## Chapter 1.5

### Construction Methods.  Desymmetrizations

By considering and comparing the development of construction methods for the derivation of ornamental structures in art and geometry, one can note a few common approaches. After considering regularities on which the simplest ornamental motifs (rosettes, friezes) are based, mostly on originals existing in nature, and after discovering the first elementary constructions, a way was opened for the creation of ornamental motifs. This was usually achieved beginning from "local symmetry" - from the one fundamental region and regularly arranged neighboring fundamental regions, and resulting in the "global symmetry" - complete ornamental filling in of the plane. Such a procedure represents, in fact, a series of extensions and dimensional transitions, leading directly or indirectly from the point groups - the symmetry groups of rosettes G20, over the line groups - the symmetry groups of friezes G21, to the plane groups - the symmetry groups of ornaments G2. In such a case, substructures (rosettes, friezes) are called generating substructures (Figure 1.15). A similar procedure can be traced for the similarity symmetry groups S20 and conformal symmetry groups C21 and C2, derived as extensions of isometric point groups - the symmetry groups of rosettes G20.

Figure 1.15
 Derivation of (a) frieze mm; (b) ornament pmm from generating rosette with the symmetry group D2.

Figure 1.16
 (a) Generating rosette with the symmetry group D4; (b) its external desymmetrization D4/C4; (c) antisymmetry group D4/C4.

Figure 1.17
 Weber diagrams of bands.

The first antisymmetry ornamental motifs are found in Neolithic ornamental art with the appearance of two-colored ceramics and for centuries have represented a suitable means for expressing the dualism, internal dynamism, alternation, with a distinct space component - a suggestion of the relationships "in front-behind", "above-below", "base-ground",¼

Figure 1.18
 (a) Colored symmetry group C4/C1; (b) D4/D2/C1.

By interpreting "colors" as physical polyvalent properties commuting with every transformation of the generating symmetry group, it is possible to extend considerably the domain of the application of colored ornaments treated as a way of modeling symmetry structures - subjects of natural science (Crystallography, Physics, Chemistry, Biology¼). As an element of creative artistic work, although being in use for centuries, colored symmetry can be, taking into consideration the abundance of unused possibilities, a very inspiring region. We find proof of this in the works of M.C. Escher (M.C. Escher, 1971a, b). On the other hand, the various applications of colors in ornaments, e.g., ornamental motifs based on the use of colors in a given ratio, by which harmony - balance of colors of different intensities - is achieved, have yet to find their mathematical interpretation. Accepting "color" as a geometric property, and colored transformations as geometric transformations which commute with the symmetries of the generating group, has opened up a large unexplored field for the theory of colored symmetry. This was made clear in the recent works discussing multi-dimensional symmetry groups, curvilinear symmetries, etc. (A.M. Zamorzaev, Yu.S. Karpova, A.P. Lungu, A.F. Palistrant, 1986).

The results of the theory of antisymmetry and colored symmetry can be used also for obtaining the minimal indexes of subgroups in the symmetry groups. As opposed to the finite groups, where for the index of the given subgroup there is exactly one possibility, in an infinite group the same subgroup may have different indexes. For example, considering a frieze with the symmetry group 11, generated by a translation X, and its colorings by N = 2,3,4,¼ colors, where the group of color permutations is the cyclic group CN of the order N, generated by the permutation c = (123¼N), the result of every such a color-symmetry desymmetrization is the symmetry group 11, i.e. the colored symmetry group 11/ 11. Therefore, we can conclude that the index of the subgroup 11 in the group 11 is any natural number N and that its minimal index is two. The results of computing the (minimal) indexes of subgroups in groups of symmetry, where the subgroups belong to the same category of symmetry groups as the groups discussed, based on the works of H.S.M. Coxeter and W.O.J. Moser (H.S.M. Coxeter, W.O.J. Moser 1980; H.S.M. Coxeter 1985, 1987) are completed with the results obtained by using antisymmetry and colored symmetry. They are given in the corresponding tables of (minimal) indexes of subgroups in the symmetry groups. Besides giving the evidence of all subgroups of the symmetry groups, these tables can serve as a basis for applying the desymmetrization method, because the (minimal) index is the (minimal) number of colors necessary to achieve the corresponding antisymmetry and color-symmetry desymmetrization. For denoting subgroups which are not normal, italic indexes are used (e.g., 3).

It is not necessary to set apart antisymmetry from colored symmetry, since antisymmetry is only the simplest case of colored symmetry (N = 2), but their independent analysis has its historical and methodical justification, because bivalence is the fundamental property of many natural and physical phenomena (electricity charges +, -, magnetism S, N, etc.) and of human thought (bivalent Aristotelian logic). In ornamental art, examples of antisymmetry are mainly consistent in the sense of symmetry, while consistent use of colored symmetry is very rare, especially for greater values of N.