Is it true that the limiting shape for Brillouin zones (for any lattice) is a circle?

You can find the definition and the step by step construction of Brillouin zones here. This picture is taken from the Graphics Gallery for The Mathematica GuideBook for Graphics.

It is known that all Brillouin zones have the same area, so if we have a picture with $n$ zones and we want to see it in a unit square then it is necessary to multiply it by $1/\sqrt{n}$. But they could be close to the circle of radius $c\sqrt{n}$ even without rescaling.

**edit by j.c.**:
This question is stated in 2D but makes sense in arbitrary dimensions.

Suppose we have a lattice $\Gamma$ in $d$-dimensions (i.e. a discrete subgroup of $\mathbb{R}^d$ that spans $\mathbb{R}^d$) (in crystallography terms, $\Gamma$ is the reciprocal lattice of some Bravais lattice). A

Bragg planeof $\Gamma$ is a hyperplane that perpendicularly bisects a line segment in $\mathbb{R}^d$ between the origin and any other element of $\Gamma$. Then the $n$thBrillouin zoneof $\Gamma$ is defined to be the closure of the set of points $P$ in $\mathbb{R}^d$ such that the line segment between $P$ and the origin intersects exactly $n-1$ Bragg planes.

whatconverges towhat? $\endgroup$ – Liviu Nicolaescu Oct 23 '15 at 13:542more comments