Triakis Icosahedron

Author

Eric W. Weisstein
January 5, 2024

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Polyhedra/TriakisIcosahedron.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/TriakisIcosahedron.html.

©2024 Wolfram Research, Inc. except for portions noted otherwise

Figure

With[{pname="TriakisIcosahedron"},
GraphicsRow[{
Graphics3D[{Opacity[.8],PolyhedronData[pname,"Polyhedron"]},Boxed->False],
Graphics3D[{PolyhedronData[pname,"Lines"]},Boxed->False],
PolyhedronData[pname,"Net"]
},ImageSize->700]]

Solid

PolyhedronData["TriakisIcosahedron","Orientations"]

Show[PolyhedronData[{"TriakisIcosahedron",#}],Boxed->False]&/@PolyhedronData["TriakisIcosahedron","Orientations"]

Solid and Dual

GraphicsRow[PolyhedronData["TriakisIcosahedron",#]&/@{"Graphics3D","Dual","DualCompound"}]

Net

PolyhedronData["TriakisIcosahedron","Net"]

PolyhedronData["TriakisIcosahedron","Net","Colored"]

NetPrintout["TriakisIcosahedron","TriakisTetrahedron.pdf"]

Properties

Initialization

<<MathWorld`Polyhedra`

p=PolyhedronData[pname="TriakisIcosahedron","Polyhedron"]

vol=PolyhedronData[pname,"Volume"]

Centroid

PolyhedronData["TriakisIcosahedron","Centroid"]

RegionCentroid[p]//RootReduce

PolyhedronCentroid[p]//RootReduce

Circumsphere

PolyhedronData[pname,"Circumsphere"]

Circumsphere[p]

MyCircumsphere[p]

ConvexHullOf

hullof=DeleteCases[Select[PolyhedronData[All],PolyhedronData[#,"ConvexHullName"]===pname&],pname]

GraphicsGrid[Partition[Graphics3D[{
{Green,Opacity[.9],PolyhedronData[#,"Polyhedron"]},
{Blue,Opacity[.5],PolyhedronData[#,"ConvexHull","Polyhedron"]}},Boxed->False,PlotLabel->Style[PolyhedronData[#,"Name"],Directive[14,Black,Italic,FontFamily->"Times"]]]&/@hullof,UpTo[3]],Dividers->All,ImageSize->200]

DihedralAngles

PolyhedronData[pname,"DihedralAngles"]//Tally

DihedralAngles[p]//RootReduce//Tally

EdgeLengths

PolyhedronData[pname,"EdgeLengths"]

PolyhedronData[pname,"Lines"]/.Line[l_]:>EuclideanDistance@@l//FullSimplify//Union

Faces

Insphere

PolyhedronData[pname,"Insphere"]

Insphere[p]//RootReduce//Timing

MyInsphere[p]//RootReduce//Timing

InertiaTensor

PolyhedronData[pname,"InertiaTensor"]

((MomentOfInertia[p])/(Volume[p]))//RootReduce//Timing

PolyhedronInertiaTensor[p]//RootReduce//Timing

MeanCylindricalRadius

((NIntegrate[(Sqrt[(x^2)+(y^2)]),{x,y,z}∈p])/(PolyhedronData[pname,"Volume"]))

(i=((Integrate[(Sqrt[(x^2)+(y^2)]),{x,y,z}∈p])/(PolyhedronData[pname,"Volume"])))//Timing

N[i]

MeanCylindricalSquareRadius

((NIntegrate[(x^2)+(y^2),{x,y,z}∈p])/(PolyhedronData[pname,"Volume"]))//FullSimplify

(i=((Integrate[(x^2)+(y^2),{x,y,z}∈p])/(PolyhedronData[pname,"Volume"])))//Timing

FullSimplify[i]

MeanSphericalRadius

((NIntegrate[(Sqrt[(x^2)+(y^2)+(z^2)]),{x,y,z}∈p])/(PolyhedronData[pname,"Volume"]))

(i=((Integrate[(Sqrt[(x^2)+(y^2)+(z^2)]),{x,y,z}∈p])/(PolyhedronData[pname,"Volume"])))//Timing

N[i]

MeanSphericalSquareRadius

((NIntegrate[(x^2)+(y^2)+(z^2),{x,y,z}∈p])/(PolyhedronData[pname,"Volume"]))

(i=((Integrate[(x^2)+(y^2)+(z^2),{x,y,z}∈p])/(PolyhedronData[pname,"Volume"])))//Timing

N[i]

FullSimplify[i]

Midsphere

PolyhedronData[pname,"Midsphere"]

Midsphere[p]//RootReduce//Timing

SurfaceArea

PolyhedronData[pname,"SurfaceArea"]

RootReduce[%]

SurfaceArea[p]//RootReduce//Timing

Total[Area/@PolyhedronData[pname,"Polygons"]]//RootReduce//Timing

Platonic

(h=VertexSubsetHulls[p,"Platonic","Skip"->{"Icosahedron"}])//Timing

h2=Insert[h,"Icosahedron"->{{1,5,23,24,25,26,27,28,29,30,31,32}},-2]

g=Function[poly,Graphics3D[{
{Gray,Opacity[.2],EdgeForm[Thin],PolyhedronData[p,"Faces"]},ConvexHull3D[N[v][[#]]]&/@poly},Boxed->False]]/@h2[[All,2]]

Volume

PolyhedronData[pname,"Volume"]

Volume[p]//FullSimplify

PolyhedronVolume[p,Method->"GaussLaw"]//FullSimplify

Perspective projections

Manipulate[PolyhedronProjectionGraph[PolyhedronData["TriakisIcosahedron","Polyhedron"],{a,b,c}],{a,-1,1},{b,-1,1},{c,-1,1}]

{1,0,-(1/11) (13-5 (Sqrt[5]))}

{.75,0,.125}/.75

g=PolyhedronProjectionGraph[PolyhedronData["TriakisIcosahedron","Polyhedron"],{1,0,-(1/11) (13-5 (Sqrt[5]))},VertexLabels->"Name"]

g=PolyhedronProjectionGraph[PolyhedronData["TriakisIcosahedron","Polyhedron"],{1,0,-.2},VertexLabels->"Name"]

g=PolyhedronProjectionGraph[PolyhedronData["TriakisIcosahedron","Polyhedron"],{1,0,-z},VertexLabels->"Name"]

v=GraphEmbedding[g];

eqn=FullSimplify[v[[13]]-v[[28]],z>0]

Solve[Thread[eqn==0],z]//FullSimplify

N[%]

{1,0,(1/61) (17+3 (Sqrt[5]))}

{.75,0,.29}/.75

g=PolyhedronProjectionGraph[PolyhedronData["TriakisIcosahedron","Polyhedron"],{1,0,(1/61) (17+3 (Sqrt[5]))},VertexLabels->"Name"]

g=PolyhedronProjectionGraph[PolyhedronData["TriakisIcosahedron","Polyhedron"],{1,0,z},VertexLabels->"Name"]

v=GraphEmbedding[g];

eqn=FullSimplify[CollinearPoints[v[[{23,13,24}]]],z>0]

Solve[eqn,z]//FullSimplify

N[%]

{1,0,(1/4) (3-(Sqrt[5]))}

g=PolyhedronProjectionGraph[PolyhedronData["TriakisIcosahedron","Polyhedron"],{1,0,(1/4) (3-(Sqrt[5]))},VertexLabels->"Name"]

g=PolyhedronProjectionGraph[PolyhedronData["TriakisIcosahedron","Polyhedron"],{1,0,.3},VertexLabels->"Name"]

g=PolyhedronProjectionGraph[PolyhedronData["TriakisIcosahedron","Polyhedron"],{1,0,z},VertexLabels->"Name"]

v=GraphEmbedding[g];

eqn=FullSimplify[v[[13]]-v[[18]],z>0]

Solve[Thread[eqn==0],z]//FullSimplify

N[%]

{1,(1/61) (Sqrt[2885-1102 (Sqrt[5])]),(1/61) (17+3 (Sqrt[5])),ArcTan[(1/4) (NumericalApproximation[Root, (RowDefault["1.52", FEExpressionsNumericalApproximationElider]), Root[145-65 ((#1)^2)+((#1)^4)&,3], ])]}

g=PolyhedronProjectionGraph[PolyhedronData["TriakisIcosahedron","Polyhedron"],{1,(1/61) (Sqrt[2885-1102 (Sqrt[5])]),(1/61) (17+3 (Sqrt[5])),ArcTan[(1/4) (NumericalApproximation[Root, (RowDefault["1.52", FEExpressionsNumericalApproximationElider]), Root[145-65 ((#1)^2)+((#1)^4)&,3], ])]},VertexLabels->"Name"]

v=GraphEmbedding[g];

ArcTan@@(v[[24]]-v[[18]])//FullSimplify

g=PolyhedronProjectionGraph[PolyhedronData["TriakisIcosahedron","Polyhedron"],{1,.3,.3},VertexLabels->"Name"]

g=PolyhedronProjectionGraph[PolyhedronData["TriakisIcosahedron","Polyhedron"],{1,x,y},VertexLabels->"Name"]

v=GraphEmbedding[g];

eqn=FullSimplify[v[[13]]-v[[24]],x>0&&y>0]

Solve[Thread[eqn==0],{x,y}]//FullSimplify

N[%]

{1,(NumericalApproximation[Root, (RowDefault["0.375", FEExpressionsNumericalApproximationElider]), Root[605-4325 ((#1)^2)+121 ((#1)^4)&,3], ]),(1/2) (-1+(Sqrt[5])),-ArcTan[(Sqrt[(1/85) (25-2 (Sqrt[5]))])]}

g=PolyhedronProjectionGraph[PolyhedronData["TriakisIcosahedron","Polyhedron"],{1,(NumericalApproximation[Root, (RowDefault["0.375", FEExpressionsNumericalApproximationElider]), Root[605-4325 ((#1)^2)+121 ((#1)^4)&,3], ]),(1/2) (-1+(Sqrt[5])),-ArcTan[(Sqrt[(1/85) (25-2 (Sqrt[5]))])]},VertexLabels->"Name"]

v=GraphEmbedding[g];

ArcTan@@(v[[13]]-v[[10]])//FullSimplify

g=PolyhedronProjectionGraph[PolyhedronData["TriakisIcosahedron","Polyhedron"],{1,.365,.625},VertexLabels->"Name"]

g=PolyhedronProjectionGraph[PolyhedronData["TriakisIcosahedron","Polyhedron"],{1,x,y},VertexLabels->"Name"];

v=GraphEmbedding[g];

FullSimplify[CollinearPoints[v[[#]]]&/@{{13,24,10},{20,24,29}},x>0&&y>0]

Solve[%,{x,y}]//FullSimplify

g=PolyhedronProjectionGraph[PolyhedronData["TriakisIcosahedron","Polyhedron"],{1,(NumericalApproximation[Root, (RowDefault["0.168", FEExpressionsNumericalApproximationElider]), Root[121-4325 ((#1)^2)+605 ((#1)^4)&,3], ]),(1/2) (1-(Sqrt[5]))},VertexLabels->"Name"]

{1,(1/29) (Sqrt[1765-722 (Sqrt[5])]),-(3/29) (-11+(Sqrt[5])),ArcTan[4 (Sqrt[(2/(285+127 (Sqrt[5])))])]}

g=PolyhedronProjectionGraph[PolyhedronData["TriakisIcosahedron","Polyhedron"],{1,(1/29) (Sqrt[1765-722 (Sqrt[5])]),-(3/29) (-11+(Sqrt[5])),ArcTan[4 (Sqrt[(2/(285+127 (Sqrt[5])))])]},VertexLabels->"Name"]

v=GraphEmbedding[g];

ArcTan@@(v[[5]]-v[[30]])//FullSimplify

g=PolyhedronProjectionGraph[PolyhedronData["TriakisIcosahedron","Polyhedron"],{1,x,y},VertexLabels->"Name"]

v=GraphEmbedding[g];

eqn=FullSimplify[v[[24]]-v[[10]],x>0&&y>0]

Solve[Thread[eqn==0],{x,y}]//FullSimplify

N[%]

{1,(Sqrt[5-2 (Sqrt[5])]),(1/2) (-1+(Sqrt[5])),-ArcCot[(Sqrt[85+38 (Sqrt[5])])]}

g=PolyhedronProjectionGraph[PolyhedronData["TriakisIcosahedron","Polyhedron"],{1,(Sqrt[5-2 (Sqrt[5])]),(1/2) (-1+(Sqrt[5])),-ArcCot[(Sqrt[85+38 (Sqrt[5])])]},VertexLabels->"Name"]

v=GraphEmbedding[g];

ArcTan@@(v[[1]]-v[[23]])//FullSimplify

g=PolyhedronProjectionGraph[PolyhedronData["TriakisIcosahedron","Polyhedron"],{1,.7,.61},VertexLabels->"Name"]

g=PolyhedronProjectionGraph[PolyhedronData["TriakisIcosahedron","Polyhedron"],{1,x,y},VertexLabels->"Name"]

v=GraphEmbedding[g];

Solve[Thread[FullSimplify[v[[24]]-v[[29]],x>0&&y>0]==0],{x,y}]//FullSimplify

{0, , 0,-ArcTan[(1/2)-((Sqrt[5])/2)]}

g=PolyhedronProjectionGraph[PolyhedronData["TriakisIcosahedron","Polyhedron"],{0,1,0,-ArcTan[(1/2)-((Sqrt[5])/2)]},VertexLabels->"Name"]

v=GraphEmbedding[g];

ArcTan@@(v[[1]]-v[[25]])//FullSimplify

Construction

Solid

TriakisIcosahedronExact=Dual[Polyhedron[TruncatedDodecahedron],Method->Semiregular];

Show[Graphics3D[TriakisIcosahedronExact],Boxed->False]

Net

<<ExtraPackages`UnfoldPolytope`

UnfoldPolytope[Flatten[Normal[PolyhedronData["TriakisIcosahedron","Faces"]]]]

PolyhedronData["TriakisIcosahedron","EdgeLengths"]

{1,(1/10) (15+(Sqrt[5]))}//N

PolyhedronData["TriakisIcosahedron","NetImage"]

x=(((1/10) (15+(Sqrt[5])))/2);

y=Sqrt[1-x^2]//FullSimplify

N[%]

α=ArcCos[{1,0}.Normalize[{x,y}]]//FullSimplify

RotationMatrix[2α].{2x,0}//FullSimplify

x2=(1/50) (45-19 (Sqrt[5]));

x2//N

piece={Polygon[{{0,0},{x,y},{2x,0}}],
Polygon[{{0,0},{x,y},{(3/50) (5+4 (Sqrt[5])),(Sqrt[(961/500)+(39/(50 (Sqrt[5])))])}}],
Polygon[{{2x,0},{x,y},{2x-(3/50) (5+4 (Sqrt[5])),(Sqrt[(961/500)+(39/(50 (Sqrt[5])))])}}]};

{(1/50) (60-7 (Sqrt[5])),(Sqrt[(961/500)+(39/(50 (Sqrt[5])))])}//N

bigpiece[n_:0,m_:0]:=Flatten[{piece,piece/.Polygon[v_]:>Polygon[{1,-1}#&/@v]}]/.Polygon[v_]:>Polygon[({1,0}(2x+x2) n+m{x+x2/2, (Sqrt[(961/500)+(39/(50 (Sqrt[5])))])}+#)&/@v];

Graphics[{LightYellow,EdgeForm[Black],g=Flatten[{bigpiece[#,0]&/@Range[0,4],bigpiece[#,1]&/@Range[0,4]}]}]

gc=NetFromPolygon[g];

SortBy[gc[[1]],N][[1]]

SortBy[gc[[1]],N][[7]]

{(1/10) (15+(Sqrt[5]))-(3/50) (5+4 (Sqrt[5])),2 (Sqrt[(961/500)+(39/(50 (Sqrt[5])))])}-SortBy[gc[[1]],N][[1]]//FullSimplify

FullSimplify[SortBy[gc[[1]],N][[7]]-SortBy[gc[[1]],N][[1]]]

gc[[1,7]]

Graphics[{{LightYellow,EdgeForm[Black],gc},MapIndexed[Text[#2[[1]],#,Background->White]&,SortBy[gc[[1]],N]]}]

gc=NetFromPolygon[FullSimplify[g]]

GraphicsComplex[{{0,(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])},{12-(7/(Sqrt[5])),(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])},{12-(7/(Sqrt[5])),(3/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])},{6-(7/(2 (Sqrt[5]))),(Sqrt[(961/500)+(39/(50 (Sqrt[5])))])+(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])},{6-(7/(2 (Sqrt[5]))),0},{(27/4)-(13/(4 (Sqrt[5]))),(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])+(3/10) (Sqrt[(1/10) (453-(Sqrt[5]))])},{(27/4)-(13/(4 (Sqrt[5]))),(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])+(1/100) (Sqrt[6170+1710 (Sqrt[5])])},{(15/2)-(3/(Sqrt[5])),(Sqrt[(961/500)+(39/(50 (Sqrt[5])))])+(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])},{(15/2)-(3/(Sqrt[5])),0},{(51/10)-(8/(5 (Sqrt[5]))),(Sqrt[(961/500)+(39/(50 (Sqrt[5])))])+(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])},{(51/10)-(8/(5 (Sqrt[5]))),0},{(1/100) (1035-107 (Sqrt[5])),(Sqrt[(17/40)-(3/(8 (Sqrt[5])))])+(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])},{(1/100) (1035-107 (Sqrt[5])),(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])+Root[61-340 ((#1)^2)+400 ((#1)^4)&,2]},{(1/100) (915-93 (Sqrt[5])),(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])+(3/10) (Sqrt[(1/10) (453-(Sqrt[5]))])},{(1/100) (915-93 (Sqrt[5])),(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])+(1/100) (Sqrt[6170+1710 (Sqrt[5])])},{(1/100) (795-79 (Sqrt[5])),(Sqrt[(17/40)-(3/(8 (Sqrt[5])))])+(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])},{(1/100) (795-79 (Sqrt[5])),(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])+Root[61-340 ((#1)^2)+400 ((#1)^4)&,2]},{(1/50) (615-58 (Sqrt[5])),(Sqrt[(961/500)+(39/(50 (Sqrt[5])))])+(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])},{(1/100) (555-51 (Sqrt[5])),(Sqrt[(17/40)-(3/(8 (Sqrt[5])))])+(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])},{(1/100) (555-51 (Sqrt[5])),(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])+Root[61-340 ((#1)^2)+400 ((#1)^4)&,2]},{(1/50) (555-51 (Sqrt[5])),(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])},{(1/50) (555-51 (Sqrt[5])),(3/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])},{(1/100) (435-37 (Sqrt[5])),(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])+(3/10) (Sqrt[(1/10) (453-(Sqrt[5]))])},{(1/100) (435-37 (Sqrt[5])),(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])+(1/100) (Sqrt[6170+1710 (Sqrt[5])])},{(1/50) (435-37 (Sqrt[5])),(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])},{(1/50) (435-37 (Sqrt[5])),(3/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])},{(1/100) (315-23 (Sqrt[5])),(Sqrt[(17/40)-(3/(8 (Sqrt[5])))])+(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])},{(1/100) (315-23 (Sqrt[5])),(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])+Root[61-340 ((#1)^2)+400 ((#1)^4)&,2]},{(1/50) (315-23 (Sqrt[5])),(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])},{(1/50) (315-23 (Sqrt[5])),(3/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])},{(11/100) (105-11 (Sqrt[5])),(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])+(3/10) (Sqrt[(1/10) (453-(Sqrt[5]))])},{(11/100) (105-11 (Sqrt[5])),(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])+(1/100) (Sqrt[6170+1710 (Sqrt[5])])},{(1/50) (60-7 (Sqrt[5])),(Sqrt[(961/500)+(39/(50 (Sqrt[5])))])+(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])},{(1/50) (60-7 (Sqrt[5])),0},{(1/25) (60-7 (Sqrt[5])),(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])},{(1/25) (60-7 (Sqrt[5])),(3/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])},{(2/25) (60-7 (Sqrt[5])),(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])},{(2/25) (60-7 (Sqrt[5])),(3/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])},{(3/25) (60-7 (Sqrt[5])),(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])},{(3/25) (60-7 (Sqrt[5])),(3/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])},{(4/25) (60-7 (Sqrt[5])),(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])},{(4/25) (60-7 (Sqrt[5])),(3/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])},{(11/50) (45-4 (Sqrt[5])),(Sqrt[(961/500)+(39/(50 (Sqrt[5])))])+(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])},{(11/50) (45-4 (Sqrt[5])),0},{(3/100) (65-3 (Sqrt[5])),(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])+(3/10) (Sqrt[(1/10) (453-(Sqrt[5]))])},{(3/100) (65-3 (Sqrt[5])),(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])+(1/100) (Sqrt[6170+1710 (Sqrt[5])])},{(1/50) (135-2 (Sqrt[5])),(Sqrt[(961/500)+(39/(50 (Sqrt[5])))])+(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])},{(1/50) (135-2 (Sqrt[5])),0},{(1/20) (15+(Sqrt[5])),(Sqrt[(17/40)-(3/(8 (Sqrt[5])))])+(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])},{(1/20) (15+(Sqrt[5])),(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])+Root[61-340 ((#1)^2)+400 ((#1)^4)&,2]},{(1/10) (15+(Sqrt[5])),(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])},{(1/10) (15+(Sqrt[5])),(3/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])},{-(3/50) (-65+3 (Sqrt[5])),(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])},{-(3/50) (-65+3 (Sqrt[5])),(3/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])},{(3/50) (5+4 (Sqrt[5])),(Sqrt[(961/500)+(39/(50 (Sqrt[5])))])+(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])},{(3/50) (5+4 (Sqrt[5])),0},{-(9/50) (-60+7 (Sqrt[5])),(Sqrt[(961/500)+(39/(50 (Sqrt[5])))])+(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])},{-(9/50) (-60+7 (Sqrt[5])),0},{-(7/50) (-60+7 (Sqrt[5])),(Sqrt[(961/500)+(39/(50 (Sqrt[5])))])+(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])},{-(7/50) (-60+7 (Sqrt[5])),0},{-(3/50) (-60+7 (Sqrt[5])),(Sqrt[(961/500)+(39/(50 (Sqrt[5])))])+(1/10) (Sqrt[(961/5)+(78/(Sqrt[5]))])},{-(3/50) (-60+7 (Sqrt[5])),0}},Polygon[{{1,49,51},{1,49,55},{51,49,33},{1,50,51},{1,50,56},{51,50,34},{35,27,53},{35,27,47},{53,27,61},{35,28,53},{35,28,48},{53,28,62},{37,19,29},{37,19,10},{29,19,4},{37,20,29},{37,20,11},{29,20,5},{39,16,25},{39,16,8},{25,16,59},{39,17,25},{39,17,9},{25,17,60},{41,12,21},{41,12,43},{21,12,57},{41,13,21},{41,13,44},{21,13,58},{33,45,47},{33,45,52},{47,45,36},{33,46,47},{33,46,51},{47,46,35},{61,23,10},{61,23,54},{10,23,38},{61,24,10},{61,24,53},{10,24,37},{4,6,8},{4,6,30},{8,6,40},{4,7,8},{4,7,29},{8,7,39},{59,14,43},{59,14,26},{43,14,42},{59,15,43},{59,15,25},{43,15,41},{57,31,18},{57,31,22},{18,31,3},{57,32,18},{57,32,21},{18,32,2}}]]

Graphics[{{LightYellow,EdgeForm[Black],gc},MapIndexed[Text[#2[[1]],#,Background->White]&,SortBy[gc[[1]],N]]}]

Quit

PolyhedronData["TriakisIcosahedron","NetImage"]