Abstract
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.
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[B] A. H. Boerdijk, Some remarks concerning close-packing of equal spheres,Philips Res. Reports 7 (1952), 303–313.
[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.
[CG] C. N. Campopiano and B. G. Glazer, A coherent digital amplitude and phase modulation scheme,IEEE Trans. Comm. 10 (1962), 90–95.
[CP] J. Cannon and C. Playoust,An Introduction to MAGMA, School of Mathematics, University of Sydney, 1993.
[C] T. Y. Chow, Penny-packings with minimal second moments,Combinatorica,15 (1995), 151–159.
[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.
[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.
[CFG] H. T. Croft, K. J. Falconer, and R. K. Guy,Unsolved Problems in Geometry, Springer-Verlag, New York, 1991.
[D] A. S. Drud,CONOPT User's Manual, Bagsvaerd, Denmark, 1993.
[FFR] J. Farges, M. F. de Feraudy, B. Raoult, and G. Torchet, Noncrystalline structure of argon clusters, I: Polyicosahedral structure of Ar N clusters, 20<N<50,J. Chem. Phys. 78 (1983), 5067–5080.
[F] L. Fejes Tóth,Regular Figures, Pergamon Press, Oxford, 1964.
[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.
[FGK] R. Fourer, D. M. Gay, and B. W. Kernighan,AMPL: A Modeling Language for Mathematical Programming, Scientific Press, South San Francisco, CA, 1993.
[FW] H. Freudenthal and B. L. van der Waerden, Over een bewering van Euclides,Simon Stevin 25 (1947), 115–121.
[GS] R. L. Graham and N. J. A. Sloane, Penny-packing and two-dimensional codes,Discrete Comput. Geom. 5 (1990), 1–11.
[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.
[Ha] T. C. Hales, The status of the Kepler conjecture,Math. Intelligencer 16(3) (1994), 47–58.
[HS1] R. H. Hardin and N. J. A. Sloane, New spherical 4-designs,Discrete Math. 106/107 (1992), 255–264.
[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.
[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.
[HM1] M. R. Hoare and J. McInnes, Statistical mechanics and morphology of very small atomic clusters,Faraday Discussions Chem. Soc. 61 (1976), 12–24.
[HM2] M. R. Hoare and J. McInnes, Morphology and statistical statics of simple micro-clusters,Adv. Phys. 32 (1983), 791–821.
[HP1] M. R. Hoare and P. Pal, Physical cluster mechanics: statics and energy systems for monoatomic systems,Adv. Phys. 20 (1971), 161–196.
[HP2] M. R. Hoare and P. Pal, Physical cluster mechanics—statistical thermodynamics and nucleation theory for monoatomic systems,Adv. Phys. 24 (1975), 645–678.
[Hs] W.-Y. Hsiang, On the sphere packing problem and the proof of Kepler's conjecture,Internat. J. Math. 4 (1993), 739–831.
[J] N. W. Johnson, Convex polyhedra with regular faces,Canad. J. Math. 18 (1966), 169–200.
[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.
[MF] C. D. Maranas and C. A. Floudas, A global optimization approach for Lennard-Jones microclusters,J. Chem. Phys. 97 (1992), 7667–7678.
[Me] J. B. M. Melissen, Densest packings of congruent circles in an equilateral triangle,Amer. Math. Monthly 100 (1993), 916–925.
[MP] M. Mollard and C. Payan, Some progress in the packing of equal circles in a square,Discrete Math. 84 (1990), 303–307.
[Mu] D. J. Muder, A new bound on the local density of sphere packings,Discrete Comput. Geom. 10 (1993), 351–375.
[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.
[N] J. A. Northby, Structure and binding of Lennard-Jones clusters: 13≤N≤147,J. Chem. Phys. 87 (1987), 6166–6175.
[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.
[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.
[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.
[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.
[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.
[We] W. Wefelmeier, Ein geometrisches Modell des Atomkerns,Z. Phys. 107 (1937), 332–346.
[Wi] L. T. Wille, Minimum-energy configurations of atomic clusters—new results obtained by simulated annealing,Chem. Phys. Lett. 133 (1987), 405–410.
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Sloane, N.J.A., Hardin, R.H., Duff, T.D.S. et al. Minimal-energy clusters of hard spheres. Discrete & Computational Geometry 14, 237–259 (1995). https://doi.org/10.1007/BF02570704
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DOI: https://doi.org/10.1007/BF02570704