Colorings of hyperbolic plane crystallographic patterns

Abstract

Abstract In color symmetry the basic problem has always been to classify symmetrically colored symmetrical patterns [13]. An important step in the study of color symmetry in the hyperbolic plane is the determination of a systematic approach in arriving at colored symmetrical hyperbolic patterns. For a given uncolored semi-regular tiling with symmetry group G a hyperbolic plane crystallographic group, this question can be addressed by applying a general framework for coloring symmetrical patterns and using right coset colorings as a tool to study the subgroup structure of G. In this paper, we present colored patterns that emerge from the hyperbolic 3 · 4 · 3 · 4 · 3 · 3 tiling where all the symmetries of the uncolored tiling permute the colors of the patterns.