Sándor Kabai
A cluster of 60 rhombic triacontahedron (RT) in two different arrangements.
TICO = truncated icosahedron
SDR = small rhombicosidodecahedron
It is demonstrated here that by turning over the RTs in the cluster results in a different arrangement. The groups each consisting of five RTs can be lifted, turned over, then lowered in place. Notice, that each RT has three faces in contact in the TICO arrangement, while it has four faces in the SRD arrangement. Therefore the latter must be more stable in statical terms. (One could wonder why C60 molecule does not assumes this arrangement instead of the TICO arrangement).
A cluster of 60 rhombic triacontahedron (RT) in two different arrangements.
TICO = truncated icosahedron
SDR = small rhombicosidodecahedron
It is demonstrated here that by turning over the RTs in the cluster results in a different arrangement. The groups each consisting of five RTs can be lifted, turned over, then lowered in place. Notice, that each RT has three faces in contact in the TICO arrangement, while it has four faces in the SRD arrangement. Therefore the latter must be more stable in statical terms. (One could wonder why C60 molecule does not assumes this arrangement instead of the TICO arrangement).
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Manipulate[Module[{},ϕ=1.618;α=ArcTan[ϕ];β=ArcTan[1/ϕ];y=Sin[α];u=ySin[α];p=u/Cos[2Pi/10];rt=PolyhedronData["RhombicTriacontahedron","Faces"];rd1=Translate[rt,{(2u+p),0,0}];rd5=Table[Rotate[rd1,i2Pi/5,{0,0,1}],{i,5}];up=Rotate[Translate[rd5,{0,0,el+5.68328+(el)}],turn,{0,1,0},{0,0,el+5.68328+(0.7017+el/2)}];tilt=Rotate[Rotate[up,Pi,{0,0,1}],-2β,{0,1,0}];tilt5=Table[Rotate[{Specularity[0.7],Hue[0.8^6i],tilt},i2Pi/5,{0,0,1}],{i,5}];top={tilt5,up};bot=Translate[Rotate[top,Pi,{0,1,0}],{0,0,0}];Graphics3D[{top,{RGBColor[0.1,1,0.9],bot}},ViewPoint{0,0,10},ViewAngle0.1,BoxedFalse,SphericalRegionTrue]],{{el,0,"lower - elevate"},-0.7017,2},{{turn,0,"turn"},0,Pi},TrackedSymbols{el,turn}]
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rtn=Rotate[rt,Pi,{0,0,1}];rtf=Rotate[rt,bet,{0,1,0}];srd=PolyhedronData["SmallRhombicosidodecahedron","Faces"];srdv=PolyhedronData["SmallRhombicosidodecahedron","VertexCoordinates"];p1=0.2;srdfr=Normal[srd]/.p:Polygon[__](p/.pts:{{_?NumericQ,_,_},__}Module[{cent,innervtx},cent=Mean[pts];innervtx={1-p1,p1}.{#,cent}&/@pts;Join[pts,pts[[{1}]],innervtx,innervtx[[{1}]]]])//#&;srdfrn=Scale[srdfr,3.5];rt60srd={Map[Translate[rtf,#]&,2.7527*srdv]};Graphics3D[{{rt60srd,Green,srdfrn}},ViewPoint{0,0,100},ViewAngle0.01,BoxedFalse]
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tico=PolyhedronData["TruncatedIcosahedron","Faces"];ticov=PolyhedronData["TruncatedIcosahedron","VertexCoordinates"];p1=0.14;ticofr=Normal[tico]/.p:Polygon[__](p/.pts:{{_?NumericQ,_,_},__}Module[{cent,innervtx},cent=Mean[pts];innervtx={1-p1,p1}.{#,cent}&/@pts;Join[pts,pts[[{1}]],innervtx,innervtx[[{1}]]]])//#&;ticofrn=Scale[ticofr,3.4];rt60tico={Map[Translate[rtn,#]&,2.7527*ticov]};Graphics3D[{{Yellow,rt60tico,Pink,ticofrn}},ViewPoint{0,0,100},ViewAngle0.01,BoxedFalse]
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