Symmetry groups associated with tilings on a flat torus

On uniform edge-n-colorings of tilings

An edge-n-coloring of a uniform tiling {\cal T} is uniform if for any two vertices of {\cal T} there is a symmetry of {\cal T} that preserves the colors of the edges and maps one vertex onto the other. This paper gives a method based on group theory and color symmetry theory to arrive at uniform edge-n-colorings of uniform tilings. The method is applied to give a complete enumeration of uniform edge-n-colorings of the uniform tilings of the Euclidean plane, for which the results point to a total of 114 colorings, n = 1, 2, 3, 4, 5. Examples of uniform edge-n-colorings of tilings in the hyperbolic plane and two-dimensional sphere are also presented.

Local and global color symmetries of a symmetrical pattern

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.