# Is there an nontrivial function whose 'period paralellograms' are Gosper Islands?

The Gosper island 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 derivative of the equianharmonic case: $d\wp(z;0,1)/dz$ (last image).

• Dear Deoxy, your source for learning elliptic functions is definitely not the best one. But already there it's explained that fundamental domains of elliptic functions are parallelograms (quotients of $\mathbb C$ by lattices), no snowflakes. More exotic fundamental domains can be constructed for automorphic rather than elliptic functions. I wonder whether one can get fractals but tiling pictures of more regular geometric structure can be seen for example in M.Yoshida's "Hypergeometric functions, my love" (1997). – Wadim Zudilin May 23 '10 at 11:54
• I don't think the question makes sense. An elliptic function has a period lattice $\Omega$, which is a subgroup of $\mathbb{C}$. There is nothing unique about “the” period parallellogram, though; it is just a convenient labeling of the equivalence classes $\mathbb{C}/\Omega$ by taking one element from each equivalence class. There is no reason, apart from convenience, why it has to be a parallellogram. Gosper islands should do just fine, though to go and pick a fractal one seems slightly perverse. – Harald Hanche-Olsen May 23 '10 at 11:56
• Harald -- the crux of my question is whether any functions exist whose 'canonical' fundamental domain is a Gosper island -- if you were to look at domain colored pictures of these hypothetical functions, one would say "aha, Gosper islands" instead of "aha, period parallelograms". – graveolensa May 23 '10 at 13:55
• The point of Harald's comment is that there is no 'canonical' fundamental domain, at least until you provide extra data. – Qiaochu Yuan May 23 '10 at 20:09
• I wonder how the choice of fundamental domain alters the discussion of elliptic curves – john mangual Jul 10 '13 at 11:47

The Gosper islands are a fundamental domain for the translation action of the Eisenstein integers $\mathbb{Z}[\frac{1+\sqrt{-3}}{2}]$, since the shapes can be constructed by deforming a Voronoi decomposition of the plane with respect to that lattice. It is therefore reasonable to look for functions in the field generated by the Weierstrass $\wp$ function for the Eisenstein integers and its first derivative $\wp'$, since this field comprises all of the meromorphic functions that are periodic with respect to the lattice.
I suppose you may want to find a function $f(z)$ such that the boundary configuration is equal to $\{ z : |f(z)| = 1 \}$, but I am somewhat doubtful that such a function exists, simply by degree considerations.