30 cubes on great rhombicosidodecahedron ​
​By Sándor Kabai
Structures are shown in relation to the 30 cubes placed on a great rhombicosidodecahedron (grd).
The centres of cubes are at the vertices of an icosidodecahedron (id) having size equivalent to golden ration.
The relationship to rhombic triacontahedron (rt) and small rhombicosidodecahedron (srd) is also shown.
The envelope of the assembly is a golden grd.
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fi=1.618;​​alf=ArcTan[fi];​​bet=ArcTan[1/fi];​​bet1=ArcTan[1/fi^2];​​y=0.5+0.5fi;​​x=y/fi;​​rte=Sqrt[x^2+y^2];​​el=yfi;​​cub=GraphicsComplex[PolyhedronData["Cube","VertexCoordinates"],Polygon[PolyhedronData["Cube","FaceIndices"]]];​​rt=GraphicsComplex[PolyhedronData["RhombicTriacontahedron","VertexCoordinates"],Polygon[PolyhedronData["RhombicTriacontahedron","FaceIndices"]]];​​rtnn=Scale[rt,2.72];​​id=GraphicsComplex[PolyhedronData["Icosidodecahedron","VertexCoordinates"],Polygon[PolyhedronData["Icosidodecahedron","FaceIndices"]]];​​idn=Rotate[id,Pi,{0,0,1}];​​srd=GraphicsComplex[PolyhedronData["SmallRhombicosidodecahedron","VertexCoordinates"],Polygon[PolyhedronData["SmallRhombicosidodecahedron","FaceIndices"]]];​​srdn=Rotate[Rotate[srd,bet,{0,1,0}],Pi,{0,0,1}];​​grd=GraphicsComplex[PolyhedronData["GreatRhombicosidodecahedron","VertexCoordinates"],Polygon[PolyhedronData["GreatRhombicosidodecahedron","FaceIndices"]]];​​dod=GraphicsComplex[PolyhedronData["Dodecahedron","VertexCoordinates"],Polygon[PolyhedronData["Dodecahedron","FaceIndices"]]];​​c1=Translate[cub,{0,0,2el}];​​c2=Rotate[c1,Pi,{1,0,0}];​​c3=Rotate[Rotate[{c1,c2},Pi/2+0.001,{1,0,0}],Pi/2+0.001,{0,0,1}];​​c4=Rotate[Rotate[{c1,c2},Pi/2+0.001,{0,1,0}],Pi/2+0.001,{0,0,1}];​​cub30p=Table[Rotate[Rotate[{c1,c2,c3,c4},bet,{1,0,0}],i2Pi/5-Pi/2,{0,0,1}],{i,5}];​​grdn=Rotate[Rotate[grd,alf,{1,0,0}],Pi/2,{0,0,1}];​​​​p3=0.3;​​grdnfr=Normal[grdn]/.p:Polygon[__](p/.pts:{{_?NumericQ,_,_},__}Module[{cent,innervtx},cent=Mean[pts];​​innervtx={1-p3,p3}.{#,cent}&/@pts;​​Join[pts,pts[[{1}]],innervtx,innervtx[[{1}]]]])//#&;​​​​sp=Sphere[{0,0,0},0.6];​​idv=PolyhedronData["Icosidodecahedron"][[1,1]];​​spid={Map[Translate[sp,#]&,2.618idv]};​​ide=Scale[id,2.618];​​p3=0.1;​​idefr=Normal[ide]/.p:Polygon[__](p/.pts:{{_?NumericQ,_,_},__}Module[{cent,innervtx},cent=Mean[pts];​​innervtx={1-p3,p3}.{#,cent}&/@pts;​​Join[pts,pts[[{1}]],innervtx,innervtx[[{1}]]]])//#&;​​​​Graphics3D[{Red,cub30p,Cyan,grdn},BoxedFalse,SphericalRegionTrue,ImageSize300,ViewPoint{0,0,100},ViewAngle0.01]​​
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Graphics3D[{Red,cub30p,Cyan,grdnfr},BoxedFalse,SphericalRegionTrue,ImageSize300,ViewPoint{0,0,100},ViewAngle0.01]
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Graphics3D[{rtnn,Red,cub30p,Cyan,grdnfr},BoxedFalse,SphericalRegionTrue,ImageSize300,ViewPoint{0,0,100},ViewAngle0.01]
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Graphics3D[{spid,Red,cub30p,Cyan,grdnfr},BoxedFalse,SphericalRegionTrue,ImageSize300,ViewPoint{0,0,100},ViewAngle0.01]
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Graphics3D[{ide,spid,Green,Red,cub30p,Cyan,grdnfr},BoxedFalse,SphericalRegionTrue,ImageSize300,ViewPoint{0,0,100},ViewAngle0.01]
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Graphics3D[{rtnn,Green,cub30p,Red,idefr,Yellow,spid,Cyan,grdnfr},BoxedFalse,SphericalRegionTrue,ImageSize300,ViewPoint{0,0,100},ViewAngle0.01]

Golden GRD elveloping cub30