A polar zonohedron (PZ) having five zones and consisting of two kinds of rhombuses (golden ratio squared and square root 2) can be attached to each face of dodecahedron. Such assembly of 12 PZs can support a cluster of 30 rhombic dodecahedron (RD) with face to face connection. By adding various polyhedra a space filling arrangement can be assembled.
Base pz
Base pz
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pz={pz1,pz2};pzin={pz,pz5,pz6};as1={{Opacity[0.5],dod},pol1};as2={dod,pol1,pol2};as3={dod,pol1,pol2,pz1};as4={dod,pol1,pol2,pz1,pz2};as5={dod,pol1,pol2,pz1,pz2,pz5};Row[Graphics3D[#,BoxedFalse,ImageSize130]&/@{as1,as2,as3,as4,as5,pzin}]
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A PZ is constructed in a dod.
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Graphics3D[{pol1,Yellow,Opacity[0.4],dod},BoxedFalse,SphericalRegionTrue,ImageSize300,ViewPoint{100,0,30},ViewAngle0.01]
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Graphics3D[{pzin,Red,Opacity[0.4],dod},BoxedFalse,SphericalRegionTrue,ImageSize300,ViewPoint{0,100,0},ViewAngle0.01]
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ico=GraphicsComplex[PolyhedronData["Icosahedron","VertexCoordinates"],Polygon[PolyhedronData["Icosahedron","FaceIndices"]]];ico21=PolyhedronData["Icosahedron","Edges"];icov=PolyhedronData["Icosahedron","VertexCoordinates"];icocyl=Map[Cylinder[icov[[#]],0.02]&,ico21];icofr=Rotate[Scale[icocyl,7],Pi,{0,0,1}];Graphics3D[{pzin,Green,icofr},BoxedFalse,SphericalRegionTrue,ImageSize300,ViewPoint{0,0,100},ViewAngle0.01]
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Row[Graphics3D[#,BoxedFalse,ImageSize130]&/@{p2align55,rd5up5,new5up,new5up5,p25up,p25up5}]
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Square root rhombohedra touch the vertices of dodecahedron.
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Graphics3D[{p2align55,dod},BoxedFalse,SphericalRegionTrue,ImageSize300,ViewPoint{0,0,-100},ViewAngle0.01]
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In physical models the square root rhombohedra can used to secure a cluster of 30 RD, in which the RDs are connected with their edges
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Graphics3D[{rd5up5,p2align55,dod},BoxedFalse,SphericalRegionTrue,ImageSize300,ViewPoint{0,0,-100},ViewAngle0.01]
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thick