Sándor Kabai
Golden pair of rhombic triacontahedron (RT) and rhombic hexecontahedron (RH)
RT and RH are connected with five of their faces. RT has 30, while RH has 60 golden rhombus faces. The distance between the centers of RT and RH is golden ratio squared. RT has a smaller RH in it, and RH has a smaller RT in it. The size ratio of the larger to small RT and RH equals to the golden ratio. RT can be built from 10 acute golden rhombohedra (AR) and 10 obtuse golden rhombohedra (OR), while RH consists of 20 AR.
Golden pair of rhombic triacontahedron (RT) and rhombic hexecontahedron (RH)
RT and RH are connected with five of their faces. RT has 30, while RH has 60 golden rhombus faces. The distance between the centers of RT and RH is golden ratio squared. RT has a smaller RH in it, and RH has a smaller RT in it. The size ratio of the larger to small RT and RH equals to the golden ratio. RT can be built from 10 acute golden rhombohedra (AR) and 10 obtuse golden rhombohedra (OR), while RH consists of 20 AR.
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rh=PolyhedronData["RhombicHexecontahedron","Faces"];rhn=Scale[rh,0.999];(*Degreeofperforationcanbesethere*)perf=0.76;fi=GoldenRatio;alf=ArcTan[fi];bet=ArcTan[1/fi];bet1=ArcTan[1/fi^2];e=1/Sin[alf];vA={0,1,0};vB={1/fi,0,0};vC={0,-1,0};vD={-1/fi,0,0};ss1={vA,vB,vC,vD,vA};ss2=perf{vA,vB,vC,vD,vA};rr1=Polygon[Join[ss1->ss2]];rh1=Translate[rr1,{0,fi,1}];rh2=Translate[rr1,{0,-fi,1}];rh3=Translate[{rh1,rh2},{0,0,-2}];rh4=Rotate[Rotate[{rh1,rh2,rh3},Pi/2,{1,0,0}],Pi/2,{0,1,0}];rh5=Rotate[Rotate[{rh1,rh2,rh3},Pi/2,{0,1,0}],Pi/2,{1,0,0}];rhp=Table[Rotate[Rotate[{rh1,rh2,rh3,rh4,rh5},bet,{1,0,0}],i2Pi/5-Pi/2,{0,0,1}],{i,5}];rhp0=Scale[rhp,1/e];Graphics3D[{Green,rhn,Yellow,rhp0},Boxed->False,SphericalRegionTrue,ImageSize200]
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rt=PolyhedronData["RhombicTriacontahedron","Faces"];rtn=Scale[Rotate[rt,0Pi,{0,0,1}],0.999];rt1=Translate[rr1,{0,0,fi}];rt2=Translate[rr1,{0,0,-fi}];rt3=Rotate[Rotate[{rt1,rt2},Pi/2,{1,0,0}],Pi/2,{0,1,0}];rt4=Rotate[Rotate[{rt1,rt2},Pi/2,{0,1,0}],Pi/2,{1,0,0}];rtp=Table[Rotate[Rotate[{rt1,rt2,rt3,rt4},bet,{1,0,0}],i2Pi/5-Pi/2,{0,0,1}],{i,5}];rhp0=Scale[rhp,1/e];sp=Sphere[];rtp0=Scale[rtp,1/e];rtup=Translate[rt,{0,0,fi^2}];rtsm=Scale[rt,1/fi];rtp0up=Translate[rtp0,{0,0,fi^2}];rhsm=Scale[rh,1/fi];rhsmup=Translate[rhsm,{0,0,fi^2}];Graphics3D[{sp,Cyan,rtp0,Green,rtn},Boxed->False,SphericalRegionTrue,ImageSize200]
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a1={Cyan,rtup,Yellow,rh};Graphics3D[{a1},Boxed->False,SphericalRegionTrue,ViewPoint{100,100,0},ImageSize300]
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a2={Yellow,rhsmup,{Cyan,rtsm},rhp0,Cyan,rtp0up};Graphics3D[{a2},Boxed->False,SphericalRegionTrue,ViewPoint{10,7,6},ImageSize300]
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a3={Cyan,rtsm,Yellow,rhp0,Cyan,rtup};Graphics3D[{a3},Boxed->False,SphericalRegionTrue,ViewPoint{10,7,6},ImageSize300]
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a4={Yellow,rh,rhsmup,Cyan,rtp0up};Graphics3D[{a4},Boxed->False,SphericalRegionTrue,ViewPoint{10,7,6},ImageSize300]
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a2={Yellow,rhsmup,{Cyan,rtsm},rhp0};Graphics3D[{a2},Boxed->False,SphericalRegionTrue,ViewPoint{10,7,6},ImageSize300]
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