Penny-packings with minimal second moments

Abstract

We consider the problem of packingn disks of unit diameter in the plane so as to minimize the second moment about their centroid. Our main result is an algorithm which constructs packings that are optimal among hexagonal packings. Using the algorithm, we prove that, except forn=212, then-point packings obtained by Graham and Sloane [1] are optimal among hexagonal packings. We also prove a result that makes precise the intuition that the “greedy algorithm” of Graham and Sloane produces approximately circular packings.

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References

  1. [1]

    R. L. Graham andN. J. A. Sloane: Penny-packing and two-dimensional codes,Discrete Comput. Geom. 5 (1990), 1–11.

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  2. [2]

    R. P. Feynman, R. B. Leighton, andM. Sands:The Feynman Lectures on Physics, 19–6, Vol. 1, Addison-Wesley, Reading, Massachusetts, 1963.

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Chow, T.Y. Penny-packings with minimal second moments. Combinatorica 15, 151–158 (1995). https://doi.org/10.1007/BF01200751

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Mathematics Subject Classification (1991)

  • 52
  • 05