CITE THIS NOTEBOOK: Cluster of 12 rhombic triacotahedra by Sándor Kabai. Wolfram Community, SEP 2 2022.
See reference YouTube video: https://youtu.be/Z7PwXm8fjYY
RTs are placed around a rhombic hexecontahedron (RH) with face-to face connection. The centres of RTs are at vertices of an icosahedron. This arrangement was discovered in quasicrystals. Various methods and polyhedra are used to show the structure. A frame consisting of 80 golden rhombohedra is attached to the cluster. See also: https://demonstrations.wolfram.com/12RhombicTriacontahedra/ and https://demonstrations.wolfram.com/80GoldenRhombohedra/
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fi=GoldenRatio;​​(*Anglesofgoldenrhombus*)​​alf=ArcTan[fi];​​bet=ArcTan[1/fi];​​bet1=ArcTan[1/fi^2];
Half diameters of golden rhombus
rtx=Cos[alf];​​rty=Sin[alf];​​(*DistanceoftwoparallelfacesofRT*)​​m=2rtyfi;​​m1=mfi^2;
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rh=GraphicsComplex[PolyhedronData["RhombicHexecontahedron","VertexCoordinates"],Polygon[PolyhedronData["RhombicHexecontahedron","FaceIndices"]]];​​rhn=Rotate[rh,Pi,{0,0,1}];​​ico=GraphicsComplex[PolyhedronData["Icosahedron","VertexCoordinates"],Polygon[PolyhedronData["Icosahedron","FaceIndices"]]];​​icov=PolyhedronData["Icosahedron","VertexCoordinates"];​​icov=ico[[1]];​​dod=GraphicsComplex[PolyhedronData["Dodecahedron","VertexCoordinates"],Polygon[PolyhedronData["Dodecahedron","FaceIndices"]]];
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rt=GraphicsComplex[PolyhedronData["RhombicTriacontahedron","VertexCoordinates"],Polygon[PolyhedronData["RhombicTriacontahedron","FaceIndices"]]];​​rtn=Rotate[rt,Pi,{0,0,1}];​​Graphics3D[{rt},ViewPoint{0,0,100},ImageSize200,ViewAngle0.01,BoxedFalse,SphericalRegionTrue]​​
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rtline={AbsoluteThickness[2],GraphicsComplex[PolyhedronData["RhombicTriacontahedron","VertexCoordinates"],Line[PolyhedronData["RhombicTriacontahedron","FaceIndices"]]]};​​Graphics3D[{rtline},ViewPoint{0,0,100},ImageSize200,ViewAngle0.01,BoxedFalse,SphericalRegionTrue]
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rttube=GraphicsComplex[PolyhedronData["RhombicTriacontahedron","VertexCoordinates"],Tube[PolyhedronData["RhombicTriacontahedron","FaceIndices"]]];​​Graphics3D[{Green,rttube},ViewPoint{0,0,100},ImageSize200,ViewAngle0.01,BoxedFalse,SphericalRegionTrue]
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rttube1=rt/.Polygon->Tube;​​Graphics3D[{Yellow,rttube1},ViewPoint{0,0,100},ImageSize200,ViewAngle0.01,BoxedFalse,SphericalRegionTrue]
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Graphics3D[{rt,Green,rttube},ViewPoint{0,0,100},ImageSize200,ViewAngle0.01,BoxedFalse,SphericalRegionTrue]
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Assembling perforated RT from faces
rtvA={rtx,0,0};​​rtvB={0,rty,0};​​rtvC={-rtx,0,0};​​rtvD={0,-rty,0};​​rts1={rtvA,rtvB,rtvC,rtvD,rtvA};​​(*setperforatehere*)​​perf=0.8;​​rts2=perf{rtvA,rtvB,rtvC,rtvD,rtvA};​​r1=Polygon[Join[rts1->rts2]];​​r2=Translate[r1,{0,0,firty}];​​r3=Translate[r1,{0,0,-firty}];​​r4=Rotate[Rotate[{r2,r3},Pi/2,{1,0,0}],Pi/2,{0,1,0}];​​r5=Rotate[Rotate[{r2,r3},Pi/2,{0,1,0}],Pi/2,{1,0,0}];​​rtp=Table[Rotate[Rotate[{r2,r3,r4,r5},bet,{1,0,0}],i2Pi/5-Pi/2,{0,0,1}],{i,5}];​​Graphics3D[{rt,Green,Scale[rtp,1.001]},BoxedFalse,SphericalRegionTrue,ImageSize200]
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Another perforated RT
d=0.4;​​rtp1=Normal[rt]/.p:Polygon[__](p/.pts:{{_?NumericQ,_,_},__}Module[{cent,innervtx},cent=Mean[pts];​​innervtx={1-d,d}.{#,cent}&/@pts;​​Join[pts,pts[[{1}]],innervtx,innervtx[[{1}]]]])//#&;​​Graphics3D[{rt,Cyan,Scale[rtp1,1.001]},BoxedFalse,SphericalRegionTrue,ImageSize200]​​
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12 RT placed at the vertices of ico having edge length m
Glow[White] - Vizible edges are shown
Opacity[0] - All edges are shown
Opacity[0.2],EdgeForm[Thick] - All edges are shown, thick