CURVED HEXAGONAL PACKINGS OF EQUAL DISKS IN A CIRCLE

For each k greater than or equal to 1 and corresponding hexagonal numb er h(k) = 3k(k + 1) + 1, we introduce m(k) = max{(k - 1)!/2, 1} packin gs of h(k) equal disks inside a circle which we call the curved hexago nal 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 pre viously 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) = 1 2) were the densest we obtained on a computer using a so-called ''bill iards'' simulation algorithm. A curved hexagonal packing pattern is in variant under a 60 degrees rotation. For k --> infinity, 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 i n the infinite plane. We found disk configurations that are denser tha n 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 pa ckings we found for h(k) + 1 and h(k) - 1 disks for k up to 5, i.e., f or 36, 38, 60, 62, 90, and 92 disks. The additional packings show the ''tightness'' of the curved hexagonal pattern for k 4 5: deleting a di sk does not change the optimum packing and its quality significantly, but adding a disk causes a substantial rearrangement in the optimum pa cking and substantially decreases the quality.