Great Rhombicosidodecahedron

Author

Eric W. Weisstein
December 27, 2009

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

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

©2009 Wolfram Research, Inc. except for portions noted otherwise

Initialization

poly=Polyhedron[GreatRhombicosidodecahedron];

Net

Show[Graphics[Net[GreatRhombicosidodecahedron]],AspectRatio->Automatic]

Show[Graphics[Net[GreatRhombicosidodecahedron,Colored->True]],AspectRatio->Automatic]

NetPrintout[GreatRhombicosidodecahedron,"GreatRhombicosidodecahedron.pdf"]

Graphic

Show[Graphics3D[poly],Boxed->False]

LiveGraphics3D

<<Utilities`LiveGraphics3D`

WriteLiveForm["a02.m",Show[Graphics3D[poly]//N,opts]]

Solid and Dual

Show[GraphicsArray[Graphics3D[#,Boxed->False]&/@{
poly,
dual=Polyhedron[PolyhedronDual[GreatRhombicosidodecahedron]],{dual,poly}
}]]

Radii

<<Utilities`Simplify`

r

{#,N[#,10]}&@(r=SqrtSimplify@FullSimplify[Inradius[dual]])

ρ=(Subscript[ρ, d])

{#,N[#,10]}&@(ρR=Together//@FullSimplify[Midradius[poly]])

{#,N[#,10]}&@(ρr=Together//@FullSimplify[Midradius[dual]])

R

{#,N[#,10]}&@(R=SqrtSimplify[Together//@FullSimplify[Circumradius[poly]]])

Check

FullSimplify[Subtract@@@({R r,#}&/@{((ρr)^2),((ρR)^2)})]

Faces

PolyhedronFaceList[poly]

Show[GraphicsArray[partition[PolyhedronFaces[poly//N],15]]]

Faces

PolyhedronFaceList[poly]

Show[GraphicsArray[partition[PolyhedronFaces[poly],15]]]

Edge Lengths

n=Norm[poly[[1,1,1]]-poly[[1,1,2]]]

RootReduce[n]

<<NumberTheory`Recognize`

Union[PolyhedronEdgeLengths[poly//N]]

RootReduce[PolyhedronEdgeLengths[poly]]

Midradius

ρ=RootReduce[Norm[(poly[[1,1,1]]+poly[[1,1,2]])/2]]

NRoots[-1+20 (x^2)-112 (x^4)+64 (x^6)==0,x]

N[%]

Circumradius

R=RootReduce[Norm[poly[[1,1,1]]]]

N[%]

Inradius

RootReduce[ρ^2/R]

N[%]

Triangular Faces

RootReduce[Norm[Centroid[Select[poly,Length[#[[1]]]==3&][[1,1]]][[1]]]]

N[%,20]

Pentagonal Faces

RootReduce[Norm[Centroid[Select[poly,Length[#[[1]]]==4&][[1,1]]][[1]]]]

N[%,20]

Dihedral Angles

Show[Graphics3D[{poly,
MapIndexed[Text[#2[[1]],#1]&,Centroid/@poly//N]
}],Boxed->False]

{4,6}

Show[Graphics3D[{poly[[{17,3}]]}],Boxed->False]

FullSimplify[π-DihedralAngle@@poly[[{17,3}]]]

N[%]180/Pi

{4,10}

Show[Graphics3D[{poly[[{17,12}]]}],Boxed->False]

FullSimplify[π-DihedralAngle@@poly[[{17,12}]]]

N[%]180/Pi

{6,10}

Show[Graphics3D[{p=poly[[{3,12}]]}],Boxed->False]

FullSimplify[π-DihedralAngle@@p]

InputForm[%]

N[%]180/Pi

Surface Area and Volume

RootReduce[{PolyhedronSymmetricSurfaceArea[poly],PolyhedronSymmetricVolume[poly]}]//FullSimplify//Timing

N[Last[%]]

Surface Area

FullSimplify[FunctionExpand[PolyhedronSymmetricSurfaceArea[poly]]]

Volume

FullSimplify[FunctionExpand[PolyhedronSymmetricVolume[poly]]]

RootReduce[{PolyhedronSurfaceArea[poly//N],PolyhedronVolume[poly//N]}]

poly=FunctionExpand[Graphics3D[GreatRhombicosidodecahedron][[1]]];
ToRadicals[RootReduce[{PolyhedronSurfaceArea[poly],PolyhedronVolume[poly]}]]

v6

With[{p="GreatRhombicosidodecahedron"},
PolyhedronData[p,"Volume"]/PolyhedronData[p,"EdgeLengths"][[1]]^3
]//FullSimplify

NIntegrate[Boole[PolyhedronData["GreatRhombicosidodecahedron","RegionFunction"][x,y,z]],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}]//Timing

Integrate[Boole[PolyhedronData["GreatRhombicosidodecahedron","RegionFunction"][x,y,z]],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}]//FullSimplify//Timing

...crash...

Projections

WriteLiveForm["a02fram1.m",
Show[Graphics3D[Wireframe[poly]],Boxed->False,ViewPoint->1000{-1,-3.3,2.86},
ViewVertical->{1,6.27,7.58}
]]

WriteLiveForm["a02fram2.m",
Show[Graphics3D[Wireframe[poly]],Boxed->False,ViewPoint->1000{1,-3.45,7.6},ViewVertical->{0,1,0}
]]

WriteLiveForm["a02fram3.m",
Show[Graphics3D[Wireframe[poly]],Boxed->False,ViewPoint->1000{-1,-1.03,1.5},
ViewVertical->{1,3.97,3.38}
]]

WriteLiveForm["a02fram4.m",
Show[Graphics3D[Wireframe[poly]],Boxed->False,ViewPoint->1000{-1.38,-1.01,2.7},
ViewVertical->{3.24,-1.69,1}
]]

PolyhedronEdgeLengths[poly]//Short

Graph

skel=Skeleton[UniformPolyhedron[28],1.1];

skel=Skeleton[poly];

HamiltonianQ[skel]

ShowGraph[skel]

DegreeSequence[skel]

SimpleGraphProperties[skel]

Edge Lengths

PolyhedronEdgeLengths[poly]

Length[poly]

Split[Sort[poly/.Polygon:>Length]]

PolyhedronEdges[poly]//Length

Radii

{#,N[#]}&/@(ρ=FullSimplify[Midradius[poly]])

{#,N[#]}&/@(R=FullSimplify[Circumradius[poly]])

{#,N[#]}&/@(r=FullSimplify[ρ^2/R])

Vertex Subset Hull

p="GreatRhombicosidodecahedron";
v=PolyhedronData[p,"VertexCoordinates"];

Platonic

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

Exact Construction

GreatRhombicosidodecahedron:=MiscPolyhedra`UniformPolyhedron[28]

GreatRhombicosidodecahedronExact:=Zonohedron[{{1,GoldenRatio,GoldenRatio-1},{1,-GoldenRatio,GoldenRatio-1},{1,-GoldenRatio,1-GoldenRatio},{1,GoldenRatio,1-GoldenRatio},{GoldenRatio,1-GoldenRatio,1},{GoldenRatio,1-GoldenRatio,-1},{GoldenRatio,GoldenRatio-1,-1},{GoldenRatio,GoldenRatio-1,1},{GoldenRatio-1,1,GoldenRatio},{GoldenRatio-1,-1,-GoldenRatio},{GoldenRatio-1,1,-GoldenRatio},{GoldenRatio-1,-1,GoldenRatio},{2,0,0},{0,2,0},{0,0,2}}/2]/.Polygon[l_List]:>Polygon[-{1/2+2GoldenRatio,1/2,1/2}+#&/@l]

PolyhedronData

PolyhedronDataString[{Flatten[Normal[PolyhedronData["GreatRhombicosidodecahedron","Faces"]]],"GreatRhombicosidodecahedron","great rhombicosidodecahedron"},{"Archimedean","Convex"}]