Dodecahedron, symmetry

Figure 16.2 The icosahedron (top) and dodecahedron (bottom) have identical symmetries but different shapes. Protein subunits of spherical viruses form a coat around the nucleic acid with the same symmetry arrangement as these geometrical objects. Electron micrographs of these viruses have shown that their shapes are often well represented by icosahedra. One each of the twofold, threefold, and fivefold symmetry axes is indicated by an ellipse, triangle, and pentagon, respectively.

You say that your nonlinear molecule has the high symmetiy of a regular polyhedron, such as a tetrahedron, cube, octahedron, dodecahedron, icosahedron,... sphere. If it is a sphere, it is monatomic. On the other hand, if it is not monatomic, it has the symmetry of one of the Platonic solids (see the introduction to Chapter 8). 

Rotating single-crystal measurements also permitted the extraction of the orientation of the magnetic tensor in the molecular reference frame and the experimental easy axis was found to coincide with the idealized tetragonal axis of the coordination dodecahedron of Dy. Crystal field calculations assuming idealized tetragonal symmetry permitted the reproduction of magnetic susceptibility data for gz = 19.9 and gxy 0 [121]. More elaborated calculations such as ab initio post Hartree-Fock CASSCF confirmed this simple analysis [119]. 

Td, possesses 32 symmetry, and requires a minimum of 12 asymmetric units the cube and octahedron, which belong to the point group Oh, possess 432 symmetry, and require a minimum of 24 asymmetric units and the dodecahedron and icosahedron, which belong to the point group Ih, possess 532 symmetry, and require a minimum of 60 asymmetric units. The number of asymmetric units required to generate each shell doubles if mirror planes are present in these structures. 

Finally, we turn to the pentagonal dodecahedron and the icosahedron. These two polyhedra have the same symmetry. They are related to each other as the cube and octahedron are related. The symmetry elements and operations are as follows. 

Look at Uie drawings accompanying Problem 3.6. Is it possible to superimpose the cube on the dodecahedron Castieman and coworfcers have recently detected a cation with mje = 28, identified as TigC. It b bebeved that the titanium atoms form a cube with the addition of twelve carbon atoms to complete a pentagonal dodecahedron. Draw the proposed structure. What e its point group symmetry ... 

There are extensive possibilities for the formation of geometric and optical isomers in eight-coordinate complexes. Thus far. apparently only one pair has been completely characterized The diglyme [= di(2-methoxyethyl)etherl adduct of samarium iodide. Sml rO(CH2CH2OCH3)2]2. has been isolated in both cis and irons forms. The trans complex (Fig. 12.39a) has a center of symmetry Thus, the I—Sm—1 angle is exactly 180. and the molecule is a bicapped trigonal antiprism. The cis isomer (Fig. (2.39b) has ihe lower symmetry of a distorted dodecahedron with I—Sm—I angles of 92 > ... 

At low normalized bites (Figure 87 the minimum at 6A = 0B - 45° corresponds to the dodecahedron of overall molecular symmetry fly (Figure 91). 

Note that we can construct some ( 5,10, 3)-spheres that are 10i 6- Take Dodecahedron and select a set S of its edges, such that every 5-gon is incident to exactly one edge of this set Replacing those edges by the (3,3)-polycycles A2, we obtains such spheres. Up to isomorphism, there exist five such sets in Dodecahedron and they yield five ( 5,10, 3)-spheres 10i 6 with 140 vertices and symmetry groups Dm, C2, D2, D3, 7. ... 

The case b = 6 is the classical fullerene case. Theorem 2.2.2 gives that all 7(5,6)-fulleroids, i.e. fullerenes of icosahedral symmetry, are of the form GCk,i(Dodecahedron). See on Figure 19.1 the first three of the following smallest icosahedral fullerenes besides Dodecahedron ... 

A similar series (0 )n>o of 7(5, 8)-fulleroids of symmetry 7 is obtained from the series (GC2n+, o(Dodecahedron))n>a by replacing 6-gons by (5,3)-polycycles C3 in the same pattern as shown in Figure 19.4. 

We conclude that t>o must be incident only to 5-gons. Let vi be a vertex incident to any two of these faces. The third face, incident to t>i, cannot be a 5-gon, since this 5-gon would be in the same orbit as F and thus would contain a symmetric image t>2 of vo, i.e. a vertex meeting a 3-fold rotation axis. There are three essentially different possible positions for vi. In each case, by applying all symmetries we find that the resulting structure must be Dodecahedron. 

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