  We consider the interactions of finite dipoles in a doubly-periodic domain. A
finite dipole is a pair of equal and opposite strength point vortices separated
by a finite distance. The dynamics of multiple finite dipoles in an unbounded
inviscid uid was first proposed by Tchieu, Kanso & Newton in [1] as a model
that captures the "far- field" hydrodynamic interactions in fish schools. In
this paper, we formulate the equations of motion governing the dynamics of
finite-dipoles in a doubly-periodic domain. We show that a single dipole in a
doubly-periodic domain exhibits periodic and aperiodic behavior, in contrast to
a single dipole in an unbounded domain. In the case of two dipoles in
doubly-periodic domain, we identify a number of interesting trajectories
including collision, collision avoidance, and passive synchronization of the
dipoles. We then examine two types of dipole lattices: rectangular and diamond.
We verify that these lattices are in a state of relative equilibrium and show
that the rectangular lattice is unstable while the diamond lattice is linearly
stable for a range of perturbations. We conclude by commenting on the insights
these models provide in the context of fish schooling.
