Ring of ten golden rhombohedra

Sándor Kabai
A ring formed by ten golden rhombohedra together with associated polyhedra is used to create intersecting patterns.
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fi=GoldenRatio;​​alf=ArcTan[fi];​​bet=ArcTan[1/fi];​​dia=N[Sqrt[1+(1+fi)^2]];​​e=N[Sqrt[1+1/fi^2]];​​bet1=ArcTan[1/fi^2];​​rtr=Sqrt[1+fi^2];​​vA={1/fi,1,0};​​vB={0,2,0};​​vC={-fi^(-1),1,0};​​vD={0,0,0};​​vE={1/fi,fi,1};​​vF={0,fi+1,1};​​vG={-fi^(-1),fi,1};​​vH={0,1/fi,1};​​r1=Polygon[{vD,vC,vB,vA}];​​r2=Polygon[{vE,vF,vG,vH}];​​r3=Polygon[{vD,vA,vE,vH}];​​r4=Polygon[{vA,vB,vF,vE}];​​r5=Polygon[{vB,vC,vG,vF}];​​r6=Polygon[{vC,vD,vH,vG}];​​prGR={r1,r2,r3,r4,r5,r6};​​h=dia*(Sin[bet-bet1]/2);​​q=N[1/fi/Tan[2*(Pi/10)]];​​p=N[1/fi/Sin[2*(Pi/10)]];​​side=2*q+p;​​k=side/2/Sin[2*(Pi/20)];​​h1=Rotate[prGR,Pi,{0,1,0}];​​h2=Rotate[h1,bet,{1,0,0}];rr1=Rotate[h2,2*(Pi/5),{0,0,1}];rr2=Rotate[h2,-2*(Pi/5),{0,0,1}];ringp={Table[Rotate[{rr1,rr2},i*2*(Pi/5),{0,0,1},{0,k,0}],{i,5}]};​​ringc=Translate[ringp,{0,-k,0}];​​ringc1=Scale[ringc,fi];​​ringc2=Scale[ringc,fi^2];​​ringc3=Scale[ringc,fi^3];​​Graphics3D[{ringc,ringc1,ringc2,ringc3},Boxed->False,ViewPoint->{0,0,100},ImageSize->500]​​
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rh=Scale[Rotate[GraphicsComplex[PolyhedronData["RhombicHexecontahedron","VertexCoordinates"],Polygon[PolyhedronData["RhombicHexecontahedron","FaceIndices"]]],Pi/2,{0,0,1}],e*{1,1,1}];rha=Table[Rotate[rh,i*2*(Pi/5),{0,0,1},{0,k,0}],{i,5}];rhb=Rotate[rha,Pi,{1,0,0},{0,k,h}];ico=Scale[Rotate[GraphicsComplex[PolyhedronData["Icosahedron","VertexCoordinates"],Polygon[PolyhedronData["Icosahedron","FaceIndices"]]],-Pi/2,{0,0,1}],(2/fi)*{1,1,1}];icoa=Table[Rotate[ico,i*2*(Pi/5),{0,0,1},{0,k,0}],{i,5}];​​icob=Rotate[icoa,Pi,{1,0,0},{0,k,h}];tico=Scale[Rotate[GraphicsComplex[PolyhedronData["TruncatedIcosahedron","VertexCoordinates"],Polygon[PolyhedronData["TruncatedIcosahedron","FaceIndices"]]],-Pi/2,{0,0,1}],(1/fi)*{1,1,1}];ticoa=Table[Rotate[tico,i*2*(Pi/5),{0,0,1},{0,k,0}],{i,5}];​​ticob=Rotate[ticoa,Pi,{1,0,0},{0,k,h}];rt=Translate[Scale[Rotate[GraphicsComplex[PolyhedronData["RhombicTriacontahedron","VertexCoordinates"],Polygon[PolyhedronData["RhombicTriacontahedron","FaceIndices"]]],Pi/2,{0,0,1}],e*{1,1,1}],{0,0,rtr}];rta=Table[Rotate[rt,i*2*(Pi/5),{0,0,1},{0,k,0}],{i,5}];rtb=Rotate[rta,Pi,{1,0,0},{0,k,h}];Graphics3D[{ringp,rha,rhb,rh,icoa,icob,ticoa,ticob,rta,rtb},Boxed->False,ViewPoint->{0,0,10},ImageSize->400]
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Graphics3D[{ringp},Boxed->False,ViewPoint->{0,0,100},ImageSize->400]
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Graphics3D[{Pink,rha,Yellow,rhb},Boxed->False,ViewPoint->{0,0,100},ImageSize->400]
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Graphics3D[{ringp,rh,Yellow,icoa,Cyan,icob,Red,Opacity[0.05],rha},Boxed->False,ViewPoint->{0,0,100},ImageSize->400]
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