PENNY-PACKINGS WITH MINIMAL SECOND MOMENTS

We consider the problem of packing n disks of unit diameter in the plane so as to minimize the second moment about their centroid. Our main result is an algorithm which constructs packings that are optimal among hexagonal packings. Using the algorithm, we prove that, except for n = 212, the n-point packings obtained by Graham and Sloane [1] are optimal among hexagonal packings. We also prove a result that makes precise the intuition that the "greedy algorithm " of Graham and Sloane produces approximately circular packings. 1