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I am dealing with 2D lattices only. Some solutions to a PDE I am interested in consists of functions the minima of which organise themselves in an ordered periodic way: for instance they might arrange in a square lattice, or triangular lattice, or oblique with a given angle. The lattices might even be non-point lattices (but still periodic): example honeycomb or kagome lattice.

I was wondering whether it exists,given a periodic lattice, a transformation that can be done on the lattice which allows for the detection of the structure of the lattice or helps its classification. More specifically I was thinking whether one has in mind a clever way to use Fourier transforms or something of the sort to obtain a parameter which describes such lattices.

Otherwise any relevant reference on the topic? There must be someone studying these systems even though I could not find much in the literature.

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    $\begingroup$ There are four tags including "lattice" in their name. Are you sure your question is about "lattice-orders"? Perhaps you would be more likely to get an answer with some other tag... $\endgroup$ – amrsa Dec 14 '16 at 11:07
  • $\begingroup$ @amrsa it is not about any of the pre existing tags but unfortunately I cannot create a new one. Anyways I really wouldn't know what other tag to include. Maybe crystallography; apart from that I really wouldn't know. That's probably also why I struggled to find something in the literature. $\endgroup$ – semola Dec 15 '16 at 14:24
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I think you're defining "lattice" as "isomorphic to $\mathbb Z ^ n$ (or particularly $\mathbb Z ^ 2$).

If that's the case, the crystallographic restriction theorem places very strong restrictions on the types of rotational symmetries your lattice can have. Namely, the group of rotational symmetries can only have order 1, 2, 3, 4 or 6.

To see which of these your lattice has, it is probably simple enough to just use basic trigonometry and see if the lattice structure is preserved by rotations of the given order.

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