Local and global color symmetries of a symmetrical pattern

This study addresses the problem of arriving at transitive perfect colorings of a symmetrical pattern consisting of disjoint congruent symmetric motifs. The pattern 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 ignoring local symmetries and considering only elements of G. A framework is presented to systematically arrive at what Roth [Geom. Dedicata (1984), 17, 99–108] defined as a coordinated coloring of , 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 , the symmetry of that is both a global and local symmetry, effects the same permutation of the colors used to color and the corresponding motif, respectively.