Discrete & Computational Geometry

, Volume 14, Issue 3, pp 237–259 | Cite as

Minimal-energy clusters of hard spheres

  • N. J. A. Sloane
  • R. H. Hardin
  • T. D. S. Duff
  • J. H. Conway


What is the tightest packing ofN equal nonoverlapping spheres, in the sense of having minimal energy, i.e., smallest second moment about the centroid? The putatively optimal arrangements are described forN≤32. A number of new and interesting polyhedra arise.


Hard Sphere Discrete Comput Geom Optimal Arrangement Convex Polyhedron Pentagonal Bipyramid 
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.


  1. [B] A. H. Boerdijk, Some remarks concerning close-packing of equal spheres,Philips Res. Reports 7 (1952), 303–313.MathSciNetzbMATHGoogle Scholar
  2. [CS] 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. [CG] 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. [CP] J. Cannon and C. Playoust,An Introduction to MAGMA, School of Mathematics, University of Sydney, 1993.Google Scholar
  5. [C] T. Y. Chow, Penny-packings with minimal second moments,Combinatorica,15 (1995), 151–159.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [CSW1] T. Coleman, D. Shalloway, and Z. Wu, Isotropic effective simulated annealing searches for low energy molecular cluster states,Comput. Optim. Applic. 2 (1993), 145–170.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [CSW2] T. Coleman, D. Shalloway, and Z. Wu, A parallel build-up algorithm for global energy minimizations of molecular clusters using effective energy simulated annealing,J. Global Optim. 4 (1994), 171–185.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [CFG] H. T. Croft, K. J. Falconer, and R. K. Guy,Unsolved Problems in Geometry, Springer-Verlag, New York, 1991.CrossRefzbMATHGoogle Scholar
  9. [D] A. S. Drud,CONOPT User's Manual, Bagsvaerd, Denmark, 1993.Google Scholar
  10. [FFR] J. Farges, M. F. de Feraudy, B. Raoult, and G. Torchet, Noncrystalline structure of argon clusters, I: Polyicosahedral structure of ArN clusters, 20<N<50,J. Chem. Phys. 78 (1983), 5067–5080.CrossRefGoogle Scholar
  11. [F] L. Fejes Tóth,Regular Figures, Pergamon Press, Oxford, 1964.zbMATHGoogle Scholar
  12. [FGW] 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.CrossRefGoogle Scholar
  13. [FGK] R. Fourer, D. M. Gay, and B. W. Kernighan,AMPL: A Modeling Language for Mathematical Programming, Scientific Press, South San Francisco, CA, 1993.Google Scholar
  14. [FW] H. Freudenthal and B. L. van der Waerden, Over een bewering van Euclides,Simon Stevin 25 (1947), 115–121.MathSciNetzbMATHGoogle Scholar
  15. [GS] R. L. Graham and N. J. A. Sloane, Penny-packing and two-dimensional codes,Discrete Comput. Geom. 5 (1990), 1–11.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [GW] P. Gritzmann and J. M. Wills, Finite packing and covering, inHandbook of Convex Geometry, P. M. Gruber and J. M. Wills, eds., North-Holland, Amsterdam, 1993, pp. 861–898.Google Scholar
  17. [Ha] T. C. Hales, The status of the Kepler conjecture,Math. Intelligencer 16(3) (1994), 47–58.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [HS1] R. H. Hardin and N. J. A. Sloane, New spherical 4-designs,Discrete Math. 106/107 (1992), 255–264.MathSciNetCrossRefGoogle Scholar
  19. [HS2] R. H. Hardin and N. J. A. Sloane, A new approach to the construction of optimal designs,J. Statist. Plan. Inference 37 (1993), 339–369.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [HS3] R. H. Hardin and N. J. A. Sloane, Expressing (a 2+b 2+c 2+d 2)3 as a sum of 23 sixth powers,J. Combin. Theory Ser. A,68 (1994), 481–485.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [HM1] M. R. Hoare and J. McInnes, Statistical mechanics and morphology of very small atomic clusters,Faraday Discussions Chem. Soc. 61 (1976), 12–24.CrossRefGoogle Scholar
  22. [HM2] M. R. Hoare and J. McInnes, Morphology and statistical statics of simple micro-clusters,Adv. Phys. 32 (1983), 791–821.MathSciNetCrossRefGoogle Scholar
  23. [HP1] M. R. Hoare and P. Pal, Physical cluster mechanics: statics and energy systems for monoatomic systems,Adv. Phys. 20 (1971), 161–196.CrossRefGoogle Scholar
  24. [HP2] M. R. Hoare and P. Pal, Physical cluster mechanics—statistical thermodynamics and nucleation theory for monoatomic systems,Adv. Phys. 24 (1975), 645–678.CrossRefGoogle Scholar
  25. [Hs] W.-Y. Hsiang, On the sphere packing problem and the proof of Kepler's conjecture,Internat. J. Math. 4 (1993), 739–831.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [J] N. W. Johnson, Convex polyhedra with regular faces,Canad. J. Math. 18 (1966), 169–200.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [KN] R. B. Kearfott and M. Novoa III, Algorithm 681: INTBIS, a portable interval Newton/bisection package,ACM Trans. Math. Software 16 (1990), 152–157.CrossRefzbMATHGoogle Scholar
  28. [MF] C. D. Maranas and C. A. Floudas, A global optimization approach for Lennard-Jones microclusters,J. Chem. Phys. 97 (1992), 7667–7678.CrossRefGoogle Scholar
  29. [Me] J. B. M. Melissen, Densest packings of congruent circles in an equilateral triangle,Amer. Math. Monthly 100 (1993), 916–925.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [MP] M. Mollard and C. Payan, Some progress in the packing of equal circles in a square,Discrete Math. 84 (1990), 303–307.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [Mu] D. J. Muder, A new bound on the local density of sphere packings,Discrete Comput. Geom. 10 (1993), 351–375.MathSciNetCrossRefzbMATHGoogle Scholar
  32. [MS] B. A. Murtagh and M. A. Saunders, MINOS 5.1 User's Guide, Technical Report SOL 83-20R, Department of Operations Research, Stanford University, Stanford, CA, 1987.Google Scholar
  33. [N] J. A. Northby, Structure and binding of Lennard-Jones clusters: 13≤N≤147,J. Chem. Phys. 87 (1987), 6166–6175.CrossRefGoogle Scholar
  34. [PWM] R. Peikert, D. Würtz, M. Monogan, and C. de Groot, Packing circles in a square: a review and new results,Maple Technical Newsletter 6 (1991), 28–34.Google Scholar
  35. [RFF] B. Raoult, J. Farges, M. F. de Feraudy, and G. Torchet, Comparison between icosahedral, decahedral, and crystalline Lennard-Jones models containing 500 to 6000 atoms,Philos. Mag. B 60 (1989) 881–906.CrossRefGoogle Scholar
  36. [S] D. Shalloway, Packet annealing: a deterministic method for global minimization: application to molecular conformation, inRecent Advances in Global Optimization, C. A. Flanders and P. M. Pardalos, eds., Princeton University Press, Princeton, NJ, 1992, pp. 433–477.Google Scholar
  37. [ST] 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
  38. [Wa] M. Walter, Constructing polyhedra without being told how to!, inShaping Space: A Polyhedral Approach, M. Senechal and G. Fleck, eds., Birkhäuser, Boston, MA, 1988, pp. 44–51.Google Scholar
  39. [We] W. Wefelmeier, Ein geometrisches Modell des Atomkerns,Z. Phys. 107 (1937), 332–346.CrossRefGoogle Scholar
  40. [Wi] L. T. Wille, Minimum-energy configurations of atomic clusters—new results obtained by simulated annealing,Chem. Phys. Lett. 133 (1987), 405–410.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1995

Authors and Affiliations

  • N. J. A. Sloane
    • 1
  • R. H. Hardin
    • 1
  • T. D. S. Duff
    • 1
  • J. H. Conway
    • 2
  1. 1.Mathematical Sciences Research CenterAT & T Bell LaboratoriesMurray HillUSA
  2. 2.Mathematics DepartmentPrinceton UniversityPrincetonUSA

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