Great dodecahedron

Figure 2-56. The four regular star polyhedra [93], From the left, the small stellated dodecahedron great dodecahedron great stellated dodecahedron and the great icosahedron. Used by permission from Oxford University Press.

Kif urt 2-66. Die lour regular star polyhedra la) Small stellated dodecahedron (b) the great dodecahedron (c) great stellated dodecahedron (d) the great icosahedron. From H. M. Cundy and A. P Rollett 2-64. Used by permission of Oxford University F ress. 

If the definition of regular polyhedra is not restricted to convex figures, their number rises from five to nine [92], The additional four are depicted in Figure 2-56 they are called by the common name of regular star polyhedra. One of them, viz., the great stellated dodecahedron, is illustrated by the decoration at the top of the Sacristy of St. Peter s Basilica in Vatican City in Figure 2-57. 

Figure 2-57. Great stellated dodecahedron as decoration at the top of the Sacristy of St. Peter s Basihca, Vatican City (photograph by the authors).

In the Preface to the Third Edition of his Regular Polytopes [20], the great geometer H. S. M. Coxeter calls attention to the icosahe-dral structure of a boron compound in which twelve boron atoms are arranged like the vertices of an icosahedron. It had been widely believed that there would be no inanimate occurrence of an icosahedron, or of a regular dodecahedron either. 

There are many compounds of lanthanides exhibiting coordination polyhedra close to a dodecahedron with triangular faces. There remains a great interest in the various environments and ligand combinations for which the dodecahedral arrangement is favourable. 

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. 

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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