Bigyrate Diminished Rhombicosidodecahedron

Author

Eric W. Weisstein
September 6, 2023

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

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

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

Polyhedron

PolyhedronData

PolyhedronData["BigyrateDiminishedRhombicosidodecahedron"]

PolyhedronData["BigyrateDiminishedRhombicosidodecahedron","VertexCoordinates"]

PolyhedronData (V12.1)

PolyhedronData["BigyrateDiminishedRhombicosidodecahedron"]

PolyhedronData["BigyrateDiminishedRhombicosidodecahedron","VertexCoordinates"]

Netlib

PolyhedronName[JohnsonSolid[79]]

Show[Graphics3D[Polyhedron[JohnsonSolid[79]]],Boxed->False]

Net

PolyhedronData

PolyhedronData["BigyrateDiminishedRhombicosidodecahedron","Net"]

PolyhedronData["BigyrateDiminishedRhombicosidodecahedron","Net","Colored"]

Netlib

Show[Graphics[Net[JohnsonSolid[79]]],AspectRatio->Automatic]

Properties

Initialization

<<MathWorld`Polyhedra`

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

vol=PolyhedronData[pname,"Volume"]

AdjacentFaceIndices

AdjacentFaceIndices[p]

Centroid

PolyhedronData["BigyrateDiminishedRhombicosidodecahedron","Centroid"]

RegionCentroid[p//N]

PolyhedronCentroid[p//N,vol]

(xbar1=RegionCentroid[p])//N//Timing

ByteCount[xbar1]

(xbar2=PolyhedronCentroid[p,1,Identity,Identity])/vol//N//Timing

ByteCount[xbar2]

(xbar3=PolyhedronCentroid[p,vol])//N//Timing

ByteCount[xbar3]

FullSimplify[xbar3]

N[%]

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

dih0=PolyhedronData["BigyrateDiminishedRhombicosidodecahedron","DihedralAngles"]

ByteCount[dih=SimplifyNumericallyDistinct[DihedralAngles[p],Debug->True]]//Timing

SortBy[dih,N]-SortBy[dih0,N]//Chop//Union

dih

GeneralizedDiameter

PolyhedronData["BigyrateDiminishedRhombicosidodecahedron","GeneralizedDiameter"]

GeneralizedDiameter[p]//FullSimplify

N[%]

InertiaTensor

MomentOfInertia[p//N,{0,0,0}]/vol//Chop//Timing

PolyhedronInertiaTensor[p//N,vol,Identity,Identity]//Chop//Timing

ByteCount[i1=PolyhedronInertiaTensor[p,vol,Identity,Identity]]//Timing

N[i1]//Chop

ByteCount[i2=PolyhedronInertiaTensor[p,vol]]//Timing

N[i2]//Chop

FullSimplify[i2]

N[%]

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]

MeanCylindricalSquareRadius

((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"]))//FullSimplify)//Timing

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]

MeanSphericalSquareRadius

((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"]))//FullSimplify)//Timing

N[i]

RegionFunction

reg=PolyhedronData["BigyrateDiminishedRhombicosidodecahedron","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

SimplePolyhedron

SimplePolyhedronQ[p]

SkeletonCoordinates

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

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

r=RecognizeGraph[g]

GraphData[r,"Graph","All"]

ToCommonEdges[GraphData[r],g]

SurfaceArea

PolyhedronData[pname,"SurfaceArea"]

FullSimplify[%]

RootReduce[%]

FullSimplify[%]

(s1=SurfaceArea[p]//ToAlgebraicRoot)//Timing

(s2=Total[Area/@PolyhedronData[pname,"Polygons"]]//ToAlgebraicRoot)//Timing

VertexSubsetHulls

PolyhedronData[pname,"VertexSubsetHulls"]

Platonic

(h=VertexSubsetHulls[p,"Platonic","Skip"->{}])//Timing

Volume

vol=PolyhedronData["BigyrateDiminishedRhombicosidodecahedron","Volume"]

N[vol]

Volume[p//N]

PolyhedronVolume[p//N]

(v1=Volume[p])//N//Timing

(v2=PolyhedronVolume[p])//N//Timing

(v3=PolyhedronVolume[p,Method->"GaussLaw"])//N//Timing

ByteCount/@{v1,v2,v3}

(vol=FullSimplify[v1])//Timing

N[vol]

Construction

Construct by rotating two pentagons about their central axes

PolyhedronData/@{"DiminishedRhombicosidodecahedron","BigyrateDiminishedRhombicosidodecahedron"}

f=PolyhedronData["DiminishedRhombicosidodecahedron","FaceIndices"]

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

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

Graphics3D[{PolyhedronData["DiminishedRhombicosidodecahedron","GraphicsComplex"],MapIndexed[Text[#2[[1]],#]&,Mean[v[[#]]]&/@f],
ClosedLine[v[[f[[#]]]]]&/@{47,48}
}]

pentagons=f[[{47,48}]]

fnew=Select[PolyhedronData["DiminishedRhombicosidodecahedron","FaceIndices"],MatchQ[Length[Intersection[Flatten[pentagons],#]],0|5]&]

g1=Mean[v[[f[[47]]]]]//FullSimplify

rot1=RotationMatrix[π/5,g1].#&/@v[[f[[47]]]]//FullSimplify;

g2=Mean[v[[f[[48]]]]]//FullSimplify

rot2=RotationMatrix[π/5,g2].#&/@v[[f[[48]]]]//FullSimplify;

vnew=ReplacePart[v,Join[Thread[pentagons[[1]]->rot1],Thread[pentagons[[2]]->rot2]]]

Length[vnew]

Length[fnew]

PolyhedronData["DiminishedRhombicosidodecahedron",{"V","F"}]

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

faddl={{13,1,3},{13,3,11,14},{14,11,41},{14,41,45,53},{53,45,49},{53,49,47,55},{55,47,43},{55,43,39,52},{52,39,9},{52,9,1,13},{26,35,20},{26,20,32,30},{30,32,31},{30,31,19,29},{29,19,34},{29,34,21,25},{25,21,5},{25,5,6,33},{33,6,22},{33,22,35,26}};

f2=Sort[SmallestRotation/@Join[fnew,faddl]]

Graphics3D[{p=Polyhedron[vnew,f2],MapIndexed[Text[#2[[1]],#,Background->White]&,vnew]}]

ConvexPolyhedronQ[p]

SimplePolyhedronQ[p]