Truncated Icosahedral Graph

Author

Eric W. Weisstein
June 21, 2023

This notebook downloaded from https://mathworld.wolfram.com/notebooks/GraphTheory/TruncatedIcosahedralGraph.nb.

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

©2023 Wolfram Research, Inc. except for portions noted otherwise

Embeddings

Names

GraphData["TruncatedIcosahedralGraph","StandardName"]

GraphData["TruncatedIcosahedralGraph","StandardNames"]

Primary

GraphData["TruncatedIcosahedralGraph"]//StyleGraphs[#,ImageSize->Small]&

Construct from Read and Wilson

g=Graph[Range[60],{{1,2},{1,5},{1,15},{2,3},{2,4},{3,8},{3,10},{4,6},{4,9},{5,9},{5,56},{6,7},{6,12},{7,8},{7,13},{8,11},{9,20},{10,17},{10,18},{11,14},{11,18},{12,19},{12,21},{13,16},{13,19},{14,16},{14,22},{15,17},{15,46},{16,23},{17,32},{18,25},{19,24},{20,21},{20,40},{21,29},{22,27},{22,28},{23,26},{23,27},{24,26},{24,30},{25,28},{25,31},{26,33},{27,34},{28,35},{29,30},{29,41},{30,36},{31,32},{31,37},{32,44},{33,36},{33,38},{34,38},{34,39},{35,37},{35,39},{36,42},{37,43},{38,45},{39,47},{40,41},{40,52},{41,49},{42,48},{42,49},{43,50},{43,51},{44,46},{44,51},{45,47},{45,48},{46,60},{47,50},{48,54},{49,55},{50,53},{51,58},{52,56},{52,57},{53,54},{53,58},{54,55},{55,57},{56,60},{57,59},{58,59},{59,60}},VertexCoordinates->(10^(-4)){{3059,-6514},{4759,-5945},{6757,-2482},{7017,-9858},{7933,-14959},{8189,-9181},{8437,-6820},{8813,-3669},{9276,-13772},{9673,2570},{10271,-1143},{10358,-10147},{10759,-6326},{11245,-1706},{11315,7790},{11493,-4067},{11671,6033},{11728,1383},{11946,-8383},{13273,-13772},{13273,-11397},{13413,-740},{13815,-4561},{14268,-8876},{14268,3283},{15002,-6618},{15002,-2505},{15002,1025},{16188,-11397},{16188,-10272},{16188,4679},{16188,6033},{17375,-6618},{17375,-2505},{17375,1025},{18109,-8876},{18109,3283},{18562,-4561},{18964,-740},{19104,-13772},{19104,-11397},{20431,-8383},{20649,1383},{20706,6033},{20884,-4067},{21062,7790},{21132,-1706},{21618,-6326},{22019,-10147},{22106,-1143},{22704,2570},{23101,-13772},{23564,-3669},{23940,-6820},{24188,-9181},{24444,-14959},{25360,-9858},{25620,-2482},{27618,-5945},{29318,-6514}}]

RecognizeGraph@g

All

GraphData["TruncatedIcosahedralGraph","Graph","All"]//StyleGraphs

Inexact

Select[GraphData["TruncatedIcosahedralGraph","Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData["TruncatedIcosahedralGraph","EdgeList"]]

Default embedding

Graph[GraphData["TruncatedIcosahedralGraph","EdgeList"]]

g=RotateGraph[LabelGraph@Graph[GraphData["TruncatedIcosahedralGraph","EdgeList"]],{29,56}]

Degenerate

GraphData["TruncatedIcosahedralGraph","Graph","Degenerate"]//StyleGraphs

Bilateral

GraphData["TruncatedIcosahedralGraph","Graph","Bilateral"]//StyleGraphs

Circular

GraphData["TruncatedIcosahedralGraph","Graph","Circular"]//StyleGraphs

Planar

GraphData["TruncatedIcosahedralGraph","Graph","Planar"]//StyleGraphs

Tutte embeddings

faces=ReverseSort@SymmetricallyDistinctFaces[GraphData["TruncatedIcosahedralGraph"]]

Tally[ReverseSort[Length/@faces]]

IGLayoutTutte[GraphData["TruncatedIcosahedralGraph"],"OuterFace"->#]&/@faces

gs=Annotate[#,VertexLabels->None]&/@{}

Tutte embedding construction

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

LCF

GraphData["TruncatedIcosahedralGraph","LCFSignature"]

GraphData["TruncatedIcosahedralGraph","Graph","LCF"]//StyleGraphs

Tally[(LCFNotation/@%)[[All,-1]]]

LCF compute all

Length[FindHamiltonianCycle[GraphData["TruncatedIcosahedralGraph"],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData["TruncatedIcosahedralGraph"],All])[[All,2]]]//Timing

lcf

LCFGraph@@@lcf//StyleGraphs

(gs=Flatten[{
(* all LCF's with order >1, sorted by order and with bilateral graphs first *)
Join[DeleteCases[#,{_Missing,_}][[All,1]],Cases[#,{_Missing,_}][[All,2]]]&/@Map[{BilateralLCFGraph@#,#}&,Map[LCFGraph@@#&,SplitBy[DeleteCases[lcf,{_,1}],Last],{2}],{2}],
(* order-1 bilateral only *)
DeleteMissing[BilateralLCFGraph@@@Cases[lcf,{_,1}]]}])//Timing

MinimalCrossing

GraphData["TruncatedIcosahedralGraph","Graph","MinimalCrossing"]//StyleGraphs

Perspective

GraphData["TruncatedIcosahedralGraph","Graph","Perspective"]//StyleGraphs

UnitDistance

GraphData["TruncatedIcosahedralGraph","UnitDistance"]

UnitDistanceGraphQ[GraphData["TruncatedIcosahedralGraph"],Debug->True]//Timing

GraphData["TruncatedIcosahedralGraph","Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData["TruncatedIcosahedralGraph","Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

gs={,}

FindUnitDistanceEmbedding[GraphData["TruncatedIcosahedralGraph"],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

UnitDistance FindSubgraphUnitDistanceEmbedding

{"TruncatedIcosahedralGraph",l=FindSubgraphUnitDistanceEmbedding[GraphData["TruncatedIcosahedralGraph"],TimeConstraint->300,"Results"->All]//Timing}

gs=UniqueEmbeddings[l]

3D default

Graph3D[GraphData["TruncatedIcosahedralGraph","Edges"]]

3D All

GraphData["TruncatedIcosahedralGraph","Graph",{"3D","All"}]

3D UnitDistance

GraphData["TruncatedIcosahedralGraph","Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData["TruncatedIcosahedralGraph","Polyhedral"]

GraphData["TruncatedIcosahedralGraph","Graph",{"3D","Polyhedron"}]

Graphics3D[{FaceForm[Green,Red],GraphData["TruncatedIcosahedralGraph","Polyhedron"]}]

3D Canonical

GraphData["TruncatedIcosahedralGraph","Graph",{"3D","Canonical"}]

Graphics3D[{Opacity[.5],Red,GraphData["TruncatedIcosahedralGraph","Polyhedron","Canonical"],Yellow,Sphere[]}]

3D Canonical construction

Graphics3D[{FaceForm[Green,Red],poly=GraphPolyhedron[GraphData["TruncatedIcosahedralGraph"],1]}]

(can=FindCanonicalPolyhedron[poly,WorkingPrecision->30,MaxIterations->50000,PrecisionGoal->20,"ProgressInterval"->1000,"TangentFactor"->1/5,"PlanarFactor"->1/5,Debug->True])//Timing

SimplePolyhedronQ[can]

Graphics3D[{Opacity[.5],Red,can,Yellow,Sphere[]}]

(lens=Mean/@Split[Sort[PolyhedronEdgeLengths[can]],Abs[#1-#2]<1.*^-10&])//N

v=PolyhedronCoordinates[can];

colors=With[{n=Length[lens]},Table[Hue[i/n],{i,0,n-1}]]

Graphics3D[{Thick,Sequence@@Nearest[Thread[lens->colors],EuclideanDistance@@v[[#]]],Line[v[[#]]]}&/@PolyhedronEdges[can],Boxed->False]

g=PolyhedronGraph[can]

Faces

GraphPlot[Graph[g="TruncatedIcosahedralGraph"],Method->None,VertexRenderingFunction->({White,EdgeForm[Black],Disk[#,.25],Black,Text[#2,#1]}&)]

e=GraphData["TruncatedIcosahedralGraph","EdgeIndices"]

CompleteCycle[e_]:=CycleFromEdgeList[Append[e,First/@Cases[Tally[Flatten[e]],{_,1}]]]

Pentagons

Length[s4=Developer`ToPackedArray[Subsets[e,{4}]]]

Length[p3=Select[s4,Length[Union[Flatten[#]]]==5&]]

f5=Select[p3,Length[Intersection[Subsets[Union[Flatten[#]],{2}],e]]==5&]

faces5=Union[CompleteCycle/@f5]

Hexagons

((90!)/(5!85!))

Length[s2=Developer`ToPackedArray[Subsets[e,{2}]]]

Take[s2,10]

Take[p3,10]

Length[l6=Flatten[Outer[Join,p3,s2,1],1]]

Tally[Flatten[#]]&/@Take[l6,5]

Select[l6,Union[Last/@Tally[Flatten[#]]]==={2}&]

faces6=Union[CycleFromEdgeList/@%54]

Length/@{faces5,faces6}

2-GraphData[g,"V"]+GraphData[g,"M"]

Sort[Join[faces5,faces6]]

Cayley graph generators

GraphData["TruncatedIcosahedralGraph","CayleyGraphGeneratingGroupNames"]

Projections

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

{(1/4) (3-(Sqrt[5])+(Sqrt[10-2 (Sqrt[5])])),(1/4) (3-(Sqrt[5])+(Sqrt[10-2 (Sqrt[5])])),1,π-ArcTan[(NumericalApproximation[Root, (RowDefault["0.343", FEExpressionsNumericalApproximationElider]), Root[656-12976 ((#1)^2)+68136 ((#1)^4)-44676 ((#1)^6)+1681 ((#1)^8)&,6], ])]}

g=PolyhedronProjectionGraph[PolyhedronData["TruncatedIcosahedron","Polyhedron"],{(1/4) (3-(Sqrt[5])+(Sqrt[10-2 (Sqrt[5])])),(1/4) (3-(Sqrt[5])+(Sqrt[10-2 (Sqrt[5])])),1,π-ArcTan[(NumericalApproximation[Root, (RowDefault["0.343", FEExpressionsNumericalApproximationElider]), Root[656-12976 ((#1)^2)+68136 ((#1)^4)-44676 ((#1)^6)+1681 ((#1)^8)&,6], ])]},VertexLabels->"Name"]

v=GraphEmbedding[g];

π-ArcTan@@(v[[15]]-v[[1]])//RootReduce

{.185,.18,.225}/.225

g=PolyhedronProjectionGraph[PolyhedronData["TruncatedIcosahedron","Polyhedron"],{0.82,.82,1},VertexLabels->"Name"]

g=PolyhedronProjectionGraph[PolyhedronData["TruncatedIcosahedron","Polyhedron"],{a,a,1},VertexLabels->"Name"]

v=Lookup[Options[g],VertexCoordinates];

eqn1=FullSimplify[CollinearPoints[v[[{45,33,60}]]],a>0]

Solve[eqn1,a]//FullSimplify

N[%]

{1,(1/11) (Sqrt[185-82 (Sqrt[5])]),-(2/11) (-4+(Sqrt[5])),ArcTan[(Sqrt[2-((2 (Sqrt[5]))/3)])]}

g=PolyhedronProjectionGraph[PolyhedronData["TruncatedIcosahedron","Polyhedron"],{1,(1/11) (Sqrt[185-82 (Sqrt[5])]),-(2/11) (-4+(Sqrt[5])),ArcTan[(Sqrt[2-((2 (Sqrt[5]))/3)])]},VertexLabels->"Name"]

v=GraphEmbedding[g];

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

g=PolyhedronProjectionGraph[PolyhedronData["TruncatedIcosahedron","Polyhedron"],{1,.115,.26},VertexLabels->"Name"]

g=PolyhedronProjectionGraph[PolyhedronData["TruncatedIcosahedron","Polyhedron"],{1,a,b},VertexLabels->"Name"]

v=Lookup[Options[g],VertexCoordinates];

eqn1=FullSimplify[CollinearPoints[v[[{1,19,9}]]],a>0&&b>0]

eqn2=FullSimplify[CollinearPoints[v[[{54,45,40}]]],a>0&&b>0]

Solve[{eqn1,eqn2},{a,b}]//FullSimplify

N[%]

{1,(Sqrt[85-38 (Sqrt[5])]),-2+(Sqrt[5]),π-ArcTan[(NumericalApproximation[Root, (RowDefault["-0.368", FEExpressionsNumericalApproximationElider]), Root[25-190 ((#1)^2)+41 ((#1)^4)&,2], ])]}

g=PolyhedronProjectionGraph[PolyhedronData["TruncatedIcosahedron","Polyhedron"],{1,(Sqrt[85-38 (Sqrt[5])]),-2+(Sqrt[5]),π-ArcTan[(NumericalApproximation[Root, (RowDefault["-0.368", FEExpressionsNumericalApproximationElider]), Root[25-190 ((#1)^2)+41 ((#1)^4)&,2], ])]},VertexLabels->"Name"]

v=GraphEmbedding[g];

ArcTan@@(v[[9]]-v[[59]])//FullSimplify

g=PolyhedronProjectionGraph[PolyhedronData["TruncatedIcosahedron","Polyhedron"],{1,.165,.27},VertexLabels->"Name"]

g=PolyhedronProjectionGraph[PolyhedronData["TruncatedIcosahedron","Polyhedron"],{1,a,b},VertexLabels->"Name"]

v=Lookup[Options[g],VertexCoordinates];

FullSimplify[CollinearPoints[v[[{54,45,40}]]],a>0&&b>0]

FullSimplify[v[[28]]-v[[55]],a>0&&b>0]

FullSimplify[Solve[Thread[%==0],{a,b}],a>0&&b>0]

N[%]

(1/5) (Sqrt[25-2 (Sqrt[5])]),1,-(Sqrt[(2/(5 (5+(Sqrt[5]))))]),π/2-ArcTan[(NumericalApproximation[Root, (RowDefault["-0.433", FEExpressionsNumericalApproximationElider]), Root[361-1946 ((#1)^2)+109 ((#1)^4)&,2], ])]}

g=PolyhedronProjectionGraph[PolyhedronData["TruncatedIcosahedron","Polyhedron"],{(1/5) (Sqrt[25-2 (Sqrt[5])]),1,-(Sqrt[(2/(5 (5+(Sqrt[5]))))]),π/2-ArcTan[(NumericalApproximation[Root, (RowDefault["-0.433", FEExpressionsNumericalApproximationElider]), Root[361-1946 ((#1)^2)+109 ((#1)^4)&,2], ])]},VertexLabels->"Name"]

v=GraphEmbedding[g];

ArcTan@@(v[[20]]-v[[24]])//FullSimplify

{.535,.59,-.15}/.59

g=PolyhedronProjectionGraph[PolyhedronData["TruncatedIcosahedron","Polyhedron"],{0.9067796610169493`,1.`,-0.2542372881355932`},VertexLabels->"Name"]

g=PolyhedronProjectionGraph[PolyhedronData["TruncatedIcosahedron","Polyhedron"],{a,1,b},VertexLabels->"Name"]

v=Lookup[Options[g],VertexCoordinates];

Solve[Thread[FullSimplify[v[[47]]-v[[50]],a>0&&b>0]==0],{a,b},Reals]//FullSimplify

N[%]

{(1/11) (Sqrt[229-(382/(Sqrt[5]))]),1,(2/11) (Sqrt[34-(62/(Sqrt[5]))]),ArcTan[(5/2)+((7 (Sqrt[5]))/6)]}

{.1,.145,.065}/.145

g=PolyhedronProjectionGraph[PolyhedronData["TruncatedIcosahedron","Polyhedron"],{(1/11) (Sqrt[229-(382/(Sqrt[5]))]),1,(2/11) (Sqrt[34-(62/(Sqrt[5]))]),ArcTan[(5/2)+((7 (Sqrt[5]))/6)]},VertexLabels->"Name"]

v=GraphEmbedding[g];

FullSimplify[ArcTan@@(v[[31]]-v[[21]])]

PolyhedronProjectionGraph[PolyhedronData["TruncatedIcosahedron","Polyhedron"],{0.6896551724137931`,1.`,0.5},VertexLabels->"Name"]

g=PolyhedronProjectionGraph[PolyhedronData["TruncatedIcosahedron","Polyhedron"],{a,1,b},VertexLabels->"Name"]

v=Lookup[Options[g],VertexCoordinates];

Solve[Thread[FullSimplify[v[[28]]-v[[31]],a>0&&b>0]==0],{a,b}]//FullSimplify

N[%]

{1/2,(1/2) (Sqrt[5+2 (Sqrt[5])]),1,π/2-ArcTan[(1/2)-(3/(2 (Sqrt[5])))]}

{.53,-.225,.6}/.6

g=PolyhedronProjectionGraph[PolyhedronData["TruncatedIcosahedron","Polyhedron"],{1/2,(1/2) (Sqrt[5+2 (Sqrt[5])]),1,π/2-ArcTan[(1/2)-(3/(2 (Sqrt[5])))]},VertexLabels->"Name"]

v=Lookup[Options[g],VertexCoordinates];

ArcTan@@(v[[9]]-v[[23]])//FullSimplify

g=PolyhedronProjectionGraph[PolyhedronData["TruncatedIcosahedron","Polyhedron"],{a,b,1},VertexLabels->"Name"]

v=Lookup[Options[g],VertexCoordinates];

eqn1=FullSimplify[(v[[21]]-v[[31]]).(v[[52]]-v[[30]]),a>0&&b>0]

eqn2=FullSimplify[SquaredEuclideanDistance@@v[[{53,33}]]-SquaredEuclideanDistance@@v[[{29,28}]],a>0&&b>0]

Solve[Thread[{eqn1,eqn2}==0],{a,b}]//FullSimplify

N[%]

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

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

v=GraphEmbedding[g];

ArcTan@@(v[[41]]-v[[1]])//FullSimplify

g=PolyhedronProjectionGraph[PolyhedronData["TruncatedIcosahedron","Polyhedron"],{0.7,0.375`,1.`},VertexLabels->"Name"]

g=PolyhedronProjectionGraph[PolyhedronData["TruncatedIcosahedron","Polyhedron"],{a,b,1},VertexLabels->"Name"]

v=Lookup[Options[g],VertexCoordinates];

eqn1=FullSimplify[CollinearPoints[v[[{55,48,42}]]],a>0&&b>0]

eqn2=FullSimplify[SquaredEuclideanDistance@@v[[{33,53}]]-SquaredEuclideanDistance@@v[[{29,28}]],a>0&&b>0]

NSolve[FullSimplify[{eqn1,eqn2}],{a,b},Reals]//FullSimplify

Solve[FullSimplify[{eqn1,eqn2==0}],{a,b},Reals]//FullSimplify

N[%]

g=PolyhedronProjectionGraph[PolyhedronData["TruncatedIcosahedron","Polyhedron"],{-.5,-.153,1},VertexLabels->"Name"]

{1,(Sqrt[85-38 (Sqrt[5])]),3-(Sqrt[5]),ArcCot[(3/(Sqrt[2]))]}

g=PolyhedronProjectionGraph[PolyhedronData["TruncatedIcosahedron","Polyhedron"],{1,(Sqrt[85-38 (Sqrt[5])]),3-(Sqrt[5]),ArcCot[(3/(Sqrt[2]))]},VertexLabels->"Name"]

v=GraphEmbedding[g];

ArcTan@@(v[[41]]-v[[1]])//FullSimplify

g=PolyhedronProjectionGraph[PolyhedronData["TruncatedIcosahedron","Polyhedron"],{1,a,b},VertexLabels->"Name"]

v=Lookup[Options[g],VertexCoordinates];

eqn1=FullSimplify[SquaredEuclideanDistance@@v[[{29,52}]]-SquaredEuclideanDistance@@v[[{50,58}]],a>0&&b>0]

eqn2=FullSimplify[CollinearPoints[v[[{8,12,32}]]],a>0&&b>0]

Solve[FullSimplify[{eqn1==0,eqn2}],{a,b},Reals]//FullSimplify

N[%]

{1, 0, -3((Sqrt[5])-7)/22}

g=PolyhedronProjectionGraph[PolyhedronData["TruncatedIcosahedron","Polyhedron"],{1,0,-(3/22) (-7+(Sqrt[5]))},VertexLabels->"Name"]

PolyhedronProjectionGraph[PolyhedronData["TruncatedIcosahedron","Polyhedron"],{1,.04,.655},VertexLabels->"Name"]

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

v=Lookup[Options[g],VertexCoordinates];

FullSimplify[v[[33]]-v[[51]],z>0]

Solve[((3 (Sqrt[5+(Sqrt[5])])-(Sqrt[85+31 (Sqrt[5])]) z)/((Sqrt[10]) (Sqrt[1+(z^2)])))==0,z]//FullSimplify

N[%]

{1, 0, 1}

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

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

g=PolyhedronProjectionGraph[PolyhedronData["TruncatedIcosahedron","Polyhedron"],{1,(Sqrt[5-2 (Sqrt[5])]),-2,π/2-ArcTan[(1/2)+(3/(2 (Sqrt[5])))]},VertexLabels->"Name"]

g=PolyhedronProjectionGraph[PolyhedronData["TruncatedIcosahedron","Polyhedron"],{1,.72,-1.5},VertexLabels->"Name"]

g=PolyhedronProjectionGraph[PolyhedronData["TruncatedIcosahedron","Polyhedron"],{1,z,-2},VertexLabels->"Name"]

v=Lookup[Options[g],VertexCoordinates];

FullSimplify[v[[14]]-v[[19]],z>0]

Solve[(-5+(Sqrt[5 (5+2 (Sqrt[5]))]) z)==0,z]//FullSimplify

ArcTan@@(v[[32]]-v[[15]])//FullSimplify

{0, 0, 1}

SelectFirst[PolyhedronData["TruncatedIcosahedron","Polygons"][[All,1]],Length[#]==5&]//Mean//Normalize//FullSimplify

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

{0, 1, 0}

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

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

v=GraphEmbedding[g];

ArcTan@@(v[[23]]-v[[17]])//FullSimplify

{1, 1, 0}

g=PolyhedronProjectionGraph[PolyhedronData["TruncatedIcosahedron","Polyhedron"],{1,1,0,ArcTan[(Sqrt[8-2 (Sqrt[2 (5+(Sqrt[5]))])])]},VertexLabels->"Name"]

v=GraphEmbedding[g];

ArcTan@@(v[[20]]-v[[24]])//FullSimplify

{1/GoldenRatio, (NumericalApproximation[Root, (RowDefault["-0.449", FEExpressionsNumericalApproximationElider]), Root[5-25 ((#1)^2)+((#1)^4)&,2], ]), 1}

g=PolyhedronProjectionGraph[PolyhedronData["TruncatedIcosahedron","Polyhedron"],{(1/2) (-1+(Sqrt[5])),(NumericalApproximation[Root, (RowDefault["-0.449", FEExpressionsNumericalApproximationElider]), Root[5-25 ((#1)^2)+((#1)^4)&,2], ]),1,-ArcTan[(NumericalApproximation[Root, (RowDefault["-0.221", FEExpressionsNumericalApproximationElider]), Root[1-21 ((#1)^2)+9 ((#1)^4)&,2], ])]},VertexLabels->"Name"]

v=SelectFirst[PolyhedronData["TruncatedIcosahedron","Polygons"][[All,1]],Length[#]==6&]//Mean//FullSimplify

FullSimplify[v/(Sqrt[(13/8)+(29/(8 (Sqrt[5])))])]

N[%]

g=PolyhedronProjectionGraph[PolyhedronData["TruncatedIcosahedron","Polyhedron"],{(1/2) (-1+(Sqrt[5])),(NumericalApproximation[Root, (RowDefault["-0.449", FEExpressionsNumericalApproximationElider]), Root[5-25 ((#1)^2)+((#1)^4)&,2], ]),1},VertexLabels->"Name"]

v=GraphEmbedding[g];

ArcTan@@(v[[59]]-v[[27]])//FullSimplify