Sándor Kabai
​Row of icosahedra along the edges of rhombic dodecahedron (RD)​
The icosahedra are connected with their faces. Eight icosahedra having the same orientation are at the vertices of a cube.
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Manipulate​​fi=GoldenRatio;alf=ArcTan[fi];bet=ArcTan[1/fi];bet1=ArcTan[1/fi^2];​​e=N
5
50-10
5
;​​m=eCos[alf-bet1];​​gam=ArcTan[Sqrt[2]];​​rat=N[2Cos[gam]];​​rde=n2m+2m;​​cube=rderat;​​rd=PolyhedronData["RhombicDodecahedron","Faces"];​​cub=PolyhedronData["Cube","Faces"];​​cubv=PolyhedronData["Cube","VertexCoordinates"];​​cubin=Scale[cub,cube];​​ico=PolyhedronData["Icosahedron","Faces"];​​icof=Rotate[ico,alf+bet1,{0,1,0}];​​icoup=Table[Rotate[Translate[icof,i{0,0,2m}],iPi,{0,0,1}],{i,n}];​​icoe=Rotate[Rotate[ico,-bet,{0,1,0}],Pi/2,{0,0,1}];​​cub8={Map[Translate[icoe,#]&,cubecubv]};​​​​v1=Rotate[{Yellow,icoup},-alf-bet1,{0,1,0}];​​v2=Rotate[v1,-2Pi/5,{0,0,1}];​​v3=Rotate[Rotate[{{Red,ico},v1,v2},2Pi/10,{0,0,1}],-alf,{0,1,0}];​​v4=Rotate[v3,Pi,{0,0,1}];​​v5={v3,v4};​​v6=Translate[v5,{0,0,cube}];​​v7=Table[Rotate[Rotate[v6,Pi/2,{0,0,1}],i2Pi/4+0.001,{0,1,0}],{i,3}];​​v8=Rotate[Rotate[v6,Pi/2,{0,0,1}],2Pi/4+0.001,{1,0,0}];​​v9=Rotate[Rotate[v6,Pi/2,{0,0,1}],-2Pi/4+0.001,{1,0,0}];​​rdn=Rotate[Scale[rd,0.9rde],Pi/4,{0,0,1}];​​Graphics3D[{{Green,rdn},cubin,{Cyan,cub8},v6,v7,v8,v9},SphericalRegionTrue,BoxedFalse,ViewPoint{2,-4,1},ImageSize300],​​{{n,3,"Frequency"},1,9,2,RadioButton},​​TrackedSymbols{n}
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Frequency
1
3
5
7
9
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Graphics3D[{cubin,{Cyan,cub8},v6},SphericalRegionTrue,BoxedFalse,ViewPoint{2,-4,1},ImageSize400]
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Graphics3D[{{Green,rdn},cubin,{Cyan,cub8},v6},SphericalRegionTrue,BoxedFalse,ViewPoint{2,-4,1},ImageSize400]
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Graphics3D[{cubin,{Cyan,cub8},v6,v7},SphericalRegionTrue,BoxedFalse,ViewPoint{2,-4,1},ImageSize400]
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