Rhombic Triacontahedral Graph

Author

Eric W. Weisstein
August 25, 2022

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

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

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

Embeddings

<<MathWorld`Graphs`

Names

GraphData["RhombicTriacontahedralGraph","StandardName"]

GraphData["RhombicTriacontahedralGraph","StandardNames"]

Primary

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

Default embedding

GraphPlot@GraphData["RhombicTriacontahedralGraph","Edges"]

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

RecognizeGraph[g]

All

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

Embedding types

GraphDataEmbeddings[GraphData["RhombicTriacontahedralGraph","EdgeList"]]

Degenerate

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

Planar

GraphData["RhombicTriacontahedralGraph","Planar"]

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

MinimalCrossing

GraphData["RhombicTriacontahedralGraph",{"RCN","CN"}]

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

MinimalCrossing construction

g=RotateGraph[MinimalCrossingGraph[GraphData["RhombicTriacontahedralGraph"],"CrossingNumber"->0]]

RecognizeGraph[g]

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Tutte embeddings

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

Tally[ReverseSort[Length/@faces]]

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

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

Circulant

GraphData["RhombicTriacontahedralGraph","Graph","Circulant"]//StyleGraphs

Halin

GraphData["RhombicTriacontahedralGraph","Halin"]

GraphData["RhombicTriacontahedralGraph","Graph","Halin"]//StyleGraphs

LCF

GraphData["RhombicTriacontahedralGraph","LCF"]

GraphData["RhombicTriacontahedralGraph","LCFSignature"]

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

UnitDistance

GraphData["RhombicTriacontahedralGraph","UnitDistance"]

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

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

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

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData["RhombicTriacontahedralGraph","VertexList"],GraphData["RhombicTriacontahedralGraph","EdgeList"]],MaxIterations->500]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData["RhombicTriacontahedralGraph","Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

g=LabelGraph[]

UnitDistance by FindUnitDistanceEmbedding

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

UnitDistance exact construction

Picture from Ed Pegg, Jr. Sep24, 2021 (after seeing the above)

cs[q_]:={Cos[q],Sin[q]}
v={
(*1*){0,0},
(*2*){1,0},
(*3*){1,0}+cs[a+b],
(*4*)cs[a],
(*5*)cs[a+b]
(*6*)
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};

v

Manipulate[
(v={
(*1*){0,0},
(*2*){1,0},
(*3*){1,0}+cs[a+b],
(*4*)cs[a],
(*5*)cs[a+b],
(*6*)cs[a+b]+cs[a]
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};
Graph[Range[Length[v]],UnitEdges[v],VertexCoordinates->v,VertexLabels->Automatic]),{{a,30Degree},0,Pi/2},{{b,20Degree},0,Pi/2}]

UnitDistance FindSubgraphUnitDistanceEmbedding

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

gs=UniqueEmbeddings[l]

Matchstick

GraphData["RhombicTriacontahedralGraph","Matchstick"]

GraphData["RhombicTriacontahedralGraph","Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData["RhombicTriacontahedralGraph","Edges"]]

3D All

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

3D UnitDistance

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

3D Polyhedron

GraphData["RhombicTriacontahedralGraph","Polyhedral"]

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

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

3D Canonical

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

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

3D Canonical construction

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

(can=FindCanonicalPolyhedron[poly,WorkingPrecision->50,MaxIterations->50000,PrecisionGoal->20,"ProgressInterval"->1000,"TangentFactor"->1/10,"PlanarFactor"->1/10,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]

Perspective projections

PolyhedronProjectionGraph[PolyhedronData["RhombicTriacontahedron","Polyhedron"],#,"Simplification"->FullSimplify]&/@{{1,(Sqrt[5-2 (Sqrt[5])]),-(3-(Sqrt[5])),π/2-ArcTan[(1/10) (5+(Sqrt[5]))]},{1,0,1},{1,0,2},{(3/4) (-1+(Sqrt[5])),(1/2) (Sqrt[(1/2) (5+(Sqrt[5]))]),1,-ArcTan[(Sqrt[2]) (-2+(Sqrt[5]))]},{1,(Sqrt[5-2 (Sqrt[5])]),(1/4) (3-(Sqrt[5])),-ArcTan[(Sqrt[5 (9-2 (Sqrt[5]))])]},{-1,(Sqrt[5-2 (Sqrt[5])]),-2+(Sqrt[5]),-ArcTan[(1/11) (Sqrt[3]) (-4+(Sqrt[5]))]},{1,(Sqrt[5-2 (Sqrt[5])]),(1/2) (-1+(Sqrt[5])),-ArcTan[(NumericalApproximation[Root, (RowDefault["-1.18", FEExpressionsNumericalApproximationElider]), Root[5-5 ((#1)^2)+((#1)^4)&,2], ])]}}

GraphData["RhombicTriacontahedralGraph","Graph","Perspective"]

Properties

Classes

GraphData[g,"Classes"]

Matrices

GraphicsRow[GraphMatrixPlot[GraphData["RhombicTriacontahedralGraph"]]]

Properties

GraphData[g,"Missing"]

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

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

Embeddings by projection

PolyhedronData[poly]

Degenerate

SkeletonGraph[PolyhedronData["RhombicTriacontahedron","Polyhedron"],#]&/@{1.9,2.7,3,3.5,5,6}//StyleGraphs

GraphData

GraphClasses

GraphClassString[g]