A series of rings of tetrahedra with embedded icosahedra is mounted on a golden truncated icosahedron. The outside ring is framed with a small rhombicosidodecahedron. Observe where the vertices of tetrahedra are located.
fi=GoldenRatio;bet=ArcTan[1/fi];u=fi/2;s=u/Tan[2Pi/20];sp1=Sphere[{0,0,0},0.04];(*ico*)ico=GraphicsComplex[PolyhedronData["Icosahedron","VertexCoordinates"],Polygon[PolyhedronData["Icosahedron","FaceIndices"]]];ic0=Rotate[Scale[ico,1/fi],-Pi/2,{0,0,1}];ica={Yellow,Translate[Rotate[Scale[ic0,1],-4Pi/5,{0,0,1}],{-u,0,0}]};icb={RGBColor[1,0.6,0.6],Translate[ic0,{u,0,0}]};ico10={Table[Rotate[Translate[{ica,icb},{0,s,0}],i2Pi/5-Pi/10,{0,0,1}],{i,5}]};(*ticog2*)z2=1/fi^2;ti2=TruncatedPolyhedron[Icosahedron[],z2];ti2n=Rotate[Scale[ti2,fi^2],Pi,{0,0,1}];(*tet*)tet=GraphicsComplex[PolyhedronData["Tetrahedron","VertexCoordinates"],Polygon[PolyhedronData["Tetrahedron","FaceIndices"]]];tetsp=Map[Translate[sp1,#]&,tet[[1]]];tetring=Rotate[Rotate[Scale[{tet,Red,tetsp},Sqrt[2]{1,1,1},{0,0,0}],-ArcTan[Sqrt[2]],{0,1,0}],Pi/4,{0,0,1}];t0={Rotate[Rotate[Scale[{tetring},1.6179{1,1,1},{0,0,0}],bet,{1,0,0}],2Pi/5,{0,0,1}]};ta=Translate[Rotate[t0,-4Pi/5,{0,0,1}],{-u,0,0}];tb=Translate[t0,{u,0,0}];tet10={Table[Rotate[Translate[{ta,tb},{0,s,0}],i2Pi/5-Pi/10,{0,0,1}],{i,5}]};tet10ser=Table[{Hue[0.24i],Scale[{tet10,Yellow,ico10},1.9^i{1,1,1},{0,0,0}]},{i,2}];(*srd*)srd=GraphicsComplex[PolyhedronData["SmallRhombicosidodecahedron","VertexCoordinates"],Polygon[PolyhedronData["SmallRhombicosidodecahedron","FaceIndices"]]];srdn=Rotate[Rotate[Scale[srd,5.8],bet,{0,1,0}],Pi,{0,0,1}];srdnfr=Normal[srdn]/.p:Polygon[__](p/.pts:{{_?NumericQ,_,_},__}Module[{cent,innervtx},cent=Mean[pts];innervtx={1-0.1,0.1}.{#,cent}&/@pts;Join[pts,pts[[{1}]],innervtx,innervtx[[{1}]]]])//#&;Graphics3D[Rotate[{srdnfr,ti2n,ico10,tet10ser,Opacity[0.3],Red,tet10},Pi/2,{0,1,0}],BoxedFalse,SphericalRegionTrue,ViewPoint{100,0,0},ViewAngle0.01,ImageSize450]
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Graphics3D[tet10]
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