Tridiminished Rhombicosidodecahedron

Author

Eric W. Weisstein
May 17, 2021

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

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

©2021 Wolfram Research, Inc. except for portions noted otherwise

Solid

PolyhedronData

PolyhedronData["TridiminishedRhombicosidodecahedron","Orientations"]

PolyhedronData["TridiminishedRhombicosidodecahedron"]

PolyhedronData (V12.1)

PolyhedronData["TridiminishedRhombicosidodecahedron"]

PolyhedronData["TridiminishedRhombicosidodecahedron","VertexCoordinates"]

Netlib

PolyhedronName[JohnsonSolid[83]]

Show[Graphics3D[Polyhedron[JohnsonSolid[83]]],Boxed->False]

Net

PolyhedronData

PolyhedronData["TridiminishedRhombicosidodecahedron","Net"]

PolyhedronData["TridiminishedRhombicosidodecahedron","Net","Colored"]

Netlib

Show[Graphics[Net[JohnsonSolid[83]]],AspectRatio->Automatic]

Skeleton graph

PolyhedronData["TridiminishedRhombicosidodecahedron","Skeleton"]

Construction

PolyhedronData[#,"Net"]&/@{"SmallRhombicosidodecahedron","TridiminishedRhombicosidodecahedron"}

PolyhedronData/@{"SmallRhombicosidodecahedron","TridiminishedRhombicosidodecahedron"}

PolyhedronData["SmallRhombicosidodecahedron","Orientations"]

Obtain "TridiminishedRhombicosidodecahedron" by removing 3 cupolas from the "C3" orientation of "SmallRhombicosidodecahedron"

Table[With[{poly={"SmallRhombicosidodecahedron","C3"}},Graphics3D[{
MapIndexed[Text[#2[[1]],#,Background->White]&,newv=RotationMatrix[.3+2π i/3,{0,0,1}].#&/@PolyhedronData[poly,"VertexCoordinates"]],
Polyhedron[newv,PolyhedronData[poly,"FaceIndices"]]
},ImageSize->Medium]],{i,0,2}]

pentagons=Flatten[{{18,42,10,24,6},{33,34,30,40,29},{47,58,48,52,51}}];

repl=With[{poly={"SmallRhombicosidodecahedron","C3"}},Thread[Delete[Range[PolyhedronData[poly,"V"]],List/@pentagons]->Range[PolyhedronData[poly,"V"]-Length[pentagons]]]]

repl[[All,1]]

v=With[{poly={"SmallRhombicosidodecahedron","C3"}},Delete[PolyhedronData[poly,"VertexCoordinates"],List/@pentagons]]

faceindices=With[{poly={"SmallRhombicosidodecahedron","C3"}},Select[PolyhedronData[poly,"FaceIndices"],Length[Intersection[#,pentagons]]==0&]/.repl]

Max[Flatten[faceindices]]

PolyhedronData["TridiminishedRhombicosidodecahedron","V"]

Graphics3D[{
MapIndexed[Text[#2[[1]],#,Background->White]&,v],
Polyhedron[v,faceindices]}]

Length/@(decagons={{5,15,39,36,41,12,2,29,24,27},{16,17,34,43,45,38,37,44,42,33},{6,21,30,18,25,26,19,31,22,7}})

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

allfaces=Sort[SmallestRotation/@Union[decagons,faceindices]]

p=Polyhedron[v,allfaces]

Graphics3D[{FaceForm[Green,Red],p}]

Properties

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

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//Timing

PolyhedronCentroid[p]

Circumsphere

sphere=PolyhedronData[pname,"Circumsphere"]

sphere=Circumsphere[p]//FullSimplify

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

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

Compound

PolyhedronData[pname,"Compound"]

Convex

PolyhedronData[pname,"Convex"]

ConvexPolyhedronQ[p]

DihedralAngles

PolyhedronData[pname,"DihedralAngles"]//Tally

DihedralAngles[p]//FullSimplify//Tally

EdgeLengths

PolyhedronData[pname,"EdgeLengths"]

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

Faces

GeneralizedDiameter

PolyhedronData[pname,"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])//Timing

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

Oriented

Graphics3D[{FaceForm[Green,Red],p}]

Region

PolyhedronData[pname,"Region"]

RegionFunction

PolyhedronData["TridiminishedRhombicosidodecahedron","RegionFunction"]

bounds=.1{-1,1}+#&/@RegionBounds[ImplicitRegion[N[PolyhedronData["TridiminishedRhombicosidodecahedron","RegionFunction"]][x,y,z],{x,y,z}]]

ranges=Flatten/@Transpose[{{x,y,z},bounds}]

RegionPlot3D[Evaluate[N[PolyhedronData["TridiminishedRhombicosidodecahedron","RegionFunction"]][x,y,z]],Evaluate[Sequence@@ranges],BoxRatios->Automatic,PlotPoints->40]

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

ineq2=FullSimplify[ineq];//Timing

ineq2

SimplePolyhedron

SimplePolyhedronQ[p]

SkeletonCoordinates

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

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

PolyhedronData[pname,"Volume"]

Volume[p]//FullSimplify

PolyhedronVolume[p]//FullSimplify

Perspective projections

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

XXX

PolyhedronProjectionGraph[PolyhedronData["DuererSolid","Polyhedron"],]