Packing squares in a torus

Abstract

The densest packings of N unit squares in a torus are studied using analytical methods as well as simulated annealing.A rich array of dense packing solutions are found: density-one packings whenN isthe sum of two square integers; a family of ‘gapped bricklayer’ Bravais lattice solutions with densityN/(N + 1); andsome surprising non-Bravais lattice configurations, including lattices of holes as well as a configurationfor N = 23 in which not all squares share the same orientation. The entropy of some of theseconfigurations and the frequency and orientation of density-one solutions as are discussed.