, Volume 15, Issue 2, pp 151–158 | Cite as

Penny-packings with minimal second moments

  • Timothy Y. Chow


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.

Mathematics Subject Classification (1991)

52 05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    R. L. Graham andN. J. A. Sloane: Penny-packing and two-dimensional codes,Discrete Comput. Geom. 5 (1990), 1–11.Google Scholar
  2. [2]
    R. P. Feynman, R. B. Leighton, andM. Sands:The Feynman Lectures on Physics, 19–6, Vol. 1, Addison-Wesley, Reading, Massachusetts, 1963.Google Scholar

Copyright information

© Akadémiai Kiadó 1995

Authors and Affiliations

  • Timothy Y. Chow
    • 1
  1. 1.Department of MathematicsM.I.T.CambridgeUSA

Personalised recommendations