# Tilings and meromorphic functions

This question and its answer by "J.M." were quite informative and inspire other questions.

If a function is meromorphic on $\mathbb C$ and doubly periodic then, as we all learned at our mother's knee, there is a fundamental domain that is a parallelogram. So someone asked whether one could do the same with the standard tiling of the plane by regular hexagons, so that the restriction of the function to one hexagon is just a shift of its restriction to any of the other hexagons. "J.M." 's answer was that that is exactly what happens with "Dixon's elliptic functions". This in no way conflicts with the existence of a fundamental domain that is a parallelogram --- indeed a rectangle (that's an exercise whose solution may take you five seconds).

All this immediately inspires two other questions:

• 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.

• What about aperiodic tilings? 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. For which tilings does such a thing exist?
[Aperiodic tilings won't work here. That follows immediately from some basic stuff from complex variables.]

• Unfortunately, my complex geometry isn't sufficient to comment on the meromorphic properties, but I'm reminded of a notion called 'pattern-equivariance' of functions on tilings of $\mathbb{R}^n$ into coefficient groups (originally the reals, but later generalised to the integers and other locally compact groups). These functions turn out the be the correct setting to form a 'locally defined' cohomology theory for tilings, as opposed to the previously global approaches via Cech cohomology of their hulls/tiling space. – Dan Rust Aug 27 '16 at 12:51
• @DanRust : What are "coefficient groups"? Maybe a typo for "quotient groups"? $\qquad$ – Michael Hardy Aug 27 '16 at 15:48
• Sorry, it's a cohomology term, as in the universal coefficients theorem. As I said, it seems only loosely related to your question, but seems like something you may be interested in nonetheless. – Dan Rust Aug 27 '16 at 16:12