Optimal packings of congruent circles on a square flat torus

TL;DR

This paper investigates the optimal arrangements of congruent circles on a square flat torus, identifying multiple solutions for certain circle counts, with implications for image super-resolution.

Contribution

It introduces a novel computational approach to determine maximal packings of congruent circles on a square torus, revealing multiple optimal configurations for specific cases.

Findings

Optimal packings found for N=6, 7, 8 circles.

For N=7, three distinct optimal arrangements exist.

The method uses computer enumeration of contact graphs.

Abstract

We consider packings of congruent circles on a square flat torus, i.e., periodic (w.r.t. a square lattice) planar circle packings, with the maximal circle radius. This problem is interesting due to a practical reason - the problem of "super resolution of images." We have found optimal arrangements for N=6, 7 and 8 circles. Surprisingly, for the case N=7 there are three different optimal arrangements. Our proof is based on a computer enumeration of toroidal irreducible contact graphs.