60 rhombic triacontahedra (RT) are arranged with facial connection in three different clusters at the vertices of the polyhedra, truncated icosahedra (TICO), truncated dodecahedron (TD) and great rhombicosidodecahedron (GRID). Structurally the GRID assembly seems to be the most stable, because each RT has a connection with four other RTs, while in the other assemblies they are in contact with three or two RTs only. Could that be the real structure of C60 molecule?
fi=1.618;alf=ArcTan[fi];bet=ArcTan[1/fi];bet1=ArcTan[1/fi^2];
RT60 at SRID vertices
RT60 at SRID vertices
rt=GraphicsComplex[PolyhedronData["RhombicTriacontahedron","VertexCoordinates"],Polygon[PolyhedronData["RhombicTriacontahedron","FaceIndices"]]];srid=GraphicsComplex[PolyhedronData["SmallRhombicosidodecahedron","VertexCoordinates"],Polygon[PolyhedronData["SmallRhombicosidodecahedron","FaceIndices"]]];sridv=srid[[1]];rta=Rotate[rt,bet,{0,1,0}];rtsrid=Map[Translate[rta,#]&,2.7552sridv];sridenv=Scale[srid,3.5];sridfr=Normal[sridenv]/.p:Polygon[__](p/.pts:{{_?NumericQ,_,_},__}Module[{cent,innervtx},cent=Mean[pts];innervtx={1-0.1,0.1}.{#,cent}&/@pts;Join[pts,pts[[{1}]],innervtx,innervtx[[{1}]]]])//#&;Graphics3D[{sridfr,Yellow,rtsrid},SphericalRegionTrue,BoxedFalse,ImageSize350]
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RT60 at vertices of TICO
RT60 at vertices of TICO
tico=GraphicsComplex[PolyhedronData["TruncatedIcosahedron","VertexCoordinates"],Polygon[PolyhedronData["TruncatedIcosahedron","FaceIndices"]]];ticov=tico[[1]];rtn=Rotate[rt,Pi,{0,0,1}];rttico=Map[Translate[rtn,#]&,2.7552ticov];ticoenv=Scale[tico,3.42];ticofr=Normal[ticoenv]/.p:Polygon[__](p/.pts:{{_?NumericQ,_,_},__}Module[{cent,innervtx},cent=Mean[pts];innervtx={1-0.1,0.1}.{#,cent}&/@pts;Join[pts,pts[[{1}]],innervtx,innervtx[[{1}]]]])//#&;Graphics3D[{ticofr,RGBColor[0.7,1,1],rttico},SphericalRegionTrue,BoxedFalse,ImageSize350]
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RT60 at TD vertices
RT60 at TD vertices
td=GraphicsComplex[PolyhedronData["TruncatedDodecahedron","VertexCoordinates"],Polygon[PolyhedronData["TruncatedDodecahedron","FaceIndices"]]];tdv=td[[1]];rttd=Map[Translate[rt,#]&,2.7552tdv];tdenv=Scale[td,3.34];tdfr=Normal[tdenv]/.p:Polygon[__](p/.pts:{{_?NumericQ,_,_},__}Module[{cent,innervtx},cent=Mean[pts];innervtx={1-0.1,0.1}.{#,cent}&/@pts;Join[pts,pts[[{1}]],innervtx,innervtx[[{1}]]]])//#&;Graphics3D[{tdfr,RGBColor[1,0.9,0.6],rttd},SphericalRegionTrue,BoxedFalse,ImageSize350]
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Notes
Notes
1. The number 2.7552 is the distance between two parallel faces of RT.
2. Within an assembly all RTs have the same orientation.
3. Other polyhedra inscribed into RT may be used to construct many different clusters. E.g. https://demonstrations.wolfram.com/RegularClustersOfPolyhedra/
2. Within an assembly all RTs have the same orientation.
3. Other polyhedra inscribed into RT may be used to construct many different clusters. E.g. https://demonstrations.wolfram.com/RegularClustersOfPolyhedra/
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