About an Algorithmic Approach to Tilings {p, q} of the Hyperbolic Plane

Abstract: In this paper, we remind previous results about the tilings {p., q} of the hy-perbolic plane. As proved in [Margenstern and Skordev 2003a], these tilings are com-binatorie, a notion which we recall in the introduction. It turned out that in this case, most of these tilings also have the interesting property that the language of the split-ting associated to the tiling is regular. In this paper, we investigate the consequence of the regularity of the language by providing algorithms to compute the path from a tile to the root of the spanning tree as well as to compute the coordinates of the neighbouring tiles. These algorithms are linear in the coordinate of the given node.