Efficient generation of hyperbolic patterns from a single asymmetric motif

We present an efficient method of constructing hyperbolic patterns based on an asymmetric motif designed in the central hyperbolic polygon. Since there is no rotational symmetry in each hyperbolic polygon, a subset of the hyperbolic group elements has to be selected carefully so that the central hyperbolic polygon is transformed to the other polygons once and only once. An efficient labeling procedure is proved by considering the group presentation and can be easily implemented using the computer. Illustrative hyperbolic patterns are constructed from given asymmetric motifs for the symmetry group [p, q]+ which consists of all compositions of an even number of reflections.