Great icosahedron

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. 

Additionally, the outer ten A1 atoms are bonded to two bromine atoms each and saturated by a donor molecule (THF, THP). The apex and base atoms in the AI12 icosahedron are not naked they are shielded by one donor molecule each. Despite the great sensitivity of these Al22 subhalides 65 and 66, it was possible to obtain solid-state 27A1 NMR spectra and XPS measurements, which showed indeed that three electronically different types of A1 atoms are present [100]. The type of... 

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 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.

Because of the number and complexity of the boranes and their derivatives we shall not attempt to describe all their structures in detail. Apart from B2H6 most of the boranes have boron skeletons which are usually described as icosahedral fragments. (They could equally well be related to the more recently discovered carboranes B C2H +2 since the boron-carbon skeletons in these compounds are highly symmetrical triangulated polyhedra (Table 24.8) which include the icosahedron.) The great theoretical interest of the boranes stems from the fact that they are electron-deficient molecules, that is, there are not sufficient valence electrons to bond together all the atoms by normal electron-pair bonds. In these molecules the number of atomic orbitals (1 for each H and 4 for each B) is greater than the total number of valence electrons ... 

One main reason for the great interest in the study of fuUerenes is certainly to be found in their particularly appealing geometrical form. The best known fullerene is the so-called buck-minsterfuUerene that contains 60 carbon atoms (Qo) and is composed of 12 pentagonal carbon rings located around the vertices of an icosahedron and 20 hexagonal carbon rings at the centers of icosahedral faces (Kroto et al. 1985). 

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