Optimal packings of up to five equal circles on a square flat torus

Abstract

We prove that a certain lattice arrangement of five equal circles on a square flat torus (the quotient of the plane by the lattice generated by two unit perpendicular vectors) is the densest possible arrangement. Our proof uses techniques from Rigidity Theory and Topological Graph Theory. We also apply these techniques to the cases of one to four equal circles on a square flat torus and prove that the known densest arrangements are the only locally maximally dense arrangements. Additionally, we establish the existence of a locally maximally dense lattice arrangement of n = a 2 + b 2 (a > b > 0 and gcd(a, b) = 1) equal circles on a square flat torus.