Wolfram Cloud Document

Building on small rhombicuboctahedron
By Sender Kabai

Cubes are placed on 12 faces of small rhombicuboctahedron that correspond to the faces of rhombic dodecahedron. The envelope of the 12 cubes is a great rhombicuboctahedron.
Notice: great rhombicuboctahedron ~ truncated cuboctahedron

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src=GraphicsComplex[PolyhedronData["SmallRhombicuboctahedron","VertexCoordinates"],Polygon[PolyhedronData["SmallRhombicuboctahedron","FaceIndices"]]];u=1/Sqrt[2];vA={u,0,0};vB={0,1,0};vC={-u,0,0};vD={0,-1,0};r01={vA,vB,vC,vD};r02=0.75{vA,vB,vC,vD};r0=Polygon[Join[r01->r02]];r1=Translate[{r0},{0,0,1}];r2=Table[Rotate[{r1},i2Pi/4+Pi/4,{1,0,0}],{i,4}];r3=Rotate[r2,Pi/2+0.001,{0,0,1}];r4=Rotate[{r2},Pi/2+0.001,{0,1,0}];rdfr={r2,r3,r4};rdn=Scale[rdfr,1.22];cub=GraphicsComplex[PolyhedronData["Cube","VertexCoordinates"],Polygon[PolyhedronData["Cube","FaceIndices"]]];cubrd=Translate[cub,{0,0,0.707}];r1=Translate[{cubrd},{0,0,1}];r2=Table[Rotate[{r1},i2Pi/4+Pi/4,{1,0,0}],{i,4}];r3=Rotate[r2,Pi/2+0.001,{0,0,1}];r4=Rotate[{r2},Pi/2+0.001,{0,1,0}];cub12={r2,r3,r4};grc=GraphicsComplex[PolyhedronData["GreatRhombicuboctahedron","VertexCoordinates"],Polygon[PolyhedronData["GreatRhombicuboctahedron","FaceIndices"]]];p1=0.2;grcfr=Normal[grc]/.p:Polygon[__](p/.pts:{{_?NumericQ,_,_},__}Module[{cent,innervtx},cent=Mean[pts];innervtx={1-p1,p1}.{#,cent}&/@pts;Join[pts,pts[[{1}]],innervtx,innervtx[[{1}]]]])//#&;Graphics3D[{grcfr,RGBColor[0,1,1],cub12,Red,rdn,Yellow,src},SphericalRegionTrue,BoxedFalse]

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