Icosahedron Towers

Icosahedron Towers - Wolfram Demonstrations Project
Icosahedra are sequentially attached at their vertices. Each icosahedron is smaller than the last in proportion to the golden ratio. Such a tower is placed at each of the 12 vertices of an icosahedron. The assembly fits inside a medial triambic icosahedron, which is one of the stellations of the icosahedron.

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Manipulate[ϕ=GoldenRatio;α=ArcTan[ϕ];α1=ArcTan[ϕ^2];β=ArcTan[1/ϕ];β1=ArcTan[1/ϕ^2];r=N[Sqrt[0.5^2+(ϕ/2)^2]];v7={-((1+Sqrt[5])/(2*Sqrt[10-2*Sqrt[5]])),-(1/2),1/Sqrt[10-2*Sqrt[5]]};v8={-((1+Sqrt[5])/(2*Sqrt[10-2*Sqrt[5]])),1/2,1/Sqrt[10-2*Sqrt[5]]};h=N[Sum[ϕ^2*(r/ϕ^n),{n,1,1000}]];s1=Polygon[{v7,v8,{0,0,0.998*h}}];s5=Table[Rotate[s1,i*2*(Pi/5),{0,0,1}],{i,5}];ico=GraphicsComplex[PolyhedronData["Icosahedron","VertexCoordinates"],Polygon[PolyhedronData["Icosahedron","FaceIndices"]]];icoedge=Scale[GraphicsComplex[PolyhedronData["Icosahedron","VertexCoordinates"],Line[PolyhedronData["Icosahedron","EdgeIndices"]]],Sum[ϕ^2*(r/ϕ^n),{n,1,fr}]*{1,1,1},{0,0,0}];icov=Table[Rotate[Translate[Scale[ico,(1/ϕ^i)*{1,1,1},{0,0,0}],Sum[ϕ^2*(r/ϕ^n),{n,1,i}]*{0,0,1}],0*i*(Pi/5),{0,0,1}],{i,fr}];icot=Table[Rotate[Translate[Scale[ico,(1/ϕ^i)*{1,1,1},{0,0,0}],Sum[ϕ^2*(r/ϕ^n),{n,1,i}]*{0,0,1}],Pi/5,{0,0,1}],{i,fr}];icot5=Table[Rotate[Rotate[{icot},2*β,{0,1,0}],i*2*(Pi/5),{0,0,1}],{i,5}];s55=Table[Rotate[Rotate[Rotate[{s5},Pi,{0,0,1}],2*β,{0,1,0}],i*2*(Pi/5),{0,0,1}],{i,5}];bottom=Rotate[{icov,icot5},Pi,{0,1,0}];cover=Rotate[{s5,s55},Pi,{0,1,0}];Graphics3D[{{Yellow,cover},If[so,{Opacity[op],Green,s55,s1},{}],{Yellow,ico},icov,icot5,bottom},Boxed->False,SphericalRegion->True,ViewPoint->{6,-10,2},ViewAngle->Pi/35,ImageSize->380],{{fr,6,"frequency"},1,8,1,ControlType->RadioButton},{{so,True,"show cover"},{True,False}},{{op,0.5,"opacity"},0,1},TrackedSymbols->Manipulate,ControllerLinking->True]