Gyrate Bidiminished Rhombicosidodecahedron

Author

Eric W. Weisstein
September 6, 2023

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

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

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

Polyhedron

PolyhedronData

PolyhedronData["GyrateBidiminishedRhombicosidodecahedron"]

PolyhedronData["GyrateBidiminishedRhombicosidodecahedron","VertexCoordinates"]

PolyhedronData (V12.1)

PolyhedronData["GyrateBidiminishedRhombicosidodecahedron"]

PolyhedronData["GyrateBidiminishedRhombicosidodecahedron","VertexCoordinates"]

Netlib

PolyhedronName[JohnsonSolid[82]]

Show[Graphics3D[Polyhedron[JohnsonSolid[82]]],Boxed->False]

Net

PolyhedronData["GyrateBidiminishedRhombicosidodecahedron","Net"]

PolyhedronData["GyrateBidiminishedRhombicosidodecahedron","Net","Colored"]

Properties

Initialization

<<MathWorld`Polyhedra`

p=PolyhedronData[pname="GyrateBidiminishedRhombicosidodecahedron","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]//FullSimplify

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

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"]))

(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"]))

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

N[i]

MeanSphericalRadius

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

(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"]))

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

N[i]

Midsphere

PolyhedronData[pname,"Midsphere"]

sphere=Midsphere[p]//FullSimplify

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

RegionFunction

reg=PolyhedronData["GyrateBidiminishedRhombicosidodecahedron","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}]]

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

ineq2

Simple

SimplePolyhedronQ[p]

SkeletonCoordinates

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

g=Graph[Range[vc],UndirectedEdge@@@PolyhedronData["GyrateBidiminishedRhombicosidodecahedron","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["*idiminishedRhombicosidodecahedron"]

Construct by rotating the bottom pentagon of "MetabidiminishedRhombicosidodecahedron" about the central axis

PolyhedronData/@{"MetabidiminishedRhombicosidodecahedron","GyrateBidiminishedRhombicosidodecahedron"}

MetabidiminishedRhombicosidodecahedronFaceIndices=PolyhedronData["MetabidiminishedRhombicosidodecahedron","FaceIndices"]

MetabidiminishedRhombicosidodecahedronFaceIndices={{1,9,7},{3,8,11},{15,23,25},{16,26,24},{17,19,28},{18,30,21},{33,44,35},{34,37,46},{39,49,41},{40,42,50},{1,3,11,9},{1,7,17,28},{2,10,12,4},{3,30,18,8},{5,27,19,17},{6,18,21,32},{7,9,44,33},{8,34,46,11},{13,33,35,43},{14,48,37,34},{15,25,21,30},{15,28,19,23},{16,24,20,29},{16,31,22,26},{23,24,26,25},{35,44,49,39},{36,40,50,45},{37,41,49,46},{38,47,50,42},{39,41,42,40},{1,28,15,30,3},{2,4,31,16,29},{5,17,7,33,13},{6,14,34,8,18},{9,11,46,49,44},{10,45,50,47,12},{19,27,20,24,23},{21,25,26,22,32},{35,39,40,36,43},{37,48,38,42,41},{2,29,20,27,5,13,43,36,45,10},{4,12,47,38,48,14,6,32,22,31}};

v=PolyhedronData["MetabidiminishedRhombicosidodecahedron","VertexCoordinates"]

v={{-(1/2),-(1/2),-1-((Sqrt[5])/2)},{-(1/2),-(1/2),(1/2) (2+(Sqrt[5]))},{-(1/2),(1/2),-1-((Sqrt[5])/2)},{-(1/2),(1/2),(1/2) (2+(Sqrt[5]))},{-(1/2),-1-((Sqrt[5])/2),-(1/2)},{-(1/2),(1/2) (2+(Sqrt[5])),-(1/2)},{0,(1/(-3+(Sqrt[5]))),(1/4) (-5-(Sqrt[5]))},{0,(1/4) (3+(Sqrt[5])),(1/4) (-5-(Sqrt[5]))},{(1/2),-(1/2),-1-((Sqrt[5])/2)},{(1/2),-(1/2),(1/2) (2+(Sqrt[5]))},{(1/2),(1/2),-1-((Sqrt[5])/2)},{(1/2),(1/2),(1/2) (2+(Sqrt[5]))},{(1/2),-1-((Sqrt[5])/2),-(1/2)},{(1/2),(1/2) (2+(Sqrt[5])),-(1/2)},{(1/4) (-5-(Sqrt[5])),0,(1/(-3+(Sqrt[5])))},{(1/4) (-5-(Sqrt[5])),0,(1/4) (3+(Sqrt[5]))},{(1/4) (-1-(Sqrt[5])),(1/2) (-1-(Sqrt[5])),(1/(-3+(Sqrt[5])))},{(1/4) (-1-(Sqrt[5])),(1/2) (1+(Sqrt[5])),(1/(-3+(Sqrt[5])))},{(1/2) (-1-(Sqrt[5])),(1/(-3+(Sqrt[5]))),(1/4) (-1-(Sqrt[5]))},{(1/2) (-1-(Sqrt[5])),(1/(-3+(Sqrt[5]))),(1/4) (1+(Sqrt[5]))},{(1/2) (-1-(Sqrt[5])),(1/4) (3+(Sqrt[5])),(1/4) (-1-(Sqrt[5]))},{(1/2) (-1-(Sqrt[5])),(1/4) (3+(Sqrt[5])),(1/4) (1+(Sqrt[5]))},{-1-((Sqrt[5])/2),-(1/2),-(1/2)},{-1-((Sqrt[5])/2),-(1/2),(1/2)},{-1-((Sqrt[5])/2),(1/2),-(1/2)},{-1-((Sqrt[5])/2),(1/2),(1/2)},{(1/(-3+(Sqrt[5]))),(1/4) (-5-(Sqrt[5])),0},{(1/(-3+(Sqrt[5]))),(1/4) (-1-(Sqrt[5])),(1/2) (-1-(Sqrt[5]))},{(1/(-3+(Sqrt[5]))),(1/4) (-1-(Sqrt[5])),(1/2) (1+(Sqrt[5]))},{(1/(-3+(Sqrt[5]))),(1/4) (1+(Sqrt[5])),(1/2) (-1-(Sqrt[5]))},{(1/(-3+(Sqrt[5]))),(1/4) (1+(Sqrt[5])),(1/2) (1+(Sqrt[5]))},{(1/(-3+(Sqrt[5]))),(1/4) (5+(Sqrt[5])),0},{(1/4) (1+(Sqrt[5])),(1/2) (-1-(Sqrt[5])),(1/(-3+(Sqrt[5])))},{(1/4) (1+(Sqrt[5])),(1/2) (1+(Sqrt[5])),(1/(-3+(Sqrt[5])))},{(1/2) (1+(Sqrt[5])),(1/(-3+(Sqrt[5]))),(1/4) (-1-(Sqrt[5]))},{(1/2) (1+(Sqrt[5])),(1/(-3+(Sqrt[5]))),(1/4) (1+(Sqrt[5]))},{(1/2) (1+(Sqrt[5])),(1/4) (3+(Sqrt[5])),(1/4) (-1-(Sqrt[5]))},{(1/2) (1+(Sqrt[5])),(1/4) (3+(Sqrt[5])),(1/4) (1+(Sqrt[5]))},{(1/2) (2+(Sqrt[5])),-(1/2),-(1/2)},{(1/2) (2+(Sqrt[5])),-(1/2),(1/2)},{(1/2) (2+(Sqrt[5])),(1/2),-(1/2)},{(1/2) (2+(Sqrt[5])),(1/2),(1/2)},{(1/4) (3+(Sqrt[5])),(1/4) (-5-(Sqrt[5])),0},{(1/4) (3+(Sqrt[5])),(1/4) (-1-(Sqrt[5])),(1/2) (-1-(Sqrt[5]))},{(1/4) (3+(Sqrt[5])),(1/4) (-1-(Sqrt[5])),(1/2) (1+(Sqrt[5]))},{(1/4) (3+(Sqrt[5])),(1/4) (1+(Sqrt[5])),(1/2) (-1-(Sqrt[5]))},{(1/4) (3+(Sqrt[5])),(1/4) (1+(Sqrt[5])),(1/2) (1+(Sqrt[5]))},{(1/4) (3+(Sqrt[5])),(1/4) (5+(Sqrt[5])),0},{(1/4) (5+(Sqrt[5])),0,(1/(-3+(Sqrt[5])))},{(1/4) (5+(Sqrt[5])),0,(1/4) (3+(Sqrt[5]))}};

Clear[ClosedLine]

ClosedLine[l_]:={Thick,Line[Append[l,l[[1]]]]}

pentagonindices={35};

pentagons=MetabidiminishedRhombicosidodecahedronFaceIndices[[pentagonindices]]

Graphics3D[{,
MapIndexed[Text[#2[[1]],#]&,Mean[v[[#]]]&/@MetabidiminishedRhombicosidodecahedronFaceIndices],
ClosedLine[v[[MetabidiminishedRhombicosidodecahedronFaceIndices[[#]]]]]&/@pentagonindices
}]

fnew=Select[MetabidiminishedRhombicosidodecahedronFaceIndices,MatchQ[Length[Intersection[Flatten[pentagons],#]],0|5]&]

rotaxes=Mean[v[[MetabidiminishedRhombicosidodecahedronFaceIndices[[#]]]]]&/@pentagonindices//FullSimplify

Transpose[{rotaxes,pentagonindices}]

rotpents=(Function[vv,RotationMatrix[π/5,#].vv]/@v[[MetabidiminishedRhombicosidodecahedronFaceIndices[[#2]]]])&@@@Transpose[{rotaxes,pentagonindices}]//FullSimplify

vnew=ReplacePart[v,Rule@@@Flatten[Thread/@Transpose[{pentagons,rotpents}],1]]

Show[hull=ConvexHullRegion[vnew],Boxed->True]

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

newf=Sort[SmallestRotation/@MeshCells[hull,2][[All,1]]]

p=Polyhedron[vnew,newf]

ConvexPolyhedronQ[p]

SimplePolyhedronQ[p]

Volume[p//N]

PolyhedronData["GyrateBidiminishedRhombicosidodecahedron","Volume"]