  We determine the topology of the moduli space of periodic tilings of the
plane by parallelograms. To each such tiling, we associate combinatorial data
via the zone curves of the tiling. We show that all tilings with the same
combinatorial data form an open subset in a suitable Euclidean space that is
homotopy equivalent to a circle. Moreover, for any choice of combinatorial
data, we construct a canonical tiling with these data.
