Symmetry icosahedron

Figure 1.11 The formation of metal clusters during the nucleation of a new phase. The co-ordination is first tetrahedral, leading to 5-fold symmetry, until the 13-atom icosahedron is formed which transforms into the cubic icosahedron of the stable phase...

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.

Any symmetric object is built up from smaller pieces that are identical and that are related to each other by symmetry. An icosahedron can therefore be divided into a number of smaller identical pieces called symmetry-related units. Protein subunits are asymmetric objects hence, a symmetry axis cannot pass through them. The minimum number of protein subunits that can form a virus shell with icosahedral symmetry is therefore equal to... 

The symmetry properties of an icosahedron are not restricted to the surface but extend through the whole volume. An asymmetric unit is therefore a part of this volume it is a wedge from the surface to the center of the icosahedron. Sixty such wedges completely fill the volume of the icosahedron. 

The asymmetric unit of an icosahedron can contain one or several polypeptide chains. The protein shell of a spherical virus with icosahedral symmetry... 

Figure 16.4 The division of the surface of an icosahedron into asymmetric units, (a) One triangular face is divided into three asymmetric units into which an object is placed. These are related by the threefold symmetry axis.

Figure 16.S Schematic illustration of the way the 60 protein subunits are arranged around the shell of safellite tobacco necrosis virus. Each subunit is shown as an asymmetric A. The view is along one of the threefold axes, as in Figure 16.3a. (a) Three subunifs are positioned on one triangular tile of an Icosahedron, in a similar way to that shown in 16.4a. The red lines represent a different way to divide the surface of the icosahedron into 60 asymmetric units. This representation will be used in the following diagrams because it is easier to see the symmetry relations when there are more than 60 subunits in the shells, (b) All subunits are shown on the surface of the virus, seen in the same orientation as 16.4a. The shell has been subdivided into 60 asymmetric units by the red lines. When the corners are joined to the center of the virus, the particle is divided into 60 triangular wedges, each comprising an asymmetric unit of the virus. In satellite tobacco necrosis virus each such unit contains one polypeptide chain...

Fullerenes are described in detail in Chapter 2 and therefore only a brief outline of their structure is presented here to provide a comparison with the other forms of carbon. The C o molecule, Buckminsterfullerene, was discovered in the mass spectrum of laser-ablated graphite in 1985 [37] and crystals of C o were fust isolated from soot formed from graphite arc electrodes in 1990 [38]. Although these events are relatively recent, the C o molecule has become one of the most widely-recognised molecular structures in science and in 1996 the codiscoverers Curl, Kroto and Smalley were awarded the Nobel prize for chemistry. Part of the appeal of this molecule lies in its beautiful icosahedral symmetry - a truncated icosahedron, or a molecular soccer ball, Fig. 4A. 

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... 

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.

A further complication is that an icosahedron sharing four faces loses its symmetry operations of the second sort and becomes chiral, D or L. The eclipsed configuration described above requires that D and l alternate. Hence, in a ring of five icosahedra one DD or LL bond must occur, with a strain in the ring of 40°, which is a rotation of 4 per shared face in each icosahedron. The space-group symmetry is thus changed from that of diamond to a subgroup. 

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. 

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). 

The dodecahydrododecaborate anion, B H 2-, is termed unique with considerable justification. This ion and its perhalo derivatives, e.g., Bi2C1i22-, are the most symmetrical molecular aggregates known. The boron atoms occupy the vertices of a regular icosahedron and each is bonded terminally to a hydrogen atom all boron atoms are environmentally equivalent.7,8 This anion is the only known example of the 7 symmetry group.8 General spectral, physical, and chemical properties of Bx2Hi22-are detailed in a paper by Muetterties et al.9... 

The molecular structure of the closo 12-vertex 1-SBnHn cluster (17) has been studied by electron diffraction methods augmented by ab initio calculations. Substantial distortions away from the regular icosahedron occurred by expansion of the pentagonal belt adjacent to sulfur.73 The UV-PES spectra of 17 has also been reported,71 and its microwave spectrum has been investigated and demonstrated that the molecule had C5v symmetry.74... 

The effect of ligand-field symmetries is less easily understood. In Fe3(CO)12, two Fe-Fe distances of 2.56 A and 2.68 A are observed. It may be argued that these distances reflect the packing of a triangle within the icosahedron, each polyhedron undergoing distortion until a common C2v symmetry is reached. 

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