Metabidiminished Rhombicosidodecahedron

Author

Eric W. Weisstein
September 8, 2023

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

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

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

Graphic

PolyhedronData["MetabidiminishedRhombicosidodecahedron"]

PolyhedronData["MetabidiminishedRhombicosidodecahedron","VertexCoordinates"]

PolyhedronData (V12.1)

PolyhedronData["MetabidiminishedRhombicosidodecahedron"]

PolyhedronData["MetabidiminishedRhombicosidodecahedron","VertexCoordinates"]

Netlib

PolyhedronName[JohnsonSolid[81]]

Show[Graphics3D[Polyhedron[JohnsonSolid[81]]],Boxed->False]

Net

PolyhedronData

PolyhedronData["MetabidiminishedRhombicosidodecahedron","Net"]

PolyhedronData["MetabidiminishedRhombicosidodecahedron","Net","Colored"]

Netlib

Show[Graphics[Net[JohnsonSolid[81]]],AspectRatio->Automatic]

Properties

Initialization

<<MathWorld`Polyhedra`

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

vol=PolyhedronData[pname,"Volume"]

AdjacentFaceIndices

AdjacentFaceIndices[p]

Centroid

Graphics3D[{AbsolutePointSize[10],
Point[{0,0,0}],{Blue,Point[PolyhedronData[pname,"Centroid"]]},Opacity[.2],PolyhedronData[pname,"GraphicsComplex"]}]

PolyhedronData[pname,"Centroid"]

RegionCentroid[p]//FullSimplify

PolyhedronCentroid[p]//FullSimplify

Circumsphere

PolyhedronData[pname,"Circumsphere"]

Circumsphere[p]//FullSimplify

(sphere=MyCircumsphere[p])//Timing

Graphics3D[{{Opacity[.2],Yellow,sphere},p},Boxed->False]

Convex

ConvexPolyhedronQ[p]

DehnInvariant

PolyhedronData[pname,"DehnInvariant"]

(dehn=DehnInvariant[p,Debug->True]//FullSimplify)//Timing

ToConwayBasis[dehn]

Complement[Select[PolyhedronData[All],PolyhedronData[#,"DehnInvariant"]==dehn&&PolyhedronData[#,"Volume"]==vol&],{pname}]

DihedralAngles

PolyhedronData[pname,"DihedralAngles"]//Tally

DihedralAngles[p]//FullSimplify//Tally

SimplifyNumericallyDistinct[DihedralAngles[p],Debug->True]

GeneralizedDiameter

PolyhedronData["MetabidiminishedRhombicosidodecahedron","GeneralizedDiameter"]

GeneralizedDiameter[p]//FullSimplify//Quiet

InertiaTensor

PolyhedronData[pname,"InertiaTensor"]

((MomentOfInertia[p,{0,0,0}])/(Volume[p]))//RootReduce//Timing

PolyhedronInertiaTensor[p]//Timing

Insphere

PolyhedronData[pname,"Insphere"]

Insphere[p]

sphere=MyInsphere[p]

MeanCylindricalRadius

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

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

N[i]

MeanSquareCylindricalRadius

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

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

N[i]

MeanSphericalRadius

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

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

N[i]

MeanSquareSphericalRadius

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

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

N[i]

Midsphere

sphere=PolyhedronData[pname,"Midsphere"]

(sphere=Midsphere[p]//FullSimplify)//Timing

Graphics3D[{{Opacity[.5],p},{Red,Midsphere[p]}},Boxed->False]

RegionFunction

reg=PolyhedronData["MetabidiminishedRhombicosidodecahedron","RegionFunction"]

bounds=RegionBounds[ImplicitRegion[Evaluate[N[reg[x,y,z]]],{x,y,z}]]

ranges=Flatten/@Transpose[{{x,y,z},.1{-1,1}+#&/@bounds}]

RegionPlot3D[Evaluate[N[reg[x,y,z]]],Evaluate[Sequence@@ranges],BoxRatios->Automatic,PlotPoints->40]

ByteCount[ineq=PolyhedronInequalities[p,{#1,#2,#3}]]

ineq2=FullSimplify[ineq];//Timing

ByteCount[ineq2]

ineq2

SkeletonCoordinates

vc=PolyhedronData["MetabidiminishedRhombicosidodecahedron","V"]

g=Graph[Range[vc],UndirectedEdge@@@PolyhedronData["MetabidiminishedRhombicosidodecahedron","EdgeIndices"]]

r=RecognizeGraph[g]

GraphData[r,"Graph","All"]

ToCommonEdges[GraphData[r],g]

SurfaceArea

PolyhedronData[pname,"SurfaceArea"]

RootReduce[%]

SurfaceArea[p]//RootReduce//Timing

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

VertexSubsetHulls

PolyhedronData[pname,"VertexSubsetHulls"]

Platonic

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

Volume

vol=PolyhedronData["MetabidiminishedRhombicosidodecahedron","Volume"]

PolyhedronVolume[p]//FullSimplify

vol=(5/3) (11+5 (Sqrt[5]))

N[%]

Construction

Construct from C2 embedding of SmallRhombicosidodecahedron

PolyhedronData[{"SmallRhombicosidodecahedron","C2"}]

Graphics3D[{PolyhedronData[{"SmallRhombicosidodecahedron","C2"},"GraphicsComplex"],MapIndexed[Text[#2[[1]],#,Background->White]&,PolyhedronData[{"SmallRhombicosidodecahedron","C2"},"VertexCoordinates"]]}]

pentagons={{10,24,6,18,42},{12,44,20,8,26}};

oldfaces=Select[PolyhedronData[{"SmallRhombicosidodecahedron","C2"},"FaceIndices"],Length[Intersection[#,pentagons[[1]]]]==0&&Length[Intersection[#,pentagons[[2]]]]==0&]

decagons={{57,48,58,19,7,40,30,39,4,16},{46,55,14,2,37,28,35,5,17,53}};

Graphics3D[{FaceForm[Green,Red],
Polyhedron[PolyhedronData[{"SmallRhombicosidodecahedron","C2"},"VertexCoordinates"],faces=Join[oldfaces,decagons]],
MapIndexed[Text[#2[[1]],#,Background->White]&,PolyhedronData[{"SmallRhombicosidodecahedron","C2"},"VertexCoordinates"]]}]

v=PolyhedronData[{"SmallRhombicosidodecahedron","C2"},"VertexCoordinates"][[Flatten[faces]//Union]]

SmallestRotation[l_]:=Sort[NestList[RotateRight,l,Length[l]-1]][[1]]

newfaces=Sort[SmallestRotation/@(faces/.Thread[(u=Union[Flatten[faces]])->Range[Length[u]]])]

p=Polyhedron[v,newfaces]

PolyhedronData["MetabidiminishedRhombicosidodecahedron",{"V","F"}]

Graphics3D[p]

Volume[N[p]]

PolyhedronData["MetabidiminishedRhombicosidodecahedron","Volume"]