Icosahedral symmetric

Fig. 9 The Nii2 icosahedron inside the AS20 dodecahedron in the icosahedrally symmetric...

Figure 7-19 Schematic icosahedrally symmetric structure with 180 subunits. The quasi-equivalent units A, B, and C are necessarily somewhat differently positioned with respect to their neighbors and must therefore assume different conformations in order to fit together tightly. From Harrison.68...

In the cooperative case, the two molecular transitions are separately allowed under well-known two-photon selection rules, since each molecule absorbs one laser photon and either emits or aosorbs a virtual photon. In the same way, the distributive case provides for excitation through three-and one-photon allowed transitions, and may thus lead to excitation of states that are formally two-photon forbidden. (In general, it is sufficient to stipulate that both transitions involved in the distributive mechanism are one-photon allowed since, with the rare exception of icosahedrally symmetric molecules, all transitions which are one-photon allowed are of necessity also three-photon allowed (Andrews and Wilkes 1985).)... 

Supramolecular chirality is widely manifested in nature for example, the DNA double helix,the protein single helices and, in humans, rhinovirus 14, a member of the major rhinoviras receptor class, possesses a protein capsid that is composed of 60 protomers arranged in an icosahedrally symmetric arrayJ As shown previously in this book, at the molecular level, chirality is very important in asymmetric catalysis for the creation of novel chiral molecules however, more recently an increasing amount of attention has been drawn to chiral supramolecular assemblies. On the supramolecular level, chirality involves the nonsymmetric arrangement of molecular subunits in a noncovalent assembly via weak interactions such as hydrogen bonding, metal coordination and n-n interaction. 

In CMAs, the majority of atom coordinations are icosahedrally symmetric. Frequently, one finds a decoration of certain atomic positions with concentric shells of icosahedral polyhedra around a central position. These arrangements are referred to as cluster substructure or clusters in short. Three types of cluster substructure, based on Bergman, Mackay, and Friauf polyhedra, are found particularly often in CMAs and can be used for a classification of the latter [15]. 

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... 

FIGURE 6.44 Several possible symmetric arrays of identical protein snbnnits, inclnding (a) cyclic symmetry, (b) dihedral symmetry, and (c) cubic symmetry, inclnding examples of tetrahedral (T), octahedral (O), and icosahedral (I) symmetry. (Irving GAs)... 

Virus symmetry The nucleocapsids of viruses are constructed in highly symmetrical ways. Symmetry refers to the way in which the protein morphological units are arranged in the virus shell. When a symmetrical structure is rotated around an axis, the same form is seen again after a certain number of degrees of rotation. Two kinds of symmetry are recognized in viruses which correspond to the two primary shapes, rod and spherical. Rod-shaped viruses have helical symmetry and spherical viruses have icosahedral symmetry. 

The scheme of cluster condensation or cluster fragment condensation leads eventually to structures observed in bulk metals. Particularly through extensive condensation of tetrahedral and octahedral clusters, arrangements closely related to the hexagonal and cubic close-packed structures can be obtained. Condensation also of icosahedral five-fold symmetrical clusters may be related to crystalline and quasicrystalline metallic structures. 

It was discovered, however, that the spherical aromaticity of the icosahedral fullerenes C20, Cjq and CgQ depends on the filling of the Jt-sheUs with electrons [107]. As pointed out in Section 14.3.1 no distortion of the cage structure is expected in these fullerenes if their shells are fully filled. Closed-shell situations are realized if the fullerene contains 2(N -1-1) Jt electrons. This is closely related to the stable noble-gas configuration of atoms or atomic ions [108]. In this case the electron distribution is spherical and all angular momenta are symmetrically distributed. Correlation of the aromatic character determined by the magnetic properties is shown in Table 14.3. 

Figure 3. Polymorphism of deltahedral surface lattices. The T = 4 icosadeltahedron at the left (A80, point group Ih) is transformed to a A80 with Dbh symmetry (middle). At the right, the half-icosahedral cap defined by the h,k = 10,0 circumferential vector has been extended by adding rings of 10 V6 connectors. The bottom of this tube could be capped symmetrically (as for the A80 models) or asymmetrically using the h,k = 10,0 cap shown in figure 4 or the two other h,k = 10,0 caps listed in table 1.

Figure 2.42 Electron microscope image of the highly symmetrical icosahedral rotavirus (scale bar = 100 nm).

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