, Volume 102, Issue 1-2, pp 27-51
Date: 15 Jan 2012

Optimal packings of up to six equal circles on a triangular flat torus

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Abstract

For each n between 1 and 6, we prove that a certain arrangement of n equal circles is the unique optimally dense packing on a standard triangular flat torus (the quotient of the plane by the lattice generated by two unit vectors with a 60 angle). The packings of 1, 2, 3, 4 and 6 circles are based on either a toroidal triangular close packing or a toroidal triangular close packing with one circle removed. The packing of 5 circles is irregular. This proves two cases of a conjecture stronger than L. Fejes Toth’s conjecture about the strong solidity of the triangular close packing on the plane.

The first three authors were partially supported by National Science Foundation grant DMS-0451254. The first author would like to thank Robert Connelly inspiring, supporting, and contributing ideas to this work. The first and fourth authors were supported by a student summer scholar’s grant from Grand Valley Sate University. We would all like to thank one of the anonymous referees for his or her extremely careful and thorough checking of our work and for pointing out some important issues.