The <a href="http://mathworld.wolfram.com/GosperIsland.html">Gosper island</a> tiles the plane, so I'm curious if a nontrivial elliptic? function exists which would have a 'period gosper-island' instead of a period parallelogram. In this case, I'm using 'trivial' to denote a function that's got the right kind of lattice, but the Gosper island is overkill and unnecessary -- for example the <a href="http://www.mai.liu.se/~halun/complex/elliptic/">derivative of the equianharmonic case</a>: $d\wp(z;0,1)/dz$ (last image).