Icosahedron, symmetry elements

Figure 6.1 The icosahedron and some of its symmetry elements, (a) An icosahedron has 12 vertices and 20 triangular faces defined by 30 edges, (b) The preferred pentagonal pyramidal coordination polyhedron for 6-coordinate boron in icosahedral structures as it is not possible to generate an infinite three-dimensional lattice on the basis of fivefold symmetry, various distortions, translations and voids occur in the actual crystal structures, (c) The distortion angle 0, which varies from 0° to 25°, for various boron atoms in crystalline boron and metal borides.

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. 

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... 

Construct models of the tetrahedron, the octahedron (both with and without blades representing dielate rings), and the icosahedron (Appendix HI Find as many symmetry elements as you can. 

Figure B.2 shows polyhedra commonly encountered. The five Platonic (or regular) solids are shown at the top. Beside the octahedron and cube, the octahedron is shown inside a cube, oriented so the symmetry elements in common coincide. These solids are conjugates one formed by connecting the face centers of the other. The tetrahedron is its own conjugate, because connecting the face centers gives another tetrahedron. The icosahedron and pentagonal dodecahedron are conjugates. The square antiprism and trigonal...

The Pentagonal Dodecahedron and the Icosahedron. These bodies (Fig. A5-8) are related to each other in the same way as are the octahedron and the cube, the vertices of one defining the face centers of the other, and vice versa. Both have the same symmetry operations, a total of 120 We shall not list them in detail but merely mention the basic symmetry elements six C5 axes ten C3 axes, fifteen C2 axes, and fifteen planes of symmetry. The group of 120 operations is designated Ih and is often called the icosahedral group. 

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 regular icosahedron, the fourth of the regular polyhedra, has twenty equilateral triangular faces, thirty edges, and twelve corners. Its name is derived from the Greek eikosi, twenty, and hedra, seat or base. It has many symmetry elements, including six fivefold axes of rotational symmetry, ten threefold axes, and fifteen twofold axes. 

The pentagonal dodecahedron is the fifth regular polyhedron. It has thirty edges, twenty corners, and twelve faces, which are regular pentagons. It is closely related to the icosahedron the relation involves interchanging corners and faces. Its symmetry elements are the same as those for the icosahedron. 

Our model [25,33,37 39,48] relies on a physical picture (see Fig. 1) that (i) there exist distinct locally favored structures in any liquids and (ii) such structures are formed in a sea of normal liquid structures and its number density increases upon cooling since they are energetically more favorable by A than normal liquid structures. The specific volume and the entropy are larger and smaller for the former than the latter, respectively, by Av and Act. We identify locally favored structures as a minimum structural unit (symmetry element). It is tetrahedral order for water-type liquids, whereas icosahedron for metallic glass formers [60]. To express such short-range bond ordering in liquids, we introduce the so-called bond-orientational order parameter Qim [64,65]. 

The degree of symmetry that the point group represents is given by the order, h, whieh is simply the sum of the number of symmetry elements that the point group possesses. For Cav /i = 4 for h = 7A and for Oh / = 48. The highest (non-infinite) symmetry group encountered in chemistry is the icosahedron, 4 (order h = 120), which describes a polyhedron sometimes encountered in cluster chemistry, e.g. and... 

Boron is unique among the elements in the structural complexity of its allotropic modifications this reflects the variety of ways in which boron seeks to solve the problem of having fewer electrons than atomic orbitals available for bonding. Elements in this situation usually adopt metallic bonding, but the small size and high ionization energies of B (p. 222) result in covalent rather than metallic bonding. The structural unit which dominates the various allotropes of B is the B 2 icosahedron (Fig. 6.1), and this also occurs in several metal boride structures and in certain boron hydride derivatives. Because of the fivefold rotation symmetry at the individual B atoms, the B)2 icosahedra pack rather inefficiently and there... 

Lithium has been alloyed with gaUium and small amounts of valence-electron poorer elements Cu, Ag, Zn and Cd. like the early p-block elements (especially group 13), these elements are icosogen, a term which was coined by King for elements that can form icosahedron-based clusters [24]. In these combinations, the valence electron concentrations are reduced to such a degree that low-coordinated Ga atoms are no longer present, and icosahedral clustering prevails [25]. Periodic 3-D networks are formed from an icosahedron kernel and the icosahedral symmetry is extended within the boundary of a few shells. 

The hexagonal unit cell of the a-rhombohedral form of elemental boron (a-R12 boron) contains 36 B atoms that form three B12 icosahedra. The B12 icosahedron occupies Wyckoff position 3(a) of site symmetry 3m, so that the asymmetric unit contains two independent B atoms in position 18(/i) of site... 

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