This study addresses the problem of arriving at transitive perfect colorings of a symmetrical pattern {\cal P} consisting of disjoint congruent symmetric motifs. The pattern {\cal P} has local symmetries that are not necessarily contained in its global symmetry groupG. The usual approach in color symmetry theory is to arrive at perfect colorings of {\cal P} ignoring local symmetries and considering only elements ofG. A framework is presented to systematically arrive at what Roth [Geom. Dedicata(1984),17, 99–108] defined as a coordinated coloring of {\cal P}, a coloring that is perfect and transitive underG, satisfying the condition that the coloring of a given motif is also perfect and transitive under its symmetry group. Moreover, in the coloring of {\cal P}, the symmetry of {\cal P} that is both a global and local symmetry, effects the same permutation of the colors used to color {\cal P} and the corresponding motif, respectively.

中文翻译：

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对称图案的局部和全局颜色对称性

这项研究解决了获得由不相交全等对称图案组成的对称图案 {\cal P} 的传递完美着色的问题。模式 {\cal P} 具有不一定包含在其全局对称群中的局部对称性G。颜色对称理论中的常用方法是忽略局部对称性并仅考虑以下元素来获得 {\cal P} 的完美着色G。提出了一个框架来系统地得出罗斯[吉姆. 奉献者(1984),17 号, 99–108] 定义为 {\cal P} 的协调着色，这是一种完美且可传递的着色G，满足给定基序的着色在其对称群下也是完美且传递的条件。此外，在 {\cal P} 的着色中，{\cal P} 的对称性既是全局对称性又是局部对称性，分别影响用于着色 {\cal P} 和相应主题的颜色的相同排列。