Working with his students, Will found that some questions about circle packings on the torus required different techniques than those used on an ordinary square. For instance, on the torus, they were able to make a finite list of possible "packing graphs" (the collection of blue line segments between tangent circles shown above) that could lead to locally maximally dense packings. This allowed the team to identify the configurations above as the global maxima when four or five circles are packed on a torus and to prove that any configuration of five or fewer circles that is a local maximum of the density is also a global maximum.

Will and his students have presented their findings at several national conferences, where they have attracted considerable attention. Perhaps most impressively, a short article about this work appeared in Science. In that article, the well-known mathematician Ronald Graham, of the University of California at San Diego, said, "It is very difficult to prove that a particular packing is optimal." Working on the torus, where there is no boundary, may make certain proofs easier to find, he continues.

You may enjoy experimenting with equal circle packings using the software that Will created on his sabbatical. It is freely available at the EZ Pack website.