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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
Article

Abstract

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).

Keywords

Greedy Algorithm Discrete Comput Geom Hexagonal Lattice Optimal Packing Neighboring Element 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    C. H. Bennett, Serially deposited amorphous aggregates of hard spheres,J. Appl. Phys. 43 (1972), 2727–2734.CrossRefzbMATHGoogle Scholar
  2. 2.
    A. R. Calderbank and N. J. A. Sloane, New trellis codes based on lattices and cosets,IEEE Trans. Inform. Theory 33 (1987), 177–195.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    C. N. Campopiano and B. G. Glazer, A coherent digital amplitude and phase modulation scheme,IEEE Trans. Comm. 10 (1962), 90–95.CrossRefGoogle Scholar
  4. 4.
    J. H. Conway and N. J. A. Sloane, A fast encoding method for lattice codes and quantizers,IEEE Trans. Inform. Theory 29 (1983), 820–824.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    G. Fejes Tóth, P. Gritzmann, and J. M. Wills, Finite sphere packing and sphere covering,Discrete Comput. Geom. 4 (1989), 19–40.MathSciNetCrossRefGoogle Scholar
  6. 6.
    J. H. Folkman and R. L. Graham, A packing inequality for compact convex subsets of the plane,Canad. Math. Bull. 12 (1969), 745–752.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    G. J. Foschini, R. D. Gitlin, and S. B. Weinstein, Optimization of two-dimensional signal constellations in the presence of Gaussian noise,IEEE Trans. Comm. 22 (1974), 28–38.CrossRefzbMATHGoogle Scholar
  8. 8.
    H. Groemer, Über die Einlagerung von Kreisen in einen konvexen Bereich,Math. Z. 73 (1960), 285–294.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    M. R. Hoare, Structure and dynamics of simple microclusters,Adv. Chem. Phys. 40 (1979), 49–135.zbMATHGoogle Scholar
  10. 10.
    M. R. Hoare and J. A. McInnes, Morphology and statistical statics of simple microclusters,Adv. in Phys. 32 (1983), 791–821.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    K. Kinoshita and M. Kobayashi, Two-dimensionally arrayed optical-fibre splicing with a CO2 laser,Appl. Optics 21 (1982), 3419–3422.CrossRefzbMATHGoogle Scholar
  12. 12.
    D. McCullough,The Great Bridge, Simon and Schuster, New York, 1972.zbMATHGoogle Scholar
  13. 13.
    N. Oler, An inequality in the geometry of numbers,Acta Math. 105 (1961), 19–48.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    N. J. A. Sloane and B. K. Teo, Theta series and magic numbers for close-packed spherical clusters,J. Chem. Phys. 83 (1985), 6520–6534.MathSciNetCrossRefGoogle Scholar
  15. 15.
    Wegner, Über endliche Kreispackungen in der Ebene,Studia Sci. Math. Hungar., to appear.Google Scholar

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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