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).
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$\begingroup$ 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). $\endgroup$– Wadim ZudilinCommented May 23, 2010 at 11:54
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5$\begingroup$ 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. $\endgroup$– Harald Hanche-OlsenCommented May 23, 2010 at 11:56
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1$\begingroup$ 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". $\endgroup$– graveolensaCommented May 23, 2010 at 13:55
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$\begingroup$ The point of Harald's comment is that there is no 'canonical' fundamental domain, at least until you provide extra data. $\endgroup$– Qiaochu YuanCommented May 23, 2010 at 20:09
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$\begingroup$ I wonder how the choice of fundamental domain alters the discussion of elliptic curves $\endgroup$– john mangualCommented Jul 10, 2013 at 11:47
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1 Answer
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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.
It is not clear what selection rule you want to apply to favor one function over another. Elliptic functions do not have canonical fundamental domains, and one has to choose extra data (e.g., a basis of the lattice, and a pair of paths in the homotopy class representing the basis) to write down boundaries in the usual theory.
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.
answered May 23, 2010 at 14:21