# Dense packings of 3k(k+1)+1 equal disks in a circle for k = 1, 2, 3, 4, and 5

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## Abstract

For each *k*≥1 and corresponding hexagonal number *h(k)=3k(k*+1)+1, we introduce *m(k)=(k*−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 the one of 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) = 2, *m*(4)=6, and *m*(5)=24) 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° rotation. When *k*≥3, the curved hexagonal packings are not mirror symmetric but they occur in *m(k)*/2 image-reflection pairs. For *k*→∞, the density (covering fraction) of curved hexagonal packings tends to *π*^{2}/12. The limit is smaller than the density of the known optimum disk packing in the infinite plane. We present packings that are better 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 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.

## Keywords

Dense Packing Equilateral Triangle Central Disk Covering Fraction Optimum Packing## Preview

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## References

- [CFG]H. T. Croft, K. J. Falconer and R. K. Guy,
*Unsolved Problems in Geometry*, Springer Verlag, Berlin, 1991, 107–111.Google Scholar - [FG]J. H. Folkman and R. L. Graham, A packing inequality for compact convex subsets of the plane,
*Canad. Math. Bull.***12**(1969), 745–752.Google Scholar - [GL1]R. L. Graham and B. D. Lubachevsky, Dense packings of equal disks in an equilateral triangle: from 22 to 34 and beyond,
*The Electronic Journ. of Combinatorics***2**(1995), #A1.Google Scholar - [GL2]R. L. Graham and B. D. Lubachevsky, Dense packings of equal disks in a circle: from 25 to 61 and beyond (
*In preparation*.)Google Scholar - [G]M. Goldberg, Packing of 14, 16, 17 and 20 circles in a circle,
*Math. Mag.***44**(1971), 134–139.Google Scholar - [K]S. Kravitz, Packing cylinders into cylindrical containers,
*Math. Mag.***40**(1967), 65–71.Google Scholar - [L]B.D. Lubachevsky, How to simulate billiards and similar systems,
*J. Computational Physics***94**(1991), 255–283.CrossRefGoogle Scholar - [LS]B. D. Lubachevsky and F. H. Stillinger, Geometric properties of random disk packings,
*J. Statistical Physics***60**(1990), 561–583.CrossRefGoogle Scholar - [O]
- [R]G. E. Reis, Dense packing of equal circles within a circle,
*Math. Mag.***48**(1975), 33–37.Google Scholar