SYMMETRY GROUPS IN

AND ORNAMENTAL ART

The isometric symmetry groups in the plane *E*^{2} can be
classified according to the spaces invariant with respect to the
action of transformations of the groups in question. Bohm symbols
have been used to denote the corresponding categories of symmetry
groups (J. Bohm, K. Dornberger-Schiff, 1966). In the symbol
*G*_{nst¼}, the first subscript *n* represents the maximal
dimension of space in which the transformations of the symmetry
group act, while the following subscripts *st*¼ represent
the maximal dimensions of subspaces that are invariant with
respect to the action of transformations of the symmetry group
and that are properly included in each other. These symbols
represent also the definitions of the corresponding categories
of isometric symmetry groups in *E*^{2}: the symmetry groups of
finite friezes *G*_{210}, rosettes *G*_{20}, friezes *G*_{21},
and ornaments *G*_{2}. In line with the relation
*G*_{210} Ì *G*_{20}, and to simplify things, the category *G*_{210} will not
be discussed individually but within the category *G*_{20}.

Antisymmetry and colored symmetry, the extensions of the
theory of symmetry, will be used only for a more detailed
analysis of the symmetry groups in *E*^{2}.