Fivefold symmetry

Prince E 1987 Diffraction patterns from tilings with fivefold symmetry Acfa Crystaiiogr.A AZ 393-400... 

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.

The cleft where this drug binds is inside the jelly roll barrel of subunit VPl. Most spherical viruses of known structure have the tip of one type of subunit close to the fivefold symmetry axes (Figure 16.15a). In all the picor-naviruses this position is, as we have described, occupied by the VPl subunit. Two of the four loop regions at the tip are considerably longer in VPl than in the other viral coat proteins. These long loops at the tips of VPl subunits protrude from the surface of the virus shell around its 12 fivefold axes (Figure 16.15b). 

A view down the fivefold symmetry axis of the icosahedtal structure (a) shows that the central capsomer is pentameric in shape and surrounded by five other capsomers as expected. The view down the pseudosixfold axis (h) shows, however, that the central capsomer is pentameric in shape and not hexameric as required for a T = 7 structure. (Adapted from 1. Rayment et al., Nature 295 110-115, 1982, hy copyright permission of Macmillan Magazines Limited.)... 

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

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.

Penrose tiling with a fivefold symmetry axis consisting of two kinds of rhomboid tiles... 

Unlike crystals that are packed with identical unit cells in 3D space, aperiodic crystals lack such units. So far, aperiodic crystals include not only quasiperiodic crystals, but also crystals in which incommensurable modulations or intergrowth structures (or composites) occur [14], That is to say, quasiperiodicity is only one of the aperiodicities. So what is quasiperiodicity Simply speaking, a structure is classified to be quasiperiodic if it is aperiodic and exhibits self-similarity upon inflation and deflation by tau (x = 1.618, the golden mean). By this, one recognizes the fact that objects with perfect fivefold symmetry can exist in the 3D space however, no 3D space groups are available to build or to interpret such structures. 

Cluster types. According to 1/1 AC structures, all three i-QCs types contain building blocks with local fivefold symmetry (Fig. 2). Obviously, a search for possible new QC/AC systems should look to precursors in which clusters with pseudoicosahedral symmetries are major structural motifs. 

The centered 10-vertex polyhedra are of particular interest since the shapes of the outer 10-vertex polyhedron depends on the interstitial atom and the electron count. In fact, four very different 10-vertex polyhedra (Fig. 7) have all been shown to form stable isolable species containing interstitial transition or post-transition metal atoms. These polyhedra include structures with three-, four-, or fivefold symmetry. Thus for the ions M Inio ° (M = Ni, Pd, Pt) found in the intermetal-lics KioInioM, the Injo polyhedron is a Cgv tetracapped trigonal prism [91]. 

The observed fivefold symmetry in the 1H and 13C NMR spectra even at very low temperature (— 150°C) with no line broadening leaves only two alternatives for the structure of the dication the nonclassical fivefold symmetrical, static structure 437 or... 

Opposite Several polyhedra with axes of fivefold symmetry. 

Two two-dimensional tiles that can be assembled into a tiling pattern with short-range fivefold symmetry. 

The fivefold symmetry has been reduced to symmetry through a plane. These symmetry orbitals can be found easily, merely by redoing the calculations using slightly modified values for C, (e.g. 1.01/7 for its resonance integral). 

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