A Brillouin zone of a full-rank point lattice 
is a region determined by
the rank of the Euclidean distance to the origin among the distances to
all lattice points. For a point 
, let 
count the lattice points
strictly closer to 
than the origin, and let 
count those at the same distance
from 
as the origin, including the origin
itself. The 
th
Brillouin zone, for a positive integer 
, is
 |
(1)
|
Away from ties, 
,
so 
iff exactly 
lattice points are closer
to 
than the origin. At a tie, this convention places 
in each zone numbered 
through 
.
The illustration above shows the perpendicular bisector construction of the first zone for the point
lattice 
.
The hyperplane on which a nonzero lattice point 
and the origin have equal distance
is
 |
(2)
|
where 
is the dot product and 
is the l2-norm.
In two dimensions, 
is the perpendicular
bisector of the line segment joining 
and 
. If 
lies on none of these hyperplanes,
the line segment from the origin
to 
crosses exactly 
of them iff 
, counting distinct hyperplanes
separately even when crossed at the same point.
The first zone is
 |
(3)
|
It is the Voronoi polygon of the origin in two dimensions and its Voronoi
cell in higher dimensions, and is a convex
polytope. For example, for 
, the first zone is the square ![[-1/2,1/2]^2](/images/equations/BrillouinZone/Inline28.svg)
. For the simple cubic
lattice 
,
it is the cube ![[-1/2,1/2]^3](/images/equations/BrillouinZone/Inline30.svg)
. At 
in the square example,
four lattice points tie for the smallest distance,
so 
belongs to 
,
, 
, and 
.
The illustration above compares the first three zones of the square point lattice 
(left) and a triangular lattice (right). Both
point lattices have covolume
1 and are shown at the same scale. Their blue first zones are a square
and a regular hexagon, respectively. For the triangular lattice, the orange second zone consists
of six triangles, and the purple third zone consists
of twelve triangles.
In solid-state physics, the construction is usually applied to the reciprocal lattice of the crystal's point lattice, using
the 
normalization. The unqualified term "Brillouin zone" commonly denotes the
first zone of this reciprocal lattice (Tong,
pp. 52, 55).
For each fixed 
,
the translations 
, 
, cover Euclidean space
and overlap only on their boundaries. Thus every zone
is a fundamental domain for translation
by 
,
although higher zones need not be connected. All
zones have the same 
-dimensional volume, equal to
the covolume of 
(Bieberbach 1939, pp. 509-515; Jones 1984, pp. 241-263).
