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I think of a snapshot of a single period of a doubly periodic function as one parallelogram-shaped tile in a tessellation, could a function have a period that repeats like honeycomb or some other not rectangular tessellation?

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What kind of functions are we talking about here? –  Qiaochu Yuan Apr 28 '11 at 19:38
    
C--> C, complex –  a little don Apr 28 '11 at 19:50

2 Answers 2

up vote 20 down vote accepted

(many literature searches and Mathematica experiments later...)

The usual Jacobi and Weierstrass elliptic functions have as their "repeating unit" a parallelogram (which can be made rhomboidal or square through appropriate choices of parameters). It is known that apart from parallelograms, hexagons can tile the plane by translation; so, why can't there be a doubly periodic function that has a hexagonal repeating unit?

It turns out that A.C. Dixon (the guy whose book on elliptic functions Hans linked to), in a long 1890(!) paper, studied a class of elliptic functions (now named after him) based on the inversion of the Abelian integral

$$\int\frac{\mathrm dt}{\left(1-t^3\right)^{2/3}}=t {}_2 F_1\left({{\frac13\quad \frac23}\atop{\frac43}}\mid t^3\right)$$

where ${}_2 F_1\left({{a\quad b}\atop{c}}\mid x\right)$ is a Gaussian hypergeometric function.

There are two of these Dixon elliptic functions, $\mathrm{sm}(z,0)=\mathrm{sm}(z)$ and $\mathrm{cm}(z,0)=\mathrm{cm}(z)$, corresponding to the usual sine and cosine respectively. Both functions have a real period $\pi_3=B\left(\frac13,\frac13\right)$ (where $B(a,b)$ is the beta function) and a complex period $\pi_3\exp(2i\pi/3)$, and satisfy the following relations (reminiscent of usual trigonometric identities):

$$\mathrm{sm}\left(\frac{\pi_3}{3}-z\right)=\mathrm{cm}(z)$$

$$\mathrm{sm}^3(z)+\mathrm{cm}^3(z)=1$$

$$\mathrm{sm}^\prime(z)=\mathrm{cm}^2(z)\quad \mathrm{cm}^\prime(z)=-\mathrm{sm}^2(z)$$

and, most relevant to the purposes of this question, a rotational invariance:

$$\exp(-2i\pi/3)\mathrm{sm}(z\exp(2i\pi/3))=\mathrm{sm}(z)\quad \mathrm{cm}(z\exp(2i\pi/3))=\mathrm{cm}(z)$$

Plots of the Dixon functions on the real line don't look very interesting:

Dixon elliptic functions, real line

but, as with the usual elliptic functions, the fun starts in the complex plane:

Dixon sm Dixon cm

These contour plots clearly display the hexagonal structure of the Dixon functions. Here is a single "fundamental period hexagon" for $\mathrm{sm}(z)$:

period hexagon for sm(z), contour plot period hexagon for sm(z), 3D

Note that a section of the real line (in the plots above, $\left(-\frac{\pi_3}3,\frac{2\pi_3}{3}\right)$) corresponds to a chord of the period hexagon.

Both Dixon elliptic functions possess three poles (once you've identified the congruent poles in the period hexagon) and three zeros within the fundamental hexagon. Of course, one could go the usual route and consider the "repeating unit" of the Dixon function to be a particular rhombus; this is equivalent; since the rhombus can be appropriately dissected into a regular hexagon, and vice-versa.

The Dixon elliptic functions can also be expressed in terms of Weierstrass elliptic functions:

$$\mathrm{sm}(z)=\frac{6\wp\left(z;0,\frac1{27}\right)}{1-3\wp^\prime\left(z;0,\frac1{27}\right)}$$

$$\mathrm{cm}(z)=\frac{3\wp^\prime\left(z;0,\frac1{27}\right)+1}{3\wp^\prime\left(z;0,\frac1{27}\right)-1}$$

(there are also expressions for Dixon functions in terms of Jacobi elliptic functions, but they are rather complicated.)

Finally, if you're interested in knowing more about the Dixon elliptic functions (including combinatorial applications), this paper is a good starting point.

A Mathematica notebook for those interested in exploring the topic further is available from me upon request.

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So this makes me wonder: what about the other 15 wallpaper groups? –  deoxygerbe May 2 '11 at 21:40
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@deoxy: That's an interesting question, too. Maybe you should ask it as a separate question? –  J. M. May 3 '11 at 5:11
    
Yeah, and what about tessellations in other geometries? –  Raskolnikov May 3 '11 at 7:52
    
@Raskolnikov: I wouldn't know how to define an elliptic function whose behavior would mimic other geometries, so I can't say there. Sorry. :( –  J. M. May 3 '11 at 7:55
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@J.M. this question: math.stackexchange.com/questions/36737/… –  deoxygerbe May 3 '11 at 17:45

I don't think so. If there are exactly two periods (not parallel), then we have a parallelogram, so in your case there must be at least three independent periods. But that implies that the function is constant (or multi-valued), as proved for example in this old book by Dixon on elliptic functions (see §32 on p. 19).

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On the other hand, Hans displays something almost, but not quite hexagonal here (scroll down to the equianharmonic case of the Weierstrass $\wp$ function). –  J. M. Apr 28 '11 at 19:40
    
I think it depends on how the tesilation works, I just realized that if you create you honeycomb bu translation it can be repartitioned in to parallelograms. Now if we ad a rotation, then it would not work. I think that this needs to be restricted to tesilations where you can pick up the whole plane and map it to itself after translation and rotation. --- still, it's not obvious to me that there are more than three periods... Can you expand on that? –  a little don Apr 28 '11 at 19:47
    
Well, as you say, you can have a honeycomb pattern, but that's really just a doubly periodic pattern in disguise (with periods 1 and $\exp(i\pi/3)$, say). If you really want to go beyond the case where there is a fundamental region in the shape of a parallelogram lurking somewhere, you will have to look for something with more than two periods, and that is apparently not possible. –  Hans Lundmark Apr 28 '11 at 20:59

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