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I am having trouble finding a mathematical definition of the Brouillin zone beyond the first, which are basically the Voronoi cells or Wigner-Seitz cells. We could imagine the set of point closer to a particular integer than any other:

$$ \min_{n \in \mathbb{Z}} \big|\big|x - n \big|\big|$$

While straightforward in one dimension, these shapes can be rather nuanced in higher dimensions.

$$ \min_{\vec{n} \in L.\mathbb{Z}^m} \big|\big|\vec{x} - \vec{n} \big|\big|$$

Even for Hexagons, the Brouillin zones look rather complicated.

It's pretty clear from the image that Broullin zones converge to a circular region as $n \to \infty$ but I can't prove as much without a definition.

Picture is taken from Wave Propogation in Periodic Structures by Léon Brillouin.

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marked as duplicate by Carlo Beenakker, Community Jun 28 '16 at 22:19

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    $\begingroup$ This belongs on Physics or Math StackExchange instead. Anyway, you shouldn't think of them as putting together the first, second, third etc Brillouin Zones. They are all simply different fundamental domains for the quotient in the Pontryagin Dual. You see that by the jigsaw puzzle about how the pieces of the n'th zone are translated back to the 1st. They are all parameterizing the same thing. Putting different Brillouin zone's would be redundant. $\endgroup$ – AHusain Jun 28 '16 at 21:41
  • $\begingroup$ @AHusain yes cohomology $H^\ast(\cdot, \mathbb{Z})$ is often a lattice. $\endgroup$ – john mangual Jun 28 '16 at 21:45
  • $\begingroup$ Do you want to use the n-th Brillouin Zone for some large n vs the 1st even though they parameterize the exact same thing? $\endgroup$ – AHusain Jun 28 '16 at 21:57
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    $\begingroup$ a definition and a proof that the limit $n\rightarrow\infty$ is a circle is given here: mathoverflow.net/questions/221636/… $\endgroup$ – Carlo Beenakker Jun 28 '16 at 22:16