Golden great rhombicosidodecahedra​
​By Sándor Kabai
Great rhombicosidodecahedron consists (GRD) of regular square, hexagonal and decagonal faces . If the sides of the faces relate to one another in proportion of the golden ratio, then such GRD is considered as golden GRD. Three different kinds of golden GRDs are shown here as associated with cluster of 30 cubes and cluster of 60 cubes.
1. Envelope of 30 cubes on small rhombicosidodecahedron (SRD).
2. Common edges with the cluster of 60 cubes.
3. Envelope of 60 cubes.
In[]:=
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;​​z=el+1;​​tx=1.2139;​​r=Sqrt[0.25+tx^2];​​dis=Sqrt[r^2-(1/fi/2)^2];​​srd=GraphicsComplex[PolyhedronData["SmallRhombicosidodecahedron","VertexCoordinates"],Polygon[PolyhedronData["SmallRhombicosidodecahedron","FaceIndices"]]];​​srdn={Rotate[Rotate[srd,bet,{0,1,0}],Pi,{0,0,1}]};​​cub=GraphicsComplex[PolyhedronData["Cube","VertexCoordinates"],Polygon[PolyhedronData["Cube","FaceIndices"]]];​​dod=GraphicsComplex[PolyhedronData["Dodecahedron","VertexCoordinates"],Polygon[PolyhedronData["Dodecahedron","FaceIndices"]]];
In[]:=
Graphics3D[srdn,BoxedFalse,SphericalRegionTrue]
Out[]=

cub30 solid

c1=Translate[cub,{0,0,-0.5-el}];​​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}];​​cub30=Table[Rotate[Rotate[{c1,c2,c3,c4},bet,{1,0,0}],i2Pi/5-Pi/2,{0,0,1}],{i,5}];​​Graphics3D[{cub30,Green,srdn},BoxedFalse,SphericalRegionTrue]
Out[]=

cube 30 perf

vA={0.5,-0.5,0};​​vB={0.5,0.5,0};​​vC={-0.5,0.5,0};​​vD={-0.5,-0.5,0};​​sq1={vA,vB,vC,vD};​​sq2=0.4{vA,vB,vC,vD};​​sq=Polygon[Join[sq1sq2]];​​cu1=Table[Rotate[Translate[sq,{0,0,-0.5}],i2Pi/4,{1,0,0}],{i,3}];​​cu2=Table[Rotate[Translate[sq,{0,0,-0.5}],i2Pi/4,{0,1,0}],{i,3}];​​cu=Scale[{cu1,cu2},0.9999];​​c1=Translate[cu,{0,0,0.5+el}];​​c2=Rotate[c1,Pi,{1,0,0}];​​c3=Rotate[Rotate[{c1,c2},Pi/2,{1,0,0}],Pi/2,{0,0,1}];​​c4=Rotate[Rotate[{c1,c2},Pi/2,{0,1,0}],Pi/2,{0,0,1}];​​cub30p=Table[Rotate[Rotate[{c1,c2,c3,c4},bet,{1,0,0}],i2Pi/5-Pi/2,{0,0,1}],{i,5}];​​Graphics3D[{srdn,Yellow,cub30p},BoxedFalse,SphericalRegionTrue]
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cub60

cc1=Translate[{cub},{0,0,-0.5-el}];​​cc5=Table[Rotate[Rotate[{cc1},-bet,{0,1,0}],i2Pi/5,{0,0,1}],{i,5}];​​cc6=Translate[Rotate[{cc5},0,{1,0,0}],{0,0,5.5675}];​​cc25={Table[Rotate[Rotate[Rotate[cc6,Pi,{0,0,1}],-2bet,{0,1,0}],i2Pi/5,{0,0,1}],{i,5}]};​​cc30=Rotate[{cc6,cc25},Pi,{0,1,0}];​​cub60={cc25,cc30,Red,cc6};​​Graphics3D[cub60,BoxedFalse,SphericalRegionTrue]​​
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Cubes in the cluster of 60 cubes seat on the faces of dodecahedron.
In[]:=
dod=GraphicsComplex[PolyhedronData["Dodecahedron","VertexCoordinates"],Polygon[PolyhedronData["Dodecahedron","FaceIndices"]]];​​dodn=Rotate[Scale[dod,2.4],Pi,{0,0,1}];​​Graphics3D[{cub60,Green,dodn},BoxedFalse,SphericalRegionTrue]
Out[]=
The center of cubes in the cluster of 60 cubes coincide with the vertices of a SRD.

grd

The frame of GRD seems to support the cluster of 60 cubes.

GRD golden 1
envelope of cub30

Enlvelope of cub30

GRD golden 2
support of cub60

Internal edges of cub60 seat on the edges of golden GRD 2.

GRD golden 3 out
framing cub60

GRD golden 3 out perf

Outside edges of cub60 seat on the edges of golden GRD 3.