Rhombic triacntahedra on surface of icosahedra​
​By Sender Kabai
Rhombic triacontahedra are placed on five faces of an icosahedron. The top view of the image corresponds to a Penrose tiling and the bottom view depicts another kind of Penrose tiling. By varying the frequency any size can be accomplished.
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fi=1.618;​​alf=ArcTan[fi];​​bet=ArcTan[1/fi];bet1=ArcTan[1/fi^2];y=Sin[alf];rtsize=2/2/fi/y;​​rh=GraphicsComplex[PolyhedronData["RhombicHexecontahedron","VertexCoordinates"],Polygon[PolyhedronData["RhombicHexecontahedron","FaceIndices"]]];​​rt=GraphicsComplex[PolyhedronData["RhombicTriacontahedron","VertexCoordinates"],Polygon[PolyhedronData["RhombicTriacontahedron","FaceIndices"]]];​​rtn={Rotate[Rotate[Scale[rt,rtsize],-bet1-alf,{0,1,0}],Pi/2,{0,0,1}]};​​Manipulate[​​n=m-1;​​p=Table[{2j-i,Sqrt[3]i,0},{i,0,n},{j,i,n}];​​tri1=Map[Translate[rtn,#]&,p];​​p1=Rotate[Rotate[tri1,-Pi/6,{0,0,1}],alf-bet1,{0,1,0}];​​p5=Table[Rotate[p1,i2Pi/5,{0,0,1}],{i,5}];​​​​Graphics3D[{Glow[White],p5},SphericalRegionTrue,BoxedFalse,ViewPoint{0,0,100},ViewAngle0.01],​​{{m,5,"frequency"},1,12,1,RadioButton},​​TrackedSymbols->m,SaveDefinitions->True]
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frequency
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Top view
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Graphics3D[{Glow[White],EdgeForm[Thick],p5},SphericalRegionTrue,BoxedFalse,ViewPoint{0,0,100},ViewAngle0.01]
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Bottom view
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Graphics3D[{Glow[White],EdgeForm[Thick],p5},SphericalRegionTrue,BoxedFalse,ViewPoint{0,0,-100},ViewAngle0.01]
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Bottom view
Notice how a rhombic hexacontahedron is seted into the void
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rhn=Translate[Scale[rh,0.7265],{0,0,-1.96}];​​Graphics3D[{rhn,Glow[White],EdgeForm[Thick],p5},SphericalRegionTrue,BoxedFalse,ViewPoint{0,0,-100},ViewAngle0.01]
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Bird’s eye view
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Graphics3D[{p5,Green,rhn},SphericalRegionTrue,BoxedFalse,ViewPoint{3,3,-5}]
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