Packing Squares in a Torus

We study the densest packings of N unit squares in a torus (i.e., using periodic, square boundary conditions in 2D) using both analytical methods and simulated annealing. We find a rich array of dense packing solutions: density-one packings when N is the sum of two square integers, a family of ``gapped bricklayer'' Bravais lattice solutions with density N/(N+1), and some surprising non-Bravais lattice configurations -- including lattices of holes, as well as a configuration for N=23 in which not all squares share the same orientation. We assess the entropy of some of these configurations, as well as the frequency and orientation of density-one solutions as N goes to infinity.