Given a metric space (
X,
d), a discrete set
,
and a
preferred point
,
the
n-th Brillouin zone is defined
(loosely speaking) as those points in
X which are closest to exactly
n elements of
S, including
x0. This notion is a direct
generalization of the concept introduced by Brillouin [
Br] in the
thirties, which describes some quantum mechanical properties of crystals.
We generalize a theorem of Bieberbach [Bi] asserting that the Brillouin
zones tile the underlying space, and that each zone has the same area.
We then use these ideas to discuss focusing of geodesics in orbifolds of
constant curvature. In the particular case of the Riemann surfaces
,
we count the number of geodesics
of length t that connect the origin to itself.