Compounds of uniform polyhedra

Compounds is a union of several polyhedra with common center. There exists an infinite number of ways to do make compounds. Limited amount of compounds may be obtained if we apply some restrictions. We will use the following:
1) Compounds should be "rigid": it losts its symmetry as result of any infinitesimal rotation of components. This restriction throws out compounds with one or two degree of freedom.
2) All components are symmetrically equivalent. This helps to rid of compounds of compounds.
3) Compound and components should have Oh (octahedral) or Ih (icosahedral) symmetry group. This is arbitrary decision which throws out intresting compounds with tetrahedral symmetry and those without planes of symmetry.

For each component Oh or Ih symmetry there exist only 7 rigid compounds with Oh or Ih symmetry. The table below represents these compounds when cube and dodecahedron are used as component. The obvious notation are used: symbol N P -> Q means that N component of symmetry P form compound of symmetry Q.
3 Oh -> Oh
4 Oh -> Oh
6 Oh -> Oh
5 Oh -> Ih
10 Oh -> Ih
10 Oh -> Ih
15 Oh -> Ih
2 Ih -> Oh
4 Ih -> Oh
4 Ih -> Oh
6 Ih -> Oh
5 Ih -> Ih
6 Ih -> Ih
10 Ih -> Ih
Here these compounds are arranged in component's order:
Compounds with octahedral components (17 x 7 = 119 species).
Compounds with icosahedral components (46 x 7 = 322 species).

Here is a Compounds Machine - VRML device for playing with co "VRML">Compounds Machine For additional information about compounds visit George Hart's Virtual Polyhedra.

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