Orbits and polyhedra in Ih point symmetry

The examination of coordinate transformations as local contractions and expansions of decorations about the poles of the principal rotational axes on the unit sphere for objects of Oh symmetry leads to intermediate geometries corresponding to particular Archimedean polyhedra related to the cube. In a similar manner, partial contractions and expansions of the decorations of the regular orbit of Ih point symmetry, i.e. the vertices of the great rhombicosidodecahedron, leads to the remaining polyhedra within the icosahedral family of Archimedean structures and orbits of Ih. 

The first pairwise contraction leads to the formation of the 3-valent structure of pentagons and hexagons, which is the cage structure of the Ceo fiillerene molecule and, with equal edge lengths, is the geometrical structure known as the truncated icosahedron. 

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