Clusters of rhombic triacontahedra (RT)
Sándor Kabai
RTs are placed at vertices of various polyhedra to produce clusters containing 8, 12, 14, 20, 30, 32, 60, 120 RTs. The RTs are connected with faces or vertices and are oriented equally within a cluster. There are four initial orientations.
RTs are placed at vertices of various polyhedra to produce clusters containing 8, 12, 14, 20, 30, 32, 60, 120 RTs. The RTs are connected with faces or vertices and are oriented equally within a cluster. There are four initial orientations.
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fi=GoldenRatio;(*Anglesanddigonalsofgoldenrhombus*)alf=ArcTan[fi];bet=ArcTan[1/fi];y=Sin[alf];x=Cos[alf];(*DistanceofparallelfacesorRT:"ed"*)ed=2yfi;rt=PolyhedronData["RhombicTriacontahedron","Faces"];(*Vertices*)rtv=rt[[1]];(*HereyoucanreplaceRTwithequivalentpolyhedron*)(*rt=dodn*)rtn=Rotate[rt,Pi,{0,0,1}];rtf=Rotate[rt,bet,{0,1,0}];rtb=Rotate[Rotate[rt,bet,{0,1,0}],Pi/4,{0,0,1}];dod=PolyhedronData["Dodecahedron","Faces"];dodv=dod[[1]];ico=PolyhedronData["Icosahedron","Faces"];icov=ico[[1]];tico=PolyhedronData["TruncatedIcosahedron","Faces"];ticov=tico[[1]];id=PolyhedronData["Icosidodecahedron","Faces"];idv=id[[1]];td=PolyhedronData["TruncatedDodecahedron","Faces"];tdv=td[[1]];srd=PolyhedronData["SmallRhombicosidodecahedron","Faces"];srdv=srd[[1]];grd=PolyhedronData["GreatRhombicosidodecahedron","Faces"];grdv=grd[[1]];cub=PolyhedronData["Cube","Faces"];cubv=cub[[1]];rd=PolyhedronData["RhombicDodecahedron","Faces"];rdv=rd[[1]];oct=PolyhedronData["Octahedron","Faces"];octv=oct[[1]];rt8=Map[Translate[rtf,#]&,ed*cubv];rt12=Map[Translate[rtn,#]&,ed*icov];rt14=Map[Translate[rtb,#]&,1.07ed*rdv];rt20=Map[Translate[rtn,#]&,ed*dodv];rt30=Map[Translate[rt,#]&,ed*idv];rt32=Map[Translate[rt,#]&,1.15ed*rtv];rt60=Map[Translate[rtn,#]&,ed*ticov];rt60a=Map[Translate[rtf,#]&,ed*srdv];rt60b=Map[Translate[rt,#]&,ed*tdv];rt120=Map[Translate[rtf,#]&,ed*grdv];Graphics3D[{rt12},BoxedFalse,SphericalRegionTrue,ImageSize200,ViewPoint{0,0,100}]
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(*InitialpositionsofRT*)Row[Graphics3D[#,Boxed->False,ImageSize->120,ViewPoint->{0,0,100}]&/@{rt,rtn,rtf,rtb}]
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Row[Graphics3D[#,Boxed->False,ImageSize->160,ViewPoint->{0,0,100}]&/@{rt8,rt12,rt14}]
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Row[Graphics3D[#,Boxed->False,ImageSize->160,ViewPoint->{0,0,100}]&/@{rt20,rt30,rt32}]
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Row[Graphics3D[#,Boxed->False,ImageSize->160,ViewPoint->{0,0,100}]&/@{rt60,rt60a,rt60b}]
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Row[Graphics3D[#,Boxed->False,ImageSize->160,ViewPoint->{0,0,100}]&/@{rt120}]
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Equivalent polyhedra
Fit various polyhedra into RT. Then return to the code and replace RT, e.g. rt = dodn; and evaluate again to get clusters of dod, etc.
Fit various polyhedra into RT. Then return to the code and replace RT, e.g. rt = dodn; and evaluate again to get clusters of dod, etc.
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rtt=GraphicsComplex[PolyhedronData["RhombicTriacontahedron","VertexCoordinates"],Tube[PolyhedronData["RhombicTriacontahedron","EdgeIndices"],0.04]];dodn=Rotate[Scale[dod,2x{1,1,1},{0,0,0}],Pi,{0,0,1}];Graphics3D[{dodn,Yellow,rtt},BoxedFalse,SphericalRegionTrue,ImageSize200]
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icon=Rotate[Scale[ico,2y{1,1,1},{0,0,0}],Pi,{0,0,1}];Graphics3D[{icon,Cyan,rtt},BoxedFalse,SphericalRegionTrue,ImageSize200]
Further possible investigations
Increase the distance of unit by multiplying “ed”. Then insert a large polyhedron at the middle.
Increase the distance of unit by multiplying “ed”. Then insert a large polyhedron at the middle.
Combining these clusters: see https://www.mathematica-journal.com/2012/09/11/clusters-produced-by-placing-rhombic-triacontahedra-at-the-vertices-of-polyhedra/
and
https://demonstrations.wolfram.com/ClusterOf45PolyhedraAroundARhombicHexacontahedron/
and
https://demonstrations.wolfram.com/ClusterOf117PolyhedraAroundARhombicTriacontahedron/
and
https://demonstrations.wolfram.com/3012RhombicTriacontahedra/
and
https://demonstrations.wolfram.com/DodecahedralClusterOfRTsAndRHs/
and
https://demonstrations.wolfram.com/ClusterOf45PolyhedraAroundARhombicHexacontahedron/
and
https://demonstrations.wolfram.com/ClusterOf117PolyhedraAroundARhombicTriacontahedron/
and
https://demonstrations.wolfram.com/3012RhombicTriacontahedra/
and
https://demonstrations.wolfram.com/DodecahedralClusterOfRTsAndRHs/