Cluster of 60 cubes related to great rhombicosidodecahedron​
​by Sándor Kabai
A cluster of 60 cubes is associated with SmallRhombicosidodecahedron (SRD), and GreatRhombicosidodecahedron (GRD).
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"]]];

cub30 & cub60

cub30
c1=Translate[cub,{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}];​​cub30=Table[Rotate[Rotate[{c1,c2,c3,c4},bet,{1,0,0}],i2Pi/5-Pi/2,{0,0,1}],{i,5}];
cub60
u5=Table[Rotate[Rotate[{c1},-bet,{0,1,0}],i2Pi/5,{0,0,1}],{i,5}];​​u6=Translate[Rotate[{u5},0,{1,0,0}],{0,0,5.5675}];​​u25=Table[Rotate[Rotate[Rotate[u6,Pi,{0,0,1}],-2bet,{0,1,0}],i2Pi/5,{0,0,1}],{i,5}];​​u30=Rotate[{Red,u6,Green,u25},Pi,{0,1,0}];​​cub60={u30,Yellow,u25,Red,u6};​​Graphics3D[{cub60},BoxedFalse,SphericalRegionTrue]
Out[]=

srd

srd=GraphicsComplex[PolyhedronData["SmallRhombicosidodecahedron","VertexCoordinates"],Polygon[PolyhedronData["SmallRhombicosidodecahedron","FaceIndices"]]];​​srdt=GraphicsComplex[PolyhedronData["SmallRhombicosidodecahedron","VertexCoordinates"],Tube[PolyhedronData["SmallRhombicosidodecahedron","FaceIndices"]]];​​srda=Rotate[Rotate[Scale[srd,fi],bet,{0,1,0}],Pi,{0,0,1}];​​srdta=Rotate[Rotate[Scale[srdt,fi],bet,{0,1,0}],Pi,{0,0,1}];​​sp1=Sphere[{0,0,0},0.15];​​srdv=srd[[1]];​​srdsp=Rotate[Rotate[Map[Translate[sp1,#]&,fisrdv],bet,{0,1,0}],Pi,{0,0,1}];​​Graphics3D[{{Red,srdsp},srda,Yellow,srdta,u30,Opacity[0.3],u6},BoxedFalse,SphericalRegionTrue]
Out[]=
In[]:=
Graphics3D[{{Red,srdsp},srda,Yellow,srdta,u30,Opacity[0.6],u25,u6},BoxedFalse,SphericalRegionTrue]
Out[]=
The centers of the cubes coincided with the vertices of a small rhombicosidodecahedron (SRD)

grd

grd=GraphicsComplex[PolyhedronData["GreatRhombicosidodecahedron","VertexCoordinates"],Polygon[PolyhedronData["GreatRhombicosidodecahedron","FaceIndices"]]];​​grda=Rotate[Rotate[Scale[grd,1.11],bet,{0,1,0}],Pi,{0,0,1}];​​grdt=Rotate[Rotate[Scale[grd/.PolygonTube,1.11],bet,{0,1,0}],Pi,{0,0,1}];​​Graphics3D[{grda,grdt,Yellow,u25,u30,Red,u6},BoxedFalse,SphericalRegionTrue]​​
Out[]=
A great rhombicosidodecahedron (GRT) frames cub60.
Graphics3D[{grdt,Yellow,u25,u30,Red,u6},BoxedFalse,SphericalRegionTrue]
Out[]=

grdg, with overlapping units "rup"

Vertices and edges of cubes coincide with those of a golden great rhombicosidodecahedron (GRTG.)

A smaller grdg now without overlapping

Golden decagon 12
Small square
Golden hexagon 20
Internal vertices and edges of cubes coincide with those of another golden great rhombicosidodecahedron (GRTG.)