Ten icosahedra are connected with their edges in a ring. One edge of each icosahedron is on the edge of a larger central icosahedron. The central icosahedron is truncated until the edges are equal. When the size of icosahedra in the ring is reduced by the golden ratio, then their edges are on a still larger central icosahedron, which can be truncated until their edges have the same length. In this way two kinds of golden truncated icosahedra are obtained with golden hexagons instead of regular hexagons. The difference is that the shorter edge or the longer edge is connected to the pentagon. If enveloping tetrahedra are placed on the ring of ten small icosahedra, then the vertices of tetrahedra coincide with the vertices of the golden truncated icosahedra 2.
In[]:=
fi=1.618;alf=ArcTan[fi];bet=ArcTan[1/fi];u=fi/2;s=u/Tan[2Pi/20];tet=GraphicsComplex[PolyhedronData["Tetrahedron","VertexCoordinates"],Polygon[PolyhedronData["Tetrahedron","FaceIndices"]]];tetring=Rotate[Rotate[Scale[tet,Sqrt[2]{1,1,1},{0,0,0}],-ArcTan[Sqrt[2]],{0,1,0}],Pi/4,{0,0,1}];t0={Rotate[Rotate[Scale[{tetring},1.618{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={Green,Table[Rotate[Translate[{ta,tb},{0,s,0}],i2Pi/5-Pi/10,{0,0,1}],{i,5}]};ico=GraphicsComplex[PolyhedronData["Icosahedron","VertexCoordinates"],Polygon[PolyhedronData["Icosahedron","FaceIndices"]]];Manipulate[Module[{ico0,ica,icb,ico10,z1,ti1,ti1n,ti1v,sp1,ti1sp,z2,ti2,ti2n,ti2v,sp2,ti2sp},ic0=Rotate[Scale[ico,sm/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}]};z1=1/(2+fi);ti1=TruncatedPolyhedron[Icosahedron[],tr1z1];ti1n=Rotate[Scale[ti1,2.236],Pi,{0,0,1}];ti1v=ti1[[1]];sp1={RGBColor[1,0.,0.6],Sphere[{0,0,0},0.08]};ti1sp=Rotate[Map[Translate[sp1,#]&,2.236ti1v],Pi,{0,0,1}];z2=1/fi^2;ti2=TruncatedPolyhedron[Icosahedron[],tr2z2];ti2n={RGBColor[0.8,0.9,1],Rotate[Scale[ti2,fi^2],Pi,{0,0,1}]};ti2v=ti2[[1]];sp2={Green,Sphere[{0,0,0},0.1]};ti2sp=Rotate[Map[Translate[sp2,#]&,2.618ti2v],Pi,{0,0,1}];Graphics3D[{ti1sp,ti2sp,If[tet10s,tet10,{}],If[ti2ns,ti2n,{}],ico10,ti1n},BoxedFalse,SphericalRegionTrue,ViewPoint{0,0,100},ViewAngle0.01,PlotRange3.6,ImageSize400]],{{tr1,0.001,"truncate icosahedon 1"},0.001,1},{{sm,fi,"reduce 10 icosahedron"},fi,1.01},{{ti2ns,False,"ico 2"},{True,False},ImageSize->Tiny,ControlPlacementLeft},{{tr2,0.001,"tr ico2"},0.001,1,ImageSize->Tiny,ControlPlacementLeft},{{tet10s,False,"10 tet"},{True,False},ImageSize->Tiny,ControlPlacementLeft},TrackedSymbols{tr1,tr2,sm,ti2ns,tet10s},SaveDefinitionsTrue]
Out[]=
