This question has three up-votes on m.s.e. but isn't getting any answers.
Every textbook says every doubly-periodic meromorphic function on $\mathbb C$ has a fundamental domain that is a parallelogram. So someone asked whether one could do with a tiling of the plane by regular hexagons what is thus done with parallegorams: Is there some meromorphic function whose restriction to each hexagon in the tiling is a shift of its restriction to any of the other hexagons? And indeed there is: Dixon's elliptic functions do just that. (That in no way conflicts with the existence of the aforementioned parallelogram: just find one whose four vertices are centers of four suitably located hexagons.)
two other questions another question:
What about other periodic tilings? For example, there is a periodic tiling by hexagons, squares, and triangles. Might the restriction of some doubly periodic meromorphic function to any of those be a shift or maybe a shift followed by a rotation, of the restriction to any of the others? For that tiling, rotation as well as translation becomes relevant. For which tilings does such a thing exist?
What about aperiodic tilings with a finite set of sizes and shapes of tiles? We'd want a function meromorpic on the whole plane whose restriction to any tile is a shift-plus-rotation of its restriction to any of the infinitely many tiles of the same shape.[The answer to this one is obvious.]