Packing Squares in a Torus

Details Packing Squares in a Torus Packing Squares in a Torus

2011-10-24

arxiv.org/abs/1110.5348v2

JSTAT, P01018 (2012)

Physics

Condensed Matter

Statistical Mechanics

14 pages, 9 figures; v2 reflects minor changes in published version

Scientific paper

10.1088/1742-5468/2012/01/P01018

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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