Chapter 1.5

 Construction Methods.

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.

However, an almost equal role in the formation of different ornamental motifs belongs to the reverse desymmetrization procedure, a way which mainly leads from the maximal symmetry groups generated by reflections, characterized by a high degree of visual and constructional simplicity, to their subgroups. The results obtained are subgroups belonging to the same category as a group undergoing desymmetrization, or its subgroups with invariant subspace(s) of lower dimension(s) (e.g., the symmetry groups of friezes G21 as line subgroups of the symmetry groups of ornaments G2). At first restricted to the maximal groups of symmetry generated by reflections, to the regular tessellations or Bravais lattices, the desymmetrization method in painting becomes in time, firstly thanks to the use of colors, an efficient procedure for deriving all symmetry groups as subgroups of wider groups. Under the term "desymmetrization" of certain symmetry group we understand this as the procedure beginning with the elimination of corresponding symmetries and resulting in the derivation of certain subgroup H of the given group. In line with this, every desymmetrization is defined by the group G and its subgroup H, i.e. by the group/subgroup symbol G/H. The reverse procedure, resulting in some supergroup of the given group G, is called symmetrization (group extension).

Within the desymmetrization method, we can, depending on the desymmetrization means used, distinguish classical-symmetry (non-colored), antisymmetry and color-symmetry desymmetrizations. Under the term "classical-symmetry desymmetrization" (non-colored desymmetrization) we will discuss all desymmetrizations realized, for example, by using an asymmetric figure belonging to the fundamental region, or by deleting their boundaries and joining two or more adjacent fundamental regions, etc. The term "non-colored" used as the alternative for "classical-symmetry", should not be understood literally, since it does not prohibit the use of colors or some of their equivalents (e.g., indexes), but includes as well, all other cases where colors have been used for a desymmetrization without resulting in some antisymmetry or color-symmetry group. In the same sense we will use the term "classical theory of symmetry" which denotes the theory of symmetry without its generalizations - antisymmetry and colored symmetry. The term "external desymmetrization" will be used to denote a desymmetrization achieved by varying boundaries of a fundamental region (Figure 1.16b).

Figure 1.16

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

Let e1 be an antiidentity transformation which satisfies the relations: e12 = E     e1S = Se1, where S is any symmetry transformation. The transformation S' = e1S is then called an antisymmetry transformation. As the interpretation of the transformation e1, it is possible to accept the alternating change of any bivalent quality, geometric or not, which commutes with symmetries, e.g., the color change black-white, change of electricity charges +, -, etc. A group which besides symmetry transformations contains antisymmetry transformations is called an antisymmetry group. As the basis for deriving antisymmetry groups we take some symmetry group G which we call a generating group of antisymmetry (or simply a generating group). By replacing the symmetries (generators) of the group G by antisymmetries (antigenerators) we obtain, as a result, an antisymmetry group G' which, depending upon whether the antiidentity transformation e1 is the element of the group G' or not, is called a senior (neutral, gray) or a junior (black-white) antisymmetry group respectively. Every senior antisymmetry group has the form G' = G×{e1} = G×C2, where the group generated by e1 is denoted by {e1}. All junior groups are isomorphic with their generating group G. Every junior antisymmetry group is uniquely defined by the generating group G and by its subgroup H of the index 2. From there originated the group/subgroup symbols G/H of junior antisymmetry groups, where the relationship G/H @ C2 holds (Figure 1.16c). Since all (normal) subgroups of the index 2 of the generating group G can be obtained knowing junior antisymmetry groups derived from G, antisymmetry is included in the desymmetrization method. Besides a large field of application in Physics, various interpretations of the antiidentity transformation as a geometric transformation which commutes with all the symmetries of the generating group, make possible the dimensional transition from the symmetry groups of the n- dimensional space to those of the (n+1)-dimensional space. For example, the symmetry groups of bands G321 can be derived by using antisymmetry from the symmetry groups of friezes G21, the symmetry groups of layers G32 from the symmetry groups of ornaments G2, etc. Corresponding black-white antisymmetry plane motifs (so-called Weber diagrams or antisymmetry mosaics) can be understood as adequate visual interpretations of the symmetry groups of bands G321 or layers G32, where the transformation e1 - color change black-white is identified with the plane reflection in the invariant plane of the generating frieze or ornament (Figure 1 .17).

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",

The next generalization of antisymmetry is the polyvalent, colored symmetry with the number of "colors" N 3, where each color is denoted by the corresponding index 1,2,,N. A permutation of the set {1,2,,N} is any one-to-one mapping of this set onto itself. Let PN be a subgroup of the symmetric permutation group SN (or simply symmetric group), i.e. of the group of all the permutations of the set {1,2,,N}, c PN and cS = Sc, where S is a symmetry transformation, an element of the symmetry group G. Then S* = cS is called a colored symmetry transformation. A color permutation c can be interpreted as a change of any polyvalent quality which commutes with symmetries S G. A colored symmetry group is a group which besides symmetry transformations contains colored symmetry transformations (or colored symmetries). By analogy to antisymmetry groups, the symmetry group G is called a generating group of colored symmetry. The colored symmetry group G* derived from G is called a junior colored symmetry group iff it is isomorphic with G. In this work only junior colored symmetry groups are discussed. Every junior colored symmetry group can be defined by the ordered pair (G,H) which consists of the group G and its subgroup H of the index N, i.e. [G:H] = N. Two groups of colored symmetry (G,H) and (G',H') are equal if there exists an isomorphism i(G) = G' which maps H onto H' (R.L.E. Schwarzenberger, 1984). For N = 2 and PN = C2, (G,H) is an antisymmetry group. A color permutation group PN is called regular if it does not contain any transformation, distinct from the identity permutation, which keeps invariant an element of the set {1,2,,N}. If it contains such a transformation, a color permutation group is called irregular. Depending upon whether the color permutation group PN is regular or not, we can distinguish two cases. For a regular group PN every colored symmetry group is uniquely defined by the generating group G and its normal subgroup H of index N - the symmetry subgroup of G*. This results in the group/subgroup symbols of the colored symmetry groups G/H, and [G:H] = N (Figure 1.18a). For the irregular group PN, besides G and H we must consider also the subgroup H1 of the group G*, which maintains each individual index (color) unchanged (i.e. group of stationariness of colors). In this case H is not a normal subgroup of G. The order of the group PN is NN1, where [G:H] = N, [H:H1] = N1 and quotient group G/H1 @ PN. To denote such colored symmetry groups, the symbols G/H/H1 are used (Figure 1.18b).

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, Biology). 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, colors, where the group of color permutations is the cyclic group CN of the order N, generated by the permutation c = (123N), 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.