Symmetry icosahedral

Molecules with icosahedral symmetry are not new but the discovery of the newest of them, Ceo or buckminsterfiillerene, has had such a profound effect on chemistry in recent years that 1 thought it useful fo include a discussion of fhe icosahedral poinf group fo which Ceo belongs. 

The protein shells of spherical viruses have icosahedral symmetry... 

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

In the T = 4 structure there are 240 subunits (4 x 60) in four different environments, A, B, C, and D, in the asymmetric unit. The A subunits interact around the fivefold axes, and the D subunits around the threefold axes (Figure 16.7). The B and C subunits are arranged so that two copies of each interact around the twofold axes in addition to two D subunits. For a T = 4 structure the twofold axes thus form pseudosixfold axes. The A, B, and C subunits interact around pseudothreefold axes clustered around the fivefold axes. There are 60 such pseudothreefold axes. The T = 4 structure therefore has a total of 80 threefold axes 20 with strict icosahedral symmetry and 60 with pseudosymmetry. 

The S domains form the viral shell by tight interactions in a manner predicted by the Caspar and Klug theory and shown in Figure 16.8. The P domains interact pairwise across the twofold axes and form protrusions on the surface. There are 30 twofold axes with icosahedral symmetry that relate the P domains of C subunits (green) and in addition 60 pseudotwofold axes relating the A (red) and B (blue) subunits (Figure 16.9). By this arrangement the 180 P domains form 90 dimeric protrusions. 

Small spherical viruses have a protein shell around their nucleic acid that is constructed according to icosahedral symmetry. Objects with icosahedral symmetry have 60 identical units related by fivefold, threefold, and twofold symmetry axes. Each such unit can accommodate one or severed polypeptide chains. Hence, virus shells are built up from multiples of 60 polypeptide chains. To preserve quasi-equivalent symmetry when packing subunits into the shell, only certain multiples (T = 1, 3, 4, 7...) are allowed. 

The capsids of polyoma virus and the related SV40 have icosahedral symmetry, with 72 pentameric assemblies of the major capsid protein. The pentamers are linked to their neighbors by flexible arms, with a p strand that augments a p sheet in the invaded pentamer. These flexible arms allow the pentamers to be linked together with both fivefold and sixfold symmetry. 

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. 

The fivefold symmetry discovered by Shechtman is modelled in terms of the stacking of icosahedra and the term icosahedral symmetry is sometimes used. 

A pecuhar sohd phase, which has been discovered not too long ago [172], is the quasi-crystalline phase. Quasi-crystals are characterized by a fivefold or icosahedral symmetry which is not of crystallographic type and therefore was assumed to be forbidden. In addition to dislocations which also exist in normal crystals, quasi-crystals show new types of defects called phasons. Computer simulations of the growth of quasicrystals [173] are still somewhat scarce, but an increasing number of quasi-crystalline details are studied by simulations, including dislocations and phasons, anomalous self-diffusion, and crack propagation [174,175]. 

The first complex intermetallic compound found to have large clusters of atoms with local icosahedral symmetry was Mg32Al4, which has 162 atoms in a body-centred cubic unit17. The unit cube contains 98 icosahedra, 20 Friauf polyhedra and 44 others. 

FIG. 2. A complex of twenty Friauf polyhedra, with icosahedral symmetry (Samson, Ref. 23). This complex contains 104 atoms, if the central icosahedral position is not occupied. Most of the atoms show approximate icosahedral ligan-cy twenty atoms, at the centers of the Friauf polyhedra, have ligancy 15 or 16. The complex was first identified in Mg32(Al,Zn)<9. In the cubic crystals that form the icosatwins and decatwins these complexes are packed in such a way as to approximate an icosahedral arrangement of twelve complexes about a central one, the structure being similar to that of 0-W. 

The structure factor for the 104-atom complex with almost perfect icosahedral symmetry determines the intensities of the diffraction maxima, in correspondence with the inverse relationship between intensity in reciprocal space and the atom-pair vectors in real space that was discovered fifty years ago by Patterson.27 The icosahedral nature of the clusters in the cubic crystal explains the appearance of the Fibonacci numbers and the golden ratio. 

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. 

Fig. 9.2 (a) The extended icosahedral symmetry in Ga,2 Li2o Cu,2 Ga6o and (b) packing of such units in the cubic cell (Im3) of Lin3Cu6Ga2i-... 

The geometry of the capsomeres results in their assembly into particles exhibiting one of two different architectural styles—helical or icosahedral symmetry (Fig. 3.2). 

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