Local and global color symmetries of a symmetrical pattern

Abstract

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 [Geom. Dedicata (1984), 17, 99-108] 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.

Global Symmetries, Local Symmetries and Groupoids

“…Color symmetry is encountered in the arts and crystallography [24][25][26][27][28][29]. In a recent paper [19], groupoids were used to exchange or move a number of colors without leaving any fixed color. While the groupoid approach is relevant for color symmetry, these color changes can also be achieved without groupoids, through suitable object choices, as exemplified in Section 3.2.2.…”

“…Local symmetries also appear outside the context of spacetime. They are encountered in arts [19], and have been applied in graph theory [20][21][22][23]. In crystallography and in the arts, symmetries are usually handled via subgroups of color groups [24][25][26][27][28][29], although modern approaches to color symmetry involve groupoids [19].…”

“…They are encountered in arts [19], and have been applied in graph theory [20][21][22][23]. In crystallography and in the arts, symmetries are usually handled via subgroups of color groups [24][25][26][27][28][29], although modern approaches to color symmetry involve groupoids [19]. Similarly, subgroups were used to handle local graph symmetries [20,23], and groupoids were also used to handle local symmetries in networks [21,22].…”