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 "

Derivation of (a) frieze |

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

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 "

(a) Generating rosette with the symmetry group D/_{4}C; (c)
antisymmetry group _{4}D/_{4}C.
_{4} |

Let *e*_{1} be an * antiidentity transformation* which
satisfies the relations:

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

The next generalization of antisymmetry is the
polyvalent, * colored symmetry* with the number of "colors"
N ³ 3, where each color is denoted by the corresponding index
1,2,¼,

(a) Colored symmetry group C; (b) _{1}
D/_{4}D/_{2}C.
_{1} |

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