Chapter 1.6

 Symbols of Symmetry Groups

When denoting symmetry groups and their generalizations, antisymmetry and colored symmetry groups, we always come across the unpleasant task of trying, at least to some extent, to reconcile and bring to accord the different sources and symbols used in literature. Most of symbols come from the work of crystallographers, some from the mathematicians who were engaged in studies of the theory of symmetry, while some chapters (e.g., that on conformal symmetry) demand the introduction of new symbols. Since only lately there have been attempts to make uniform the symbols of symmetry groups, positive results are mainly achieved with the symmetry groups of ornaments G2 (International symbols). In the other cases, a great number of authors, with their original results introduced together new or modified symbols. Therefore, it is unavoidable to accept the compromise solution and quote several alternative kinds of symbols. Also, this offers possibilities for the application of optimal symbols in each particular case, since for the different practical needs of the theory of symmetry, every kind of symbols has its advantages, but also, disadvantages.

For denoting the symmetry groups of friezes and ornaments, the simplified version of the International symbols (M. Senechal, 1975; H.S.M. Coxeter, W.O.J. Moser, 1980) will be used, while in other cases the non-coordinate symbols, used by Soviet authors (A.V. Shubnikov, V.A. Koptsik, 1974) will be indicated also. The symbols of antisymmetry and colored symmetry groups will be given in the group/subgroup notation (G/H, G/H/H1) (A.V. Shubnikov, V.A. Koptsik, 1974; A.M. Zamorzaev, 1976; H.S.M. Coxeter, 1985, 1987; V.A. Koptsik, J.N. Kotzev, 1974).

The International symbols are coordinate symbols of symmetry groups. For the symmetry groups of friezes and ornaments, the first coordinate denotes the translational subgroup p (c with the rhombic lattice) while the other coordinates are symbols of glide reflections g and reflections m perpendicular to the corresponding coordinate axis and symbols of the rotation axis n collinear with the corresponding coordinate axis.

The non-coordinate symbols of symmetry groups are mainly used in the works of Soviet authors, in which (a) denotes a translation, (ã) a glide reflection, n the order of a rotation, the absence of symbols between elements - collinearity (incidence) of relevant elements of symmetry (denoted in the original works by the symbol ), while the symbol : denotes perpendicularity of relevant symmetry elements. For example, the symmetry groups D4 and C4 will be denoted, respectively, by 4 and 4m.

Given at the beginning of each chapter is a survey of the geometric-algebraic characteristics of the groups of symmetry discussed: presentation, group order, group structure, reducibility, form of the fundamental region, enantiomorphism, polarity (non-polarity, bipolarity), group-subgroup relations, table of minimal indexes of subgroups in groups, Cayley diagrams. Further on are discussed the antisymmetry and color-symmetry desymmetrizations, construction methods, questions related to continuous groups and to different problems of algebraic-geometric properties of symmetry groups, which directly influence the different visual parameters.