Given a regular or uniform tessellation of hyperbolic plane, is there a way to find a group of cells that will tile the whole plane?
For example: in the $(6,6,7)$ tessellation (truncated $\{3,7\}$), the "ratio" of hexagons to heptagons is $7:3$ (as each vertex can be understood as $2\cdot 1/6$ of hexagon + $1/7$ of heptagon). A group of $7$ hexagons and $3$ heptagons could be theoretically a reptile (and any reptile must have a multiple of these numbers).
I know of one shape in this tessellation that will tile the whole hyperbolic plane -- it has 40 tiles (28 hexagons and 12 heptagons) and it's used in several ingenious ways in the game HyperRogue (http://zenorogue.blogspot.cz/2015/02/hyperrogue-60-five-new-lands.html has an explanation).
I'd like to find some alternate shapes to suggest for this game, but I'm not sure whether there has been any research done in this area.