The Brillouin zone is an essential concept in condensed matter
physics.

Recall that if a lattice is described by lattice vectors \(\vec a_1,
\vec a_2, \vec a_3\), the reciprocal lattice vectors are defined by
the following 9 constraints:
\begin{equation*}
\vec a_i \cdot \vec b_j = 2\pi \delta_{ij}
\end{equation*}

On this reciprocal lattice, the Brillouin zone is defined to be the
set of all points closer to the origin that any other lattice points.