Icosahedron point group

The point groups T, and /j. consist of all rotation, reflection and rotation-reflection synnnetry operations of a regular tetrahedron, cube and icosahedron, respectively. 

The 4 point group is that to which a regular icosahedron, illustrated in Figure 4.13(a), belongs. It contains 20 equilateral triangles arranged in a three-dimensional sttucture. This is the conformation of the anion, in which there is a boron atom, with a hydrogen atom... 

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. 

I. Croups with very high symmetry. These point groups may be defined by the large number of characteristic symmetry elements, but most readers will recognize them immediately as Platonic solids of high symmetry, a. Icosahedrd, Ik.—The icosahedron (Fig. 3.10a), typified by the B12H 2 ion (Fig. 3.10b), has six C3 axes, ten C3 axes, fifteen C2 axes, fifteen mirror... 

The point groups Td,Oh, and / are the respective symmetry group of Tetrahedron, Cube, and Icosahedron the point groups T, O, and I are their respective normal subgroup of rotations. The point group 7 is generated by T and the central symmetry inversion of the centre of the Isobarycenter of the Tetrahedron. 

The symmetry of many molecules and especially of crystals is immediately obvious. Benzene has a six-fold symmetry axis and is planar, buckminsterfullerene (or just fullerene or footballene) contains 60 carbon atoms, regularly arranged in six- and five-membered rings with the same symmetry (point group //,) as that of the Platonic bodies pentagon dodecahedron and icosahedron (Fig. 2.7-1). Most crystals exhibit macroscopically visible symmetry axes and planes. In order to utilize the symmetry of molecules and crystals for vibrational spectroscopy, the symmetry properties have to be defined conveniently. 

Figure 2.7-1 Three objects belonging to the same point group //, the pentagondodecahedron, the icosahedron, and the buckminsterfullerene.

Fig. 3.10 Point groups and molecules of high symmetry (a) icosahedron, (b) the BijHi ion. (c) octahedron, (d) sulfur hexaflucride, (e) bexacyanocoballatc(lll) amon. (f) tetrahedron, (g) ammonium cation, and (h> teirafhioroborate anion.

Figure 4.5 The five Platonic solids (a) regular tetrahedron (b) regular octahedron (c) regular cube (d) regular icosahedron (e) regular dodecahedron. The point group symbol for each solid is given below each diagram...

The two most complex Platonic solids, the icosahedron and dodecahedron, (Figure 4.5 d, e), both have 5 (five-fold inversion) axes. In these a vertex is rotated by 12°, (360/5)°, and then translated by the height of the solid. In the case of a dodecahedron a 5 axis passes through each vertex, while in the case of a regular icosahedron, a 5 axis passes through the centre of each face. The point group symbol for both of these solids is 2/m 3 5, sometimes written as 5 3 2/m. 

For example, for a cube P-C + F = 8-12 + 6 = 2, as required. Similarly for an octahedron P-C + F = 6-12 + 8=2. The octahedron, moreover, is a geometrical dual of the cube, because the role of P and F are interchanged in the two structures. Thus the 8 ppints of the cube correspond to the 8 faces of the octahedron, and the 6 faces of the cube correspond to the 6 points of the octahedron. The duals share exactly the same value C (=12), and exactly, the same point group symmetry (Oh). For any case where the Descartes-Euler formula applies, duals are defined by the interchange of P and F, C held constant. Thus, Figure 1 indicates the dodecahedron and icosahedron are duals, and the tetrahedron is self-dual. 

The point group symmetries for which JT distortions are of interest are the Di,h symmetry of the square, the Td symmetry of the tetrahedron, the O/, symmetry of the cube or octahedron, and the //, symmetry of the icosahedron, dodecahedron, or Ceo- The important features of the top-reps of these JT distortions are discussed below. 

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