Line group symmetry

Figure 2.15 line repetition symmetry groups and corresponding sequences of torsion angles for cis and trails polydienes. Position of mirror planes (m), inversion centers (i), and binary axes (2) along polymer chain are also indicated. Torsion angle of single bond, CHA-CHA, is assumed to be trails (T) in both cis and lruns polydienes. 

Fig. 54 Tiling patterns formed by self assembly of T-shaped bolaamphiphiles, their net-notations (top line) and plane group symmetries (highest symmetries, middle line) and reduced symmetries of their distorted variants (bottom line)...

The point group P C PBL and when P = PBL (which is so for a holosymmetric space group) the points and lines of symmetry mark out the basic domain il of the Brillouin zone. When... 

This treatment of symmetry demonstrates that the threshold above which significant autocorrelation is estimated to lie must be revised upward in line with the point group symmetry of the structure (van Heel et al, 2000 Orlova et al, 1997). Interestingly this results in the correct significance level for an icosahedral object being similar to the 0.5 correlation coefficient FSC criterion. 

Fi re 1.45. The distribution of the inversion centers in the triclinic space group symmetry P1. Eight independent centers are labeled from a through h . Inversion centers that are equivalent to one another are marked using symbols of the same size and shading. The invisible centers are drawn using dotted lines. 

If we plot some property of the ethane molecule, say, its potential energy, as a function of the torsion angle, we obtain a curve with threefold periodicity. In crystallographic parlance we have a repeating one-dimensional pattern with the line group pm and periodicity t = 120°. The special positions (fixed points) at a = 0° and 60° (modulo 120°) correspond to structures with special symmetry at 0° the eclipsed conformations with D31, symmetry, at 60° the staggered conformations with >3d symmetry. A general position corresponds to a chiral conformation with Z>3 symmetry only. Note that the order of D3 is half that of or 3d but the number of isometric Dj conformations is double the number of isometric 1)31, or conformations. 

Whyte [2-37] extended the definition of chirality as follows Three-dimensional forms (point arrangements, structures, displacements, and other processes) which possess non-superposable mirror images are called chiral . A chiral process consists of successive states, all of which are chiral. The two main classes of chiral forms are screws and skews. Screws may be conical or cylindrical and are ordered with respect to a line. Examples of the latter are the left-handed and right-handed helices in Figure 2-50. The skews, on the other hand, are ordered around their center. Examples are chiral molecules having point-group symmetry. 

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