TWO- AND THREE-DIMENSIONAL GROUPS OF HYPER TABLET P-SYMMETRIES AND THEIR APPLICATION. A.M.Zamorzaev and A.F.Palistrant, Geometry Department of Moldova State University. Kishinev. Moldova

One of the first extensions of the notion of symmetry is Shubnikov theory of antisymmetry, used as the basis for various applications in geometrical crystallography and its generalizations. Interpretation of antisymmetry as two-colored symmetry resulted in the idea of p-symmetry; the other generalization of antisymmetry - the multiple antisymmetry is obtained ascribing to the points of a figure not only one, but several qualitatively different sign + or -. Diverse approaches to the colored antisymmetry introduced by Pawley, Neronova and Belov, and their further generalization - cryptosymmetry of Niggli and Wondratchek are the syntesis of the both.

The mentioned generalizations of antisymmetry and colored symmetry are included in P-symmetry, using arbitrary number of colors p (not only p=2, as antisymmetry) assigned to the points of a figure, and arbitrary group P of color-permutations (not only cyclic groups, as Belov psymmetry. Belov p-symmetry is a particular case of P-symmetry with cyclic color-permutation group P={(12...p)} , and Shubnikov antisymmetry is treated as 2-symmetry.

Signs and indexes ascribed to the points of a figure posses the extra geometrical sense with regard to the space in which the symmetry group acts; in additional dimensions the signs and indexes can be interpreted geometrically. From this result the possibility to use two-dimensional and three-dimensional P-symmetry groups for modeling certain categories of multidimensional symmetry groups.

The more detailed explanation of the ideas mentioned, the recent methods of applying two-dimensional and three-dimensional symmetry groups of rosettes, tablets and hyper-tablets, as well, as crystallographic and hyper-crystallographic P-symmetry to the study of multidimensional symmetry groups will be given in this communication.