Ring of 10, 20 and 40 RH
Sándor Kabai
20 rhombic hexecontahedra (RH) are is arranged in a ring. The centres of RH are on a wavy decagon.
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fi=GoldenRatio;​​alf=ArcTan[fi];​​bet=ArcTan[1/fi];​​yt=Sin[alf];​​u=ytCos[bet];​​p=u/Cos[2Pi/10];​​dif=uTan[bet];​​x=Cos[alf];​​s=N[x/Sin[2Pi/10]];​​d=x/Tan[2Pi/10];​​pin=s+2d;​​rad=pin/Sin[2Pi/20];​​rh=PolyhedronData["RhombicHexecontahedron","Faces"];​​rr1=Translate[rh,{-pin,0,-dif}];​​rr2=Translate[rh,2{-pin,0,-dif}];​​rr3=Rotate[rr1,-4Pi/5,{0,0,1}];​​f4={rh,Green,rr1,Yellow,rr2,Cyan,rr3};​​f5=Rotate[f4,-2Pi/20,{0,0,1}];​​ring20=Table[Rotate[Translate[f5,{0,-rad,0}],i2Pi/5,{0,0,1}],{i,5}];​​Graphics3D[{ring20},ImageSize340,ViewPoint{0,0,100},BoxedFalse,SphericalRegionTrue]​​
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The unit consisting of five RH
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Graphics3D[{f5},ImageSize300,ViewPoint{0,0,100},BoxedFalse,SphericalRegionTrue]
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Checking the configuration.
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f6=Translate[Rotate[f4,-2Pi/20,{0,0,1}],{0,-rad,0}];​​yf=2pinSin[2Pi/20];​​xf=2pinCos[2Pi/20];​​vA={0,-rad,0};​​vB={-xf,-rad+yf,-2dif};​​vC={xf,-rad+yf,-2dif};​​sB=Sphere[vB,1.5];​​sA=Sphere[vA,1];​​vet=Table[Rotate[{sB,Pink,sA,Yellow,Tube[{vB,vA,vC},0.3]},i2Pi/5-0Pi/5,{0,0,1}],{i,5}];​​Graphics3D[{f6,vet},ImageSize300,ViewPoint{0,0,100},BoxedFalse,SphericalRegionTrue]​​​​
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Now use twice as large RH, and omit rr1 and rr3.
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rhten=Scale[rh,2{1,1,1},{0,0,0}];​​rr2a=Translate[rhten,2{-pin,0,-dif}];​​f4a={rhten,Cyan,rr2a};​​f5a=Rotate[f4a,-2Pi/20,{0,0,1}];​​ring10=Table[Rotate[Translate[f5a,{0,-rad,0}],i2Pi/5,{0,0,1}],{i,5}];​​Graphics3D[{ring10},ImageSize340,ViewPoint{0,0,100},BoxedFalse,SphericalRegionTrue]​​
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Graphics3D[{f4a},ImageSize200,ViewPoint{0,0,100},BoxedFalse,SphericalRegionTrue]
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Now use half as large RH.
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rhsm=Scale[rh,0.5{1,1,1},{0,0,0}];​​rr1b=Translate[rhsm,{-pin,0,-dif}];​​rr2b=Translate[rhsm,2{-pin,0,-dif}];​​rr3b=Rotate[rr1b,-4Pi/5,{0,0,1}];​​f4b={rhsm,Green,rr1b,Yellow,rr2b,Cyan,rr3b};​​f5b=Rotate[f4b,-2Pi/20,{0,0,1}];​​ring20tip=Table[Rotate[Translate[f5b,{0,-rad,0}],i2Pi/5,{0,0,1}],{i,5}];​​Graphics3D[{ring20tip},ImageSize340,ViewPoint{0,0,100},BoxedFalse,SphericalRegionTrue]
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