Local and global color symmetries of a symmetrical pattern

This study addresses the problem of arriving at transitive perfect colorings of a symmetrical pattern P consisting of disjoint congruent symmetric motifs. The pattern P has local symmetries that are not necessarily contained in its global symmetry group G. The usual approach in color symmetry theory is to arrive at perfect colorings of P ignoring local symmetries and considering only elements of G. A framework is presented to systematically arrive at what Roth defined as a coordinated coloring of P, a coloring that is perfect and transitive under G, satisfying the condition that the coloring of a given motif is also perfect and transitive under its symmetry group. Moreover, in the coloring of P, the symmetry of P that is both a global and local symmetry, effects the same permutation of the colors used to color P and the corresponding motif, respectively.