A cluster of 12 icosahedra (ico12) with edge to edge connection is created by placing at each vertex of a larger icosahedron (ICO).
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fi=GoldenRatio;alf=ArcTan[fi];bet=ArcTan[1/fi];bet1=ArcTan[1/fi^2];ico=GraphicsComplex[PolyhedronData["Icosahedron","VertexCoordinates"],Polygon[PolyhedronData["Icosahedron","FaceIndices"]]];icov=ico[[1]];ico12=Map[Translate[ico,#]&,fiicov];Graphics3D[{ico12},SphericalRegionTrue,BoxedFalse]
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ico1=Scale[ico,fi^2];Graphics3D[{ico12,Green,Scale[ico1,0.9999]},SphericalRegionTrue,BoxedFalse]
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The ico12 cluster is enveloped by a larger ICO.
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icofr1=Normal[ico1]/.p:Polygon[__](p/.pts:{{_?NumericQ,_,_},__}Module[{cent,innervtx},cent=Mean[pts];innervtx={1-0.2,0.2}.{#,cent}&/@pts;Join[pts,pts[[{1}]],innervtx,innervtx[[{1}]]]])//#&;Graphics3D[{ico12,Green,Scale[icofr1,1.0001]},SphericalRegionTrue,BoxedFalse]
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The frame of enveloping ICO allows to see the inside structure.
z2=1/fi^2;uu=0.5Sqrt[3]/3;vA={0.5-z2,-uu,0};vB={-0.5+z2,-uu,0};perf=0.65;(*Triangles*)vAp=Table[RotationTransform[i2Pi/3,{0,0,1}][vA],{i,3}];vBp=Table[RotationTransform[i2Pi/3,{0,0,1}][vB],{i,3}];pA=Polygon[vAp];pB=Polygon[vBp];iA={0.5,-uu,0};vI=Table[RotationTransform[i2Pi/3,{0,0,1}][iA],{i,3}];pI=Polygon[vI];(*Hexagon*)vA1=RotationTransform[2Pi/3,{0,0,1}][vA];vA2=RotationTransform[4Pi/3,{0,0,1}][vA];vB1=RotationTransform[2Pi/3,{0,0,1}][vB];vB2=RotationTransform[4Pi/3,{0,0,1}][vB];hexa={vA,vB1,vA1,vB2,vA2,vB};hexb=perfhexa;hex=Polygon[Join[hexa->hexb]];i1={hex};i2=Translate[Rotate[Translate[{i1},{0,uu,0}],-bet1,{1,0,0}],{0,0,fi/2}];i3=Rotate[i2,Pi,{0,0,1}];i4=Table[Rotate[Rotate[{i2,i3},alf,{1,0,0}],i2Pi/5,{0,0,1}],{i,5}];i5=Rotate[i4,Pi,{1,0,0}];(*Pentagon*)zT=Sqrt[0.25+(fi/2)^2];k=z2Sin[bet];zP=zT-k;xP=z2Cos[bet];spP=Sphere[{xP,0,zP},0.03];vP={xP,0,0};vP5a=Table[RotationTransform[i2Pi/5+Pi/2,{0,0,1}][vP],{i,5}];vP5b=perfvP5a;pent1=Translate[Polygon[Join[vP5avP5b]],{0,0,zP}];pent5=Table[Rotate[Rotate[Rotate[pent1,Pi,{0,0,1}],-2bet,{1,0,0}],i2Pi/5,{0,0,1}],{i,5}];pent6=Rotate[{pent1,pent5},Pi,{1,0,0}];ticog2=Rotate[Scale[{i4,i5,pent1,pent5,pent6},1.0001],-Pi/2,{0,0,1}];tic={i4,i5};tica=Rotate[Scale[tic,fi^2{1,1,1},{0,0,0}],-Pi/2,{0,0,1}];Graphics3D[{ico12,Red,tica},SphericalRegionTrue,BoxedFalse]
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The part of the enveloping ICO not in contact with ico12 corresponds to a golden truncated icosahedron (TICOG).
TICOG shown as a frame
One ico12 cluster is placed at each vertex of a largar ICO forming a fractal structure.
One ico12 cluster is placed at each vertex of a largar ICO forming a fractal structure.
The relationship with TICOG remains the same.