Penny-packing and two-dimensional codes

Abstract

We consider the problem of packing n equal circles (i.e., pennies) in the plane so as to minimize the second moment U about their centroid. These packings are also minimal-energy two-dimensional codes. Adding one penny at a time according to the greedy algorithm produces a unique sequence of packings for the first 75 pennies, and appears to produce optimal packings for infinitely many values of n. Several other conjectures are proposed, and a table is given of the best packings known for n≤500. For large n, U∼√3 n2/(4π). © 1990 Springer-Verlag New York Inc.