Packing Squares in a Torus

TL;DR

This paper investigates the densest arrangements of unit squares on a torus using analytical and simulated annealing methods, revealing diverse solutions including density-one packings, gapped bricklayer lattices, and surprising non-Bravais configurations.

Contribution

It introduces new dense packing solutions for squares on a torus, including non-traditional lattice arrangements and configurations with holes, expanding understanding of geometric packings.

Findings

Density-one packings when N is sum of two squares.

Existence of gapped bricklayer lattice solutions with density N/(N+1).

Discovery of non-Bravais lattice configurations, including holes and mixed orientations.

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 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. The entropy of some of these configurations and the frequency and orientation of density-one solutions as N goes to infinity are discussed.