Tiling the Hyperbolic Plane with a Single Pentagonal Tile

In this paper, we study the number of tilings of the hyperbolic plane that can be constructed, starting from a single pentagonal tile, the only permitted transformations on the basic tile being the replication by displacement along the lines of the pentagrid. We obtain that there is no such tiling with five colours, that there are exactly two of them with four colours and a single trivial tiling with one colour. For three colours, the number of solutions depends of the assortment of the colours. For half of them, there is a continuous number of such tilings, for one of them there are four solutions, for the two other ones, there is no such tiling. For two colours, there is always a continuous number of such tilings. By contrast, there is no such analog in the euclidean plane with the similar constraints.