The squares on the faces of a rhombic hexecontahedron (RH) and a median rhombic triacontahedron (MRT) are actually the faces of a cluster of 60 cubes . This can be shown when the size of RH and MR is reduced . The cubes seat on the smaller RH and MRT .
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Manipulate[Module[{fi,alf,bet,y,x,el,cub,rh,c1,u5,u6,u25,u30,tA,va,vd,vA,p1,p2,p3,p4,p5,mrt,mrt60,rh60},fi=1.618;alf=ArcTan[fi];bet=ArcTan[1/fi];y=0.5+0.5fi;x=y/fi;el=yfi;cub=GraphicsComplex[PolyhedronData["Cube","VertexCoordinates"],Polygon[PolyhedronData["Cube","FaceIndices"]]];rh={RGBColor[1,0.8,0.4],GraphicsComplex[PolyhedronData["RhombicHexecontahedron","VertexCoordinates"],Polygon[PolyhedronData["RhombicHexecontahedron","FaceIndices"]]]};c1=Translate[cub,{0,0,-0.5-el}];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[{u6,u25},Pi,{0,1,0}];tA={fi^2,0,fi};va={1,1/fi,fi};vd={1,-1/fi,fi};vA={1/fi,0,fi};p1=Polygon[{tA,va,vA,vd}];p2=Rotate[p1,Pi,{0,0,1}];p3=Rotate[{p1,p2},Pi,{0,1,0}];p4=Rotate[Rotate[{p1,p2,p3},Pi/2,{0,1,0}],Pi/2,{0,0,1}];p5=Rotate[Rotate[{p1,p2,p3},Pi/2,{1,0,0}],Pi/2,{0,0,1}];mrt=Table[Rotate[Rotate[{p1,p2,p3,p4,p5},-bet,{1,0,0}],i2Pi/5,{0,0,1}],{i,5}];mrt60=Rotate[Scale[mrt,rmrt2.116],Pi/2,{0,0,1}];rh60=Scale[rh,rrh3.05];Graphics3D[{If[mrt60s,mrt60,{}],If[rh60s,rh60,{}],Green,If[u30s,u30,{}],RGBColor[0.4,1,1],u25,Red,u6},SphericalRegionTrue,BoxedFalse,ViewPoint->{0,100,50},ViewAngle0.009,ImageSize300]],{{rmrt,1,"reduce MRT"},1,0.7},{{mrt60s,True,"show MRT"},{True,False}},{{rh60s,False,"show RH"},{True,False},ImageSize->Tiny,ControlPlacementLeft},{{rrh,1,"reduce RH"},1,0.618,ImageSize->Tiny,ControlPlacementLeft},{{u30s,True,"30 cubes"},{True,False},ImageSize->Tiny,ControlPlacementLeft},TrackedSymbols{rmrt,mrt60s,rh60s,rrh,u30s},SaveDefinitionsTrue]
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