Parabidiminished Rhombicosidodecahedron

Author

Eric W. Weisstein
May 14, 2021

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

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

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

Graphic

PolyhedronData

PolyhedronData["ParabidiminishedRhombicosidodecahedron","Orientations"]

PolyhedronData["ParabidiminishedRhombicosidodecahedron"]

PolyhedronData["ParabidiminishedRhombicosidodecahedron","VertexCoordinates"]

PolyhedronData (V12.1)

PolyhedronData["ParabidiminishedRhombicosidodecahedron"]

PolyhedronData["ParabidiminishedRhombicosidodecahedron","VertexCoordinates"]

Netlib

PolyhedronName[JohnsonSolid[80]]

Show[Graphics3D[Polyhedron[JohnsonSolid[80]]],Boxed->False]

Net

PolyhedronData

PolyhedronData["ParabidiminishedRhombicosidodecahedron","Net"]

PolyhedronData["ParabidiminishedRhombicosidodecahedron","Net","Colored"]

Netlib

Show[Graphics[Net[JohnsonSolid[80]]],AspectRatio->Automatic]

Construction

Construct from C5 embedding of SmallRhombicosidodecahedron

PolyhedronData[{"SmallRhombicosidodecahedron","C5"}]

v0=PolyhedronData[{"SmallRhombicosidodecahedron","C5"},"VertexCoordinates"];

{zmin,zmax}=MinMax[v0[[All,3]]]

N[%11]

pentagons=Flatten[Position[v0,{_,_,#}]]&/@{zmin,zmax}

oldfaces=Select[PolyhedronData[{"SmallRhombicosidodecahedron","C5"},"FaceIndices"],Length[Intersection[#,pentagons[[1]]]]==0&&Length[Intersection[#,pentagons[[2]]]]==0&]

Graphics3D[{PolyhedronData[{"SmallRhombicosidodecahedron","C5"},"GraphicsComplex"],MapIndexed[Text[#2[[1]],#,Background->White]&,PolyhedronData[{"SmallRhombicosidodecahedron","C5"},"VertexCoordinates"]]}]

Length/@(decagons={{23,27,31,33,29,25,11,15,13,9},{42,46,50,52,48,44,12,4,2,10}})

Graphics3D[{FaceForm[Green,Red],
Polyhedron[PolyhedronData[{"SmallRhombicosidodecahedron","C5"},"VertexCoordinates"],faces=Join[oldfaces,decagons]],
MapIndexed[Text[#2[[1]],#,Background->White]&,PolyhedronData[{"SmallRhombicosidodecahedron","C5"},"VertexCoordinates"]]}]

v=PolyhedronData[{"SmallRhombicosidodecahedron","C5"},"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["ParabidiminishedRhombicosidodecahedron",{"V","F"}]

Graphics3D[p]

Volume[N[p]]

PolyhedronData["ParabidiminishedRhombicosidodecahedron","Volume"]

Properties

<<MathWorld`PolyhedronOperations`

p=PolyhedronData[pname="ParabidiminishedRhombicosidodecahedron","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//Quiet//Timing

PolyhedronCentroid[p]//Quiet//Timing

Circumsphere

PolyhedronData[pname,"Circumsphere"]

Circumsphere[p]//FullSimplify

(sphere=MyCircumsphere[p])//Timing

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

Convex

ConvexPolyhedronQ[p]

DihedralAngles

PolyhedronData[pname,"DihedralAngles"]//Tally

DihedralAngles[p]//FullSimplify//Tally

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

EdgeLengths

PolyhedronData[pname,"EdgeLengths"]

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

Faces

GeneralizedDiameter

PolyhedronData["ParabidiminishedRhombicosidodecahedron","GeneralizedDiameter"]

GeneralizedDiameter[p]//FullSimplify//Quiet

InertiaTensor

PolyhedronData[pname,"InertiaTensor"]

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

r=RootApproximant[N[i,1000],8]//FullSimplify//Quiet

PolyhedronInertiaTensor[p]//Timing

Insphere

PolyhedronData[pname,"Insphere"]

Insphere[p]

(sphere=MyInsphere[p])//Timing

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"])))//Timing

N[i]

r=RootApproximant[N[i,1000],32]//FullSimplify

N[i-r,10^4]

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"])))//Timing

N[i]

r=RootApproximant[N[i,1000],32]

N[i-r,10^4]

Midsphere

sphere=PolyhedronData[pname,"Midsphere"]

(sphere=Midsphere[p,"Simplification"->FullSimplify,Debug->True])//Timing

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

RegionFunction

reg=PolyhedronData["ParabidiminishedRhombicosidodecahedron","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["ParabidiminishedRhombicosidodecahedron","V"]

g=Graph[Range[vc],UndirectedEdge@@@PolyhedronData["ParabidiminishedRhombicosidodecahedron","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//Timing

PolyhedronVolume[p,Method->"GaussLaw"]//FullSimplify//Timing