Repeated patterns of dense packings of equal disks in a square

Abstract

We examine sequences of dense packings of n congruent non-overlapping disks inside a square which follow specific patterns as n increases along certain values, n = n(1), n(2), ...n(k), .... Extending and improving previous work of Nurmela and Östergård [NO] where previous patterns for n = n(k) of the form k2, k2 - 1, k2 - 3, k(k + 1), and 4k2 + k were observed, we identify new patterns for n = k2 - 2 and n = k2 + [k/2]. We also find denser packings than those in [NO] for n =21, 28, 34, 40, 43, 44, 45, and 47. In addition, we produce what we conjecture to be optimal packings for n =51, 52, 54, 55, 56, 60, and 61. Finally, for each identified sequence n(1), n(2), ...n(k), ... which corresponds to some specific repeated pattern, we identify a threshold index k0, for which the packing appears to be optimal for k ≤ k0, but for which the packing is not optimal (or does not exist) for k > k0.