Pentagonal Hexecontahedral Graph

Author

Eric W. Weisstein
September 1, 2022

This notebook downloaded from http://mathworld.wolfram.com/notebooks/GraphTheory/PentagonalHexecontahedralGraph.nb.

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

©2022 Wolfram Research, Inc. except for portions noted otherwise

Graph

Primary

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

All

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

Planar

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

MinimalCrossing

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

Tutte embeddings

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

Tally[ReverseSort[Length/@faces]]

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

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

Tutte embedding construction 5

GraphPlot[RotateGraph[]]

GraphPlot@RotateGraph[g=,Pi]

GraphPlot@(g=RotateGraph[,Pi,VertexLabels->Automatic,VertexStyle->Automatic])

v=GraphEmbedding[g];

Mean[v]

v=(#-Mean[v])&/@v;

Mean[v]

v[[51]]

Graphics[{MapIndexed[Text[#2[[1]],#]&,pent=CirclePoints[Csc[Pi/5]/2.,5]]}]

v[[{2,84,92,51,39}]]=pent

g=Graph[Range[Length[v]],EdgeList[g],VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Default embedding

Graph@GraphData["PentagonalHexecontahedralGraph","Edges"]

Degenerate

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

LCF

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

GraphData["PentagonalHexecontahedralGraph","LCF"]

Perspective

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

UnitDistance

GraphData["PentagonalHexecontahedralGraph","UnitDistance"]

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

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

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

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

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

UnitDistance FindSubgraphUnitDistanceEmbedding

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

gs=UniqueEmbeddings[l]

3D default

Graph3D[GraphData["PentagonalHexecontahedralGraph","Edges"]]

3D All

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

3D UnitDistance

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

3D Polyhedron

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

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

3D Canonical

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

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

3D Canonical construction

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

(can=FindCanonicalPolyhedron[poly,WorkingPrecision->50,MaxIterations->10000,PrecisionGoal->20])//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]

Embeddings

GraphDataEmbeddings[GraphData["PentagonalHexecontahedralGraph","EdgeList"]]

Perspective projections

PolyhedronProjectionGraph[PolyhedronData["PentagonalHexecontahedron","Polyhedron"],#,"Simplification"->RootReduce]&/@{{0,0,1},{3+(Sqrt[5]),0,1},{-1,(Sqrt[3 (9-4 (Sqrt[5]))]),-(3-(Sqrt[5])),-ArcTan[(1/2)-(3/(2 (Sqrt[5])))]}}

Stereographic projection

Degenerate

GraphPlot[SkeletonGraph[PolyhedronData[poly,"Faces"]//Normal,#],Method->None]&/@{1.9,2.7,3,3.5,5,6}

Properties

Classes

GraphData[g,"Classes"]

Matrices

GraphicsRow[GraphMatrixPlot[GraphData["PentagonalHexecontahedralGraph"]]]

Properties

GraphData[g,"Missing"]

GraphData[g,{"V","M"}]

Quiet@TableForm[GraphDataTable[g,x],TableDepth->2]//TraditionalForm

GraphData

GraphClasses

GraphClassString[g]