Discrete & Computational Geometry

, Volume 5, Issue 1, pp 1–11 | Cite as

Penny-packing and two-dimensional codes

  • R. L. Graham
  • N. J. A. Sloane


We consider the problem of packingn equal circles (i.e., pennies) in the plane so as to minimize the second momentU about their centroid. These packings are also minimal-energy two-dimensional codes. Adding one penny at a time according to the greedy algorithm produces a unique sequence of packings for the first 75 pennies, and appears to produce optimal packings for infinitely many values ofn. Several other conjectures are proposed, and a table is given of the best packings known forn≤500. For largen, U∼√3n2/(4π).


Greedy Algorithm Discrete Comput Geom Hexagonal Lattice Optimal Packing Neighboring Element 
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Copyright information

© Springer-Verlag New York Inc. 1990

Authors and Affiliations

  • R. L. Graham
    • 1
  • N. J. A. Sloane
    • 1
  1. 1.AT&T Bell LaboratoriesMathematical Sciences Research CenterMurray HillUSA

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