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## Computer Algorithms to Determine Colorings that arise from Plane Crystallographic Groups

Ma. Louise Antonette N. De Las Penas
Department of Mathematics
Ateneo De Manila University
Loyola Heights, Quezon City, Philippines

### Abstract

Color symmetry has generated research in many areas of symmetry theory and many problems continue to be of interest today, particularly the classification and analysis of colored symmetrical patterns.

To analyze a colored symmetrical pattern, we not only consider the symmetry group G of the (uncolored) pattern but also the symmetry group K of the pattern when it is colored. Since not all elements of G permute colors, we also consider the subgroup H of elements of G which effect color permutations. This subgroup H contains K as a normal subgroup of elements of H which fixes the colors.

In this paper, we develop a computer program to assist us in our study of colored symmetrical patterns. We determine all colorings of a symmetrical pattern for which the elements of a given subgroup H of the symmetry group G of the uncolored pattern permute the colors and the elements of a given subgroup K of G fix the colors. It is often a difficult task to generate such colorings by hand especially when dealing with a coloring with a large number of colors. The computer program was conceptualized to make the enormous task of listing such colorings manageable. In this paper, we look at colored patterns whose symmetry groups are plane crystallographic groups.

Consider the colored patterns below which are assumed to repeat over the entire plane. For both the colored patterns, the symmetry group G of the patterns with the colors disregarded is a hexagonal plane crystallographic group of type p6m generated by a,b,x,y where a is a 60 degree counterclockwise rotation about the indicated point P, b is a reflection in a horizontal line through P, and x, y are translations as indicated. These colored patterns have been generated using the program by choosing the subgroups H=<a,x,y> and K=<a3,x,y> of G. H and K are hexagonal plane crystallographic groups of types p6 and p2 respectively. Moreover, the program identifies the first coloring as perfect (all elements of G permute the colors) and the second coloring as non-perfect (there exist at least one element of G that does not permute the colors). Note that for the second coloring, the reflectionb does not permute the colors.