1]={-25.5,0};
p[-2]={-23.1,0.1};
p[-3]={-23.1,1.1};
p[-4]={-23.3,2.1};
p[-5]={-23.9,2.9};
p[-6]={-24.7,3.5};
p[-7]={-25.6,3.9};
p[-8]={-26.6,4.1};
p[-9]={-27.6,4.0};
p[-10]={-28.5,3.7};
p[-11]={-29.4,3.2};
p[-12]={-30.2,2.6};
p[-13]={-30.8,1.8};
p[-14]={-31.3,0.9};
p[-15]={-31.9,-0.8};
p[-16]={-32.4,-6.7};
p[-17]={-28.9,-11.9};
p[-18]={-23.2,-14.7};
p[-19]={-16.9,-14.8};
p[-20]={-11.0,-12.6};
p[-21]={-6.1,-8.7};
p[-22]={-2.5,-3.5};
p[-23]={-2.4,2.5};
p[-24]={2.3,5.0};
p[-25]={2.9,7.5};
p[-26]={4.0,7.0};
p[-27]={-7.2,12.4};
p[-28]={12.3,23};
p[-29]={21.3,37.6};
p[-30]={24.5,41.7};
p[-31]={7.3,40.4};
p[-32]={-4.6,40.3};
p[-33]={-13.5,40.7};
p[-34]={-21.0,41.3};
p[-35]={-33.0,42.8};
p[-36]={-44.9,44.9};
p[-37]={-40.0,2.5};
p[-38]={-40.1,-7.6};
p[-39]={-40.6,-15.8};
p[-40]={-41.6,-26.3};
p[-41]={-44.9,-44.9};
p[-42]={-27.5,-42.1};
a={0,140(Sqrt[3])-85};
b={-140,-85};
c={140,-85};

Edges

edge[1]=Line[{p[1],p[2]}];
edge[2]=Line[{p[1],p[10]}];
edge[3]=Line[{p[1],p[15]}];
edge[4]=Line[{p[2],p[3]}];
edge[5]=Line[{p[3],p[4]}];
edge[6]=Line[{p[4],p[5]}];
edge[7]=Line[{p[5],p[6]}];
edge[8]=Line[{p[6],p[7]}];
edge[9]=Line[{p[7],p[8]}];
edge[10]=Line[{p[8],p[9]}];
edge[11]=Line[{p[9],p[10]}];
edge[12]=Line[{p[10],p[11]}];
edge[13]=Line[{p[11],p[12]}];
edge[14]=Line[{p[12],p[13]}];
edge[15]=Line[{p[13],p[14]}];
edge[16]=Line[{p[14],p[15]}];
edge[17]=Line[{p[15],p[16]}];
edge[18]=Line[{p[16],p[17]}];
edge[19]=Line[{p[17],p[18]}];
edge[20]=Line[{p[18],p[19]}];
edge[21]=Line[{p[19],p[20]}];
edge[22]=Line[{p[20],p[21]}];
edge[23]=Line[{p[21],p[22]}];
edge[24]=Line[{p[22],p[-24]}];
edge[25]=Line[{p[-24],p[-25]}];
edge[26]=Line[{p[-25],p[-26]}];
edge[27]=Line[{p[-26],p[-28]}];
edge[28]=Line[{p[-28],p[-29]}];
edge[29]=Line[{p[2],p[18]}];
edge[30]=Line[{p[3],p[19]}];
edge[31]=Line[{p[4],p[20]}];
edge[32]=Line[{p[5],p[21]}];
edge[33]=Line[{p[6],p[22]}];
edge[34]=Line[{p[7],p[23]}];
edge[35]=Line[{p[8],p[25]}];
edge[36]=Line[{p[9],p[27]}];
edge[37]=Line[{p[10],p[29]}];
edge[38]=Line[{p[11],p[31]}];
edge[39]=Line[{p[12],p[32]}];
edge[40]=Line[{p[13],p[33]}];
edge[41]=Line[{p[14],p[34]}];
edge[42]=Line[{p[15],p[35]}];
edge[43]=Line[{p[16],p[37]}];
edge[44]=Line[{p[17],p[38]}];
edge[45]=Line[{p[18],p[39]}];
edge[46]=Line[{p[19],p[40]}];
edge[47]=Line[{p[20],p[41]}];
edge[48]=Line[{p[21],p[42]}];
edge[49]=Line[{p[22],p[-26]}];
edge[50]=Line[{p[-24],p[24]}];
edge[51]=Line[{p[-25],p[-23]}];
edge[52]=Line[{p[-25],p[-9]}];
edge[53]=Line[{p[-25],p[-27]}];
edge[54]=Line[{p[-28],p[-27]}];
edge[55]=Line[{p[-29],p[-10]}];
edge[56]=Line[{p[27],p[25]}];
edge[57]=Line[{p[23],p[22]}];
edge[58]=Line[{p[23],p[25]}];
edge[59]=Line[{p[32],p[31]}];
edge[60]=Line[{p[32],p[33]}];
edge[61]=Line[{p[33],p[34]}];
edge[62]=Line[{p[34],p[35]}];
edge[63]=Line[{p[35],p[36]}];
edge[64]=Line[{p[36],p[37]}];
edge[65]=Line[{p[37],p[38]}];
edge[66]=Line[{p[38],p[39]}];
edge[67]=Line[{p[39],p[40]}];
edge[68]=Line[{p[40],p[41]}];
edge[69]=Line[{p[41],p[42]}];
edge[70]=Line[{p[42],p[-30]}];
edge[71]=Line[{p[-30],p[-31]}];
edge[72]=Line[{p[-29],p[-30]}];
edge[73]=Line[{p[-31],p[-11]}];
edge[74]=Line[{p[-31],p[-32]}];
edge[75]=Line[{p[-1],p[-2]}];
edge[76]=Line[{p[-2],p[-3]}];
edge[77]=Line[{p[-3],p[-4]}];
edge[78]=Line[{p[-4],p[-5]}];
edge[79]=Line[{p[-5],p[-6]}];
edge[80]=Line[{p[-6],p[-7]}];
edge[81]=Line[{p[-7],p[-8]}];
edge[82]=Line[{p[-8],p[-9]}];
edge[83]=Line[{p[-9],p[-10]}];
edge[84]=Line[{p[-10],p[-11]}];
edge[85]=Line[{p[-11],p[-12]}];
edge[86]=Line[{p[-12],p[-13]}];
edge[87]=Line[{p[-13],p[-14]}];
edge[88]=Line[{p[-14],p[-15]}];
edge[89]=Line[{p[-15],p[-16]}];
edge[90]=Line[{p[-16],p[-17]}];
edge[91]=Line[{p[-17],p[-18]}];
edge[92]=Line[{p[-18],p[-19]}];
edge[93]=Line[{p[-19],p[-20]}];
edge[94]=Line[{p[-20],p[-21]}];
edge[95]=Line[{p[-21],p[-22]}];
edge[96]=Line[{p[-22],p[24]}];
edge[97]=Line[{p[24],p[25]}];
edge[98]=Line[{p[25],p[26]}];
edge[99]=Line[{p[26],p[28]}];
edge[100]=Line[{p[-2],p[-18]}];
edge[101]=Line[{p[-3],p[-19]}];
edge[102]=Line[{p[-4],p[-20]}];
edge[103]=Line[{p[-5],p[-21]}];
edge[104]=Line[{p[-6],p[-22]}];
edge[105]=Line[{p[-7],p[-23]}];
edge[106]=Line[{p[-9],p[-27]}];
edge[107]=Line[{p[-12],p[-32]}];
edge[108]=Line[{p[-13],p[-33]}];
edge[109]=Line[{p[-14],p[-34]}];
edge[110]=Line[{p[-15],p[-35]}];
edge[111]=Line[{p[-16],p[-37]}];
edge[112]=Line[{p[-17],p[-38]}];
edge[113]=Line[{p[-18],p[-39]}];
edge[114]=Line[{p[-19],p[-40]}];
edge[115]=Line[{p[-20],p[-41]}];
edge[116]=Line[{p[-21],p[-42]}];
edge[117]=Line[{p[-22],p[26]}];
edge[118]=Line[{p[29],p[28]}];
edge[119]=Line[{p[-32],p[-33]}];
edge[120]=Line[{p[-33],p[-34]}];
edge[121]=Line[{p[-34],p[-35]}];
edge[122]=Line[{p[-35],p[-36]}];
edge[123]=Line[{p[-36],p[-37]}];
edge[124]=Line[{p[-37],p[-38]}];
edge[125]=Line[{p[-38],p[-39]}];
edge[126]=Line[{p[-39],p[-40]}];
edge[127]=Line[{p[-40],p[-41]}];
edge[128]=Line[{p[-41],p[-42]}];
edge[129]=Line[{p[-42],p[30]}];
edge[130]=Line[{p[31],p[30]}];
edge[131]=Line[{p[30],p[29]}];
edge[132]=Line[{p[27],p[28]}];
edge[133]=Line[{p[-22],p[-23]}];
edge[134]=Line[{p[-1],p[-10]}];
edge[135]=Line[{p[-1],p[-15]}];
edge[136]=Line[{p[-41],b}];
edge[137]=Line[{p[36],c}];
edge[138]=Line[{p[-36],b}];
edge[139]=Line[{p[-36],a}];
edge[140]=Line[{p[41],a}];
edge[141]=Line[{p[41],c}];
edge[142]=Line[{b,a}];
edge[143]=Line[{a,c}];
edge[144]=Line[{c,b}];

Pictures

Basic Pictures

edges=Table[edge[i],{i,1,144}];

Graphics[PrependTo[edges,RGBColor[0.35,0.5,0.78]],AspectRatio->Automatic]

edges=Table[edge[i],{i,1,135}];

Graphics[PrependTo[edges,RGBColor[0.35,0.5,0.78]],AspectRatio->Automatic]

Labels

Computations (Proofs of Lemmas 5.2, 5.3)

Basic Formulas

Angles

angle[x_,y_,z_]:=ArcCos[(((p[x]-p[y]).(p[z]-p[y]))/(Norm[p[x]-p[y]]Norm[p[z]-p[y]]))]*(180/π);

Obtuseness

obtuse[x_,y_,z_]=angle[x,y,z]-90;

Proof of Lemma 5.2

Using the “obtuse” function defined above we check that for each vertex (Subscript[v, i]) of the proposed subpath, (Subscript[v, i+1]) is the only adjacent vertex of (Subscript[v, i]) such that the angle spanned by (Subscript[v, i+1]), (Subscript[v, i]), and (Subscript[p, 1]) is greater than or equal to π/2.

Note that (Subscript[p, 2]) or p[2] has only two adjacent vertices: p[3] and p[18]. So we compute that:

{obtuse[1,2,3], obtuse[1,2,18]}

Thus the only possible choice for the successor of p[2] into our path is p[3]. Next we check vertices that are adjacent to p[3], and continue this procedure until we reach p[-41]. Then the last computation shows that the path must leave the square.

{obtuse[1,3,4], obtuse[1,3,19]}

{obtuse[1,4,5], obtuse[1,4,20]}

{obtuse[1,5,6], obtuse[1,5,21]}

{obtuse[1,6,7], obtuse[1,6,22]}

{obtuse[1,7,8], obtuse[1,7,23]}

{obtuse[1,8,9], obtuse[1,8,25]}

{obtuse[1,9,10], obtuse[1,9,27]}

{obtuse[1,10,11], obtuse[1,10,29]}

{obtuse[1,11,12], obtuse[1,11,31]}

{obtuse[1,12,13], obtuse[1,12,32]}

{obtuse[1,13,14], obtuse[1,13,33]}

{obtuse[1,14,15], obtuse[1,14,34]}

{obtuse[1,15,16], obtuse[1,15,35]}

{obtuse[1,16,17], obtuse[1,16,37]}

{obtuse[1,17,18], obtuse[1,17,38]}

{obtuse[1,18,19], obtuse[1,18,39]}

{obtuse[1,19,20], obtuse[1,19,40]}

{obtuse[1,20,21], obtuse[1,20,41]}

{obtuse[1,21,22], obtuse[1,21,42]}

{obtuse[1,22,-24], obtuse[1,22,23],obtuse[1,22,-26]}

{obtuse[1,-24,24], obtuse[1,-24,-25]}

{obtuse[1,24,25], obtuse[1,24,-22]}

{obtuse[1,25,26], obtuse[1,25,27]}

{obtuse[1,26,28], obtuse[1,26,-22]}

{obtuse[1,28,29], obtuse[1,28,27]}

{obtuse[1,29,30], obtuse[1,29,10]}

{obtuse[1,30,-42], obtuse[1,30,31]}

{obtuse[1,-42,-41], obtuse[1,-42,-21]}

{obtuse[1,-41,-40], obtuse[1,-41,-20]}

Proof of Lemma 5.3

We follows the same general procedure described in the proof of Lemma 5.2.

p[o]={0,0};

{obtuse[o,24, 25], obtuse[o,24, -22]}

{obtuse[o,25, 27], obtuse[o,25, 26]}

{obtuse[o,27, 9], obtuse[o,27, 28]}

{obtuse[o,9, 10], obtuse[o,9, 27],obtuse[o,9, 8]}

{obtuse[o,10, 11], obtuse[o,10, 29]}

{obtuse[o,11, 12], obtuse[o,11, 31]}

{obtuse[o,12, 13], obtuse[o,12, 32]}

{obtuse[o,13, 14], obtuse[o,13, 33]}

{obtuse[o,14, 15], obtuse[o,14, 34]}