Parabigyrate Rhombicosidodecahedron

Author

Eric W. Weisstein
May 14, 2021

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

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

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

Graphic

PolyhedronData

PolyhedronData["ParabigyrateRhombicosidodecahedron","Orientations"]

PolyhedronData["ParabigyrateRhombicosidodecahedron"]

PolyhedronData["ParabigyrateRhombicosidodecahedron","VertexCoordinates"]

PolyhedronData (V12.1; inexact)

PolyhedronData["ParabigyrateRhombicosidodecahedron"]

PolyhedronData["ParabigyrateRhombicosidodecahedron","VertexCoordinates"]

Netlib

PolyhedronName[JohnsonSolid[73]]

Show[Graphics3D[Polyhedron[JohnsonSolid[73]]],Boxed->False]

Net

PolyhedronData

PolyhedronData["ParabigyrateRhombicosidodecahedron","Net"]

PolyhedronData["ParabigyrateRhombicosidodecahedron","Net","Colored"]

Netlib

Show[Graphics[Net[JohnsonSolid[73]]],AspectRatio->Automatic]

Properties

<<MathWorld`PolyhedronOperations`

p=PolyhedronData[pname="ParabigyrateRhombicosidodecahedron","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]//FullSimplify//Quiet//Timing

Circumsphere

PolyhedronData[pname,"Circumsphere"]

Circumsphere[p]//RootReduce//Timing

(sphere=MyCircumsphere[p])//Timing

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

Convex

ConvexPolyhedronQ[p]

DihedralAngles

PolyhedronData[pname,"DihedralAngles"]//Tally

DihedralAngles[p]//FullSimplify//Tally

dih=DihedralAngles[p]

(dih2=SimplifyNumericallyDistinct[dih,Debug->True,"Simplification"->FullSimplify])//Timing

N[dih-dih2]//Chop

EdgeLengths

PolyhedronData[pname,"EdgeLengths"]

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

Faces

GeneralizedDiameter

PolyhedronData["ParabigyrateRhombicosidodecahedron","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])//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[r-i,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]//FullSimplify

N[r-i,10^4]

Midsphere

sphere=PolyhedronData[pname,"Midsphere"]

(sphere=Midsphere[p])//Timing

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

Oriented

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

RegionFunction

PolyhedronData["ParabigyrateRhombicosidodecahedron","RegionFunction"]

bounds=RegionBounds[ImplicitRegion[N[PolyhedronData["ParabigyrateRhombicosidodecahedron","RegionFunction"]][x,y,z],{x,y,z}]]

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

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

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

ByteCount[ineq2=FullSimplify[ineq]]//Timing

ineq2

SkeletonCoordinates

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

e=PolyhedronData["ParabigyrateRhombicosidodecahedron","EdgeIndices"]

g=Graph[Range[vc],UndirectedEdge@@@e]

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,Method->"GaussLaw"]//FullSimplify

Construction

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

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

PolyhedronData["SmallRhombicosidodecahedron","Orientations"]

Obtain GyrateRhombicosidodecahedron by rotating two opposite cupolas in the C5 orientation of SmallRhombicosidodecahedron

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

toppentagon={36,21,38,3,1};

topcupola={23,27,31,33,29,25,11,15,13,9};

botpentagon={16,14,55,60,57};

botcupola={4,2,10,42,46,50,52,48,44,12};

v=PolyhedronData["SmallRhombicosidodecahedron","VertexCoordinates"];

ClosedLine[l_]:=Append[l,l[[1]]]

Graphics3D[{
MapIndexed[Text[#2[[1]],#,Background->White]&,PolyhedronData["SmallRhombicosidodecahedron","VertexCoordinates"]],
PolyhedronData["SmallRhombicosidodecahedron","GraphicsComplex"],
Thick,ClosedLine[v[[topcupola]]],ClosedLine[v[[botcupola]]]}]

v2=ReplacePart[v,Join[
(#->(RotationMatrix[π/5,{0,0,1}].v[[#]]))&/@toppentagon,
(#->(RotationMatrix[π/5,{0,0,-1}].v[[#]]))&/@botpentagon]]//FullSimplify

faceindices=PolyhedronData["SmallRhombicosidodecahedron","FaceIndices"];

oldf=Select[faceindices,!(Length[Intersection[#,topcupola]]==2&&Length[Intersection[#,toppentagon]]>0)&&!(Length[Intersection[#,botcupola]]==2&&Length[Intersection[#,botpentagon]]>0)&]

topf=Select[faceindices,Length[Intersection[#,topcupola]]==2&&Length[Intersection[#,toppentagon]]>0&]/.Thread[topcupola->RotateLeft[topcupola]]

botf=Select[faceindices,Length[Intersection[#,botcupola]]==2&&Length[Intersection[#,botpentagon]]>0&]/.Thread[botcupola->RotateLeft[botcupola]]

faceindices=Join[oldf,topf,botf]

Graphics3D[p=Polyhedron[v2,faceindices]]

v2