Curved Hexagonal Packings of Equal Disks in a Circle

Abstract

For each k ≥ 1 and corresponding hexagonal number h(k) = 3k(k+1)+1, we introduce $m(k) = \max \{{(k-1)!}/{2}, 1\}$ packings of h(k) equal disks inside a circle which we call the curved hexagonal packings. The curved hexagonal packing of 7 disks (k = 1, m(1)=1) is well known and one of the 19 disks (k = 2, m(2)=1) has been previously conjectured to be optimal. New curved hexagonal packings of 37, 61, and 91 disks (k = 3, 4, and 5, m(3)=1, m(4)=3, and m(5)=12) were the densest we obtained on a computer using a so-called ``billiards'' simulation algorithm. A curved hexagonal packing pattern is invariant under a $60^{\circ}$ rotation. For $k \rightarrow \infty$ , the density (covering fraction) of curved hexagonal packings tends to ${\pi^2}/{12}$ . The limit is smaller than the density of the known optimum disk packing in the infinite plane. We found disk configurations that are denser than curved hexagonal packings for 127, 169, and 217 disks (k = 6, 7, and 8). In addition to new packings for h(k) disks, we present the new packings we found for h(k)+1 and h(k)-1 disks for k up to 5, i.e., for 36, 38, 60, 62, 90, and 92 disks. The additional packings show the ``tightness'' of the curved hexagonal pattern for k ≤ 5: deleting a disk does not change the optimum packing and its quality significantly, but adding a disk causes a substantial rearrangement in the optimum packing and substantially decreases the quality.