CITE THIS NOTEBOOK: Great rhombicosidodecahedron demos by Sándot Kabai. Wolfram Community NOV 3 2022.
This is a summary of exploration I made in relation to GRD
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grd=GraphicsComplex[PolyhedronData["GreatRhombicosidodecahedron","VertexCoordinates"],Polygon[PolyhedronData["GreatRhombicosidodecahedron","FaceIndices"]]];grdn=Scale[grd,0.217];grdtube=grd/.Polygon->Tube;Graphics3D[{grd,Green,grdtube},ImageSize200]
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Equivalent: Truncated icosidodecahedron
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truncatedgrd=GraphicsComplex[PolyhedronData["TruncatedIcosidodecahedron","VertexCoordinates"],Polygon[PolyhedronData["TruncatedIcosidodecahedron","FaceIndices"]]];Graphics3D[{truncatedgrd},ImageSize200]
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Common faces of GRD, ICO, DOD, RT and TET
Common faces of GRD, ICO, DOD, RT and TET
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fi=GoldenRatio;bet=ArcTan[1/fi];ico=GraphicsComplex[PolyhedronData["Icosahedron","VertexCoordinates"],Polygon[PolyhedronData["Icosahedron","FaceIndices"]]];cubt=GraphicsComplex[PolyhedronData["Cube","VertexCoordinates"],Tube[PolyhedronData["Cube","FaceIndices"]]];cubtn=Scale[cubt,fi];tet=GraphicsComplex[PolyhedronData["Tetrahedron","VertexCoordinates"],Polygon[PolyhedronData["Tetrahedron","FaceIndices"]]];dod=PolyhedronData["Dodecahedron","Faces"];rt=PolyhedronData["RhombicTriacontahedron","Faces"];Manipulate[icon=Scale[Rotate[ico,-bet,{0,1,0}],eni];tetn=Rotate[Rotate[Scale[tet,entSqrt[2]fi{1,1,1},{0,0,0}],-ArcTan[Sqrt[2]],{0,1,0}],-Pi/4,{0,0,1}];dodn=Rotate[Scale[dod,end{1,1,1},{0,0,0}],-bet,{0,1,0}];rtn=Rotate[Scale[rt,enrt{1,1,1},{0,0,0}],bet,{0,1,0}];grd=GraphicsComplex[PolyhedronData["GreatRhombicosidodecahedron","VertexCoordinates"],Polygon[PolyhedronData["GreatRhombicosidodecahedron","FaceIndices"]]];grdn=Scale[grd,0.217];Graphics3D[{rtn,dodn,LightGreen,tetn,Pink,icon,Green,cubtn,Cyan,grdn},BoxedFalse,SphericalRegionTrue,PlotRange1.6,ViewPoint{4,2,2},ViewAngle0.2],{{enrt,0.45,"enlarge rt"},0.45,0.589},{{end,0.5,"enlarge dodecahedron"},0.5,0.67},{{eni,0.8,"enlarge icosahedron"},0.8,1.053},{{ent,1,"enlarge tetrahedron"},1,1.7},TrackedSymbols->{enrt,end,eni,ent},SaveDefinitions->True]
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GRD as a truncation of icosidodecahedron
GRD as a truncation of icosidodecahedron
GRD is truncated in a manner that the square faces are maintained as opposed to the usual truncation, where no squares are produced.
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fi=GoldenRatio;alf=ArcTan[fi];bet=ArcTan[1/fi];bet1=ArcTan[1/fi^2];y=0.5+0.5fi;el=yfi;u=0.5Sqrt[3]/3;unit=Sqrt[3]/3;icofr=Normal[ico]/.p:Polygon[__]:>(p/.pts:{{_?NumericQ,_,_},__}:>Module[{cent,innervtx},cent=Mean[pts];innervtx={1-0.15,0.15}.{#,cent}&/@pts;Join[pts,pts[[{1}]],innervtx,innervtx[[{1}]]]])//#&;icofr1=Rotate[Scale[icofr,2.618{1,1,1},{0,0,0}],Pi/2,{0,0,1}];Manipulate[v=x/Cos[bet1];is=(0.5unit+v)/unit;t=2isunit;shift=0.5unit+v;sink=xTan[bet1];(*square*)s1={x,-x,0};sqv=Table[RotationTransform[i2Pi/4,{0,0,1}][s1],{i,4}];sq=Polygon[sqv];sq1=Translate[sq,{0,0,isfi-sink}];sq2=Translate[sq,{0,0,-isfi+sink}];sq3=Rotate[{sq1,sq2},Pi/2,{1,0,0}];sq4=Rotate[{sq1,sq2},Pi/2,{0,1,0}];sq30=Table[Rotate[Rotate[{sq1,sq2,sq3,sq4},bet,{1,0,0}],i2Pi/5+Pi/5,{0,0,1}],{i,5}];(*hexagon*)b1={x,-u,0};b2=RotationTransform[2Pi/3,{0,0,1}][b1];b3=RotationTransform[4Pi/3,{0,0,1}][b1];b123=Polygon[{b1,b2,b3}];c1={-x,-u,0};c2=RotationTransform[2Pi/3,{0,0,1}][c1];c3=RotationTransform[4Pi/3,{0,0,1}][c1];hex=Rotate[Polygon[{b1,c2,b2,c3,b3,c1}],Pi/3,{0,0,1}];t1=Rotate[Translate[{hex},{0,-shift,0}],bet1,{1,0,0}];t2=Rotate[t1,Pi,{0,0,1}];hexa=Table[Rotate[Rotate[Translate[{t1,t2},{0,0,isfi}],alf,{1,0,0}],i2Pi/5,{0,0,1}],{i,5}];hexb=Rotate[hexa,Pi,{1,0,0}];hex20={hexa,hexb};(*decagon*)height=isfi-sink;p0={-x,-x,height};p1=RotationTransform[bet,{0,1,0}][p0];p2=RotationTransform[2Pi/5,{0,0,1}][p1];p3=RotationTransform[4Pi/5,{0,0,1}][p1];p4=RotationTransform[6Pi/5,{0,0,1}][p1];p5=RotationTransform[8Pi/5,{0,0,1}][p1];q0={-x,x,height};q1=RotationTransform[bet,{0,1,0}][q0];q2=RotationTransform[2Pi/5,{0,0,1}][q1];q3=RotationTransform[4Pi/5,{0,0,1}][q1];q4=RotationTransform[6Pi/5,{0,0,1}][q1];q5=RotationTransform[8Pi/5,{0,0,1}][q1];dec1=Rotate[Polygon[{p1,q1,p2,q2,p3,q3,p4,q4,p5,q5}],Pi/2,{0,0,1}];dec5=Table[Rotate[Rotate[Rotate[dec1,Pi,{0,0,1}],-2bet,{1,0,0}],i2Pi/5,{0,0,1}],{i,5}];dec6=Rotate[{dec1,dec5},Pi,{1,0,0}];dec12={dec1,dec5,dec6};grdg={dec12,Red,sq30,Green,hex20};Graphics3D[{icofr1,grdg},Boxed->False,SphericalRegion->True,ImageSize->300],{{x,0.18,"ratio of edges"},0,0.5,Appearance->"Labeled"},TrackedSymbols:>{x},SaveDefinitions->True]
Normally, the truncation of ID involves the creation of rectangles, and the size does not change
Cluster of 60 cubes and cluster of 60 rhombic dodecahedra (RD) are framed by GRD
Cluster of 60 cubes and cluster of 60 rhombic dodecahedra (RD) are framed by GRD
The centres of cubes and RDs are at the vertices of small rhombicosidodecahedron (SRD)
The inner edges of the cubes are aligned on the edges of another golden GRD
RT places at the vertices of GRD
RT places at the vertices of GRD
RT can be replaced with cube, RD, ICO, DOD or OCT
Relationship of GRD to cluster of TET30 and TET60
GRD in heaxagonal lattice
Relationship of GRD with TET, TOCT, ICO, ID, DRT and golden lines
Relationship of GRD with TET, TOCT, ICO, ID, DRT and golden lines