CITE THIS NOTEBOOK: Joining two rhombic hexecontahedra (RH), and Ring of 10 RHs by Sándor Kabai. Wolfram Community SEP 2022.
Two rhombic hexecontahedra (RH) are joined in a way that one prolate golden rhombohedron (prGR) is shared by the two RHs. Since each RH consists of 20 prGRs, the new assembly thus created has 39 units, and one becomes unnecessary. The loss is 1-39/40 = 0.025 %.
Five of such pair of RHs can form a ring of 190 prGRs.
Five of such pair of RHs can form a ring of 190 prGRs.
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Manipulate[ϕ=GoldenRatio;α=ArcTan[GoldenRatio];β=ArcTan[1/GoldenRatio];y=Sin[α];x=Cos[α];u=yCos[β];p=u/Cos[2Pi/10];v=uTan[2Pi/10];dif=uTan[β];dist=(2u+p)/2/Tan[2Pi/20];rt=PolyhedronData["RhombicTriacontahedron","Faces"];rh=PolyhedronData["RhombicHexecontahedron","Faces"];rh1=Translate[rh,{(2u+p)/2,0,0}];rh2={RGBColor[0.2,1,1],Translate[rh,{-(2u+p)/2,0,-dif}]};rh2m={RGBColor[1,1,0.12],Translate[rh,{-(2u+p)/2-moveϕ,0,-dif-move}]};pair={rh1,rh2};Graphics3D[{rh1,rh2m},BoxedFalse,SphericalRegion->True,ViewPoint{0,-10,10},ViewAngle0.08,PlotRange{{-5.5,3.5},{-3,3},{-4,3}},ImageSize300],{{move,0,"move"},1.2,0},TrackedSymbols{move}]
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ten=Table[Rotate[Translate[pair,{0,dist,0}],i2Pi/5,{0,0,1}],{i,5}];Graphics3D[{ten},BoxedFalse,SphericalRegion->True,ViewPoint{0,0,10},ViewAngle0.1]
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