Packing Squares in a Torus
Don Blair, Christian D. Santangelo, Jon Machta

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.
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Taxonomy
TopicsTheoretical and Computational Physics
