# Mathematical Definition of $n$-Brouillin Zone [duplicate]

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.

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

• @AHusain yes cohomology $H^\ast(\cdot, \mathbb{Z})$ is often a lattice. – john mangual Jun 28 '16 at 21:45
• a definition and a proof that the limit $n\rightarrow\infty$ is a circle is given here: mathoverflow.net/questions/221636/… – Carlo Beenakker Jun 28 '16 at 22:16