Metabigyrate Rhombicosidodecahedron

Author

Eric W. Weisstein
May 14, 2021

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

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

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

Graphic

PolyhedronData["MetabigyrateRhombicosidodecahedron","StandardNames"]

PolyhedronData

PolyhedronData["MetabigyrateRhombicosidodecahedron","Orientations"]

PolyhedronData["MetabigyrateRhombicosidodecahedron"]

PolyhedronData["MetabigyrateRhombicosidodecahedron","VertexCoordinates"]

PolyhedronData (V12.1; inexact)

PolyhedronData["MetabigyrateRhombicosidodecahedron"]

PolyhedronData["MetabigyrateRhombicosidodecahedron","VertexCoordinates"]

Netlib

PolyhedronName[JohnsonSolid[74]]

Show[Graphics3D[Polyhedron[JohnsonSolid[74]]],Boxed->False]

Net

PolyhedronData

PolyhedronData["MetabigyrateRhombicosidodecahedron","Net"]

PolyhedronData["MetabigyrateRhombicosidodecahedron","Net","Colored"]

Netlib

Show[Graphics[Net[JohnsonSolid[74]]],AspectRatio->Automatic]

Properties

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

AdjacentFaceIndices

AdjacentFaceIndices[p]

Length[%]

Centroid

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

PolyhedronData["MetabigyrateRhombicosidodecahedron","Centroid"]

PolyhedronData[pname,"Centroid"]

RegionCentroid[p]//FullSimplify

PolyhedronCentroid[p]//FullSimplify

Circumsphere

PolyhedronData["MetabigyrateRhombicosidodecahedron","Circumsphere"]

Graphics3D[{p,Blue,Opacity[.4],Sphere[{0,0,0},(1/2) (Sqrt[11+4 (Sqrt[5])])]}]

SimplifyNumericallyDistinct[Norm/@PolyhedronCoordinates[p],Debug->True]//Union

FullSimplify[(Sqrt[(11/4)+(Sqrt[5])])-(1/2) (Sqrt[11+4 (Sqrt[5])])]

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["MetabigyrateRhombicosidodecahedron","GeneralizedDiameter"]

GeneralizedDiameter[p]//FullSimplify//Quiet

InertiaTensor

PolyhedronData["MetabigyrateRhombicosidodecahedron","InertiaTensor"]

i=MomentOfInertia[p,{0,0,0}]/vol;

i=PolyhedronInertiaTensor[p,vol];//Timing

ByteCount[i]

i

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

PolyhedronData["MetabigyrateRhombicosidodecahedron","RegionFunction"]

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

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

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

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

ineq2=FullSimplify[ineq];//Timing

ineq2

SkeletonCoordinates

PolyhedronData["MetabigyrateRhombicosidodecahedron","SkeletonGraph"]

g=Graph[Range[Length[v]],UndirectedEdge@@@PolyhedronData["MetabigyrateRhombicosidodecahedron","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,Method->"GaussLaw"]//FullSimplify

Construction

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

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

PolyhedronData["SmallRhombicosidodecahedron","Orientations"]

Obtain GyrateRhombicosidodecahedron by rotating two adjacent cupolas in the C2 orientation of SmallRhombicosidodecahedron

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

toppentagon={42,10,24,6,18};

topcupola={53,46,55,14,2,37,28,35,5,17};

botpentagon={20,8,26,12,44};

botcupola={57,48,58,19,7,40,30,39,4,16};

v=PolyhedronData[{"SmallRhombicosidodecahedron","C2"},"VertexCoordinates"];

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

Graphics3D[{
MapIndexed[Text[#2[[1]],#,Background->White]&,PolyhedronData[{"SmallRhombicosidodecahedron","C2"},"VertexCoordinates"]],
PolyhedronData[{"SmallRhombicosidodecahedron","C2"},"GraphicsComplex"],
Thick,
ClosedLine/@{v[[topcupola]],v[[botcupola]]}
}]

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

faceindices=PolyhedronData[{"SmallRhombicosidodecahedron","C2"},"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