Rhombicosidodecahedron, great

These structural details are emphasized in Figure 2.19, with, in the second and later rows Figures 2.19b-d, the local sets of 10, 6 and 4 vertices of the great rhombicosidodecahedron identified about a representative pole position on a face and, then, in the second column of diagrams, as fiilly decorated elliptical projections of the 120-vertex cage. 

There are 6 five-fold rotational symmetry elements in an object of Ih point symmetry. Thus, in Figure 2.19b the 120 vertices of the great rhombicosidodecahedron are arranged in sets of 10 about the poles of these axes on the unit sphere. That construction emphasises that uniform contractions of these sets about these axes points will return the 12-vertex Platonic solid, the icosahedron, in which each vertex has Csv site symmetry. There are 10 three-fold rotational axes and, so, in Figure 2.19c the decoration pattern is arranged to divide the 120 vertices into sets of 6 about the 20 poles of these axes on the unit sphere. Again, uniform contraction of these subsets of vertices onto these positions on the unit sphere generates the fifth Platonic solid, the dodecahedron, and the site symmetry each vertex is Csy. 

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. 

Figure 2.19 The divisions of the vertices of the regular orbit of point symmetry, defining the great rhombicosidodecahedron, into decoration sets about the rotational axes points [C5], row b, [C3], row c and [C2], row d, on the unit sphere.

Figure 2.20 Formation of the lower orbits of Iji symmetry O12, the icosahedron [row a] O20, the dodecahedron, [row b] and O30, the icosidodecahedron, [row c] of Figure 2.4 by coalescing the local sets of vertices of the great rhombicosidodecahedron onto the poles of the C5, C3 and C2 rotational axes with colour codings as in Figure 2.19.

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