Local and global color symmetries of a symmetrical pattern

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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.