Optimization Letters

, Volume 12, Issue 3, pp 585–613 | Cite as

Globally optimized packings of non-uniform size spheres in \(\mathbb {R}^{d}\): a computational study

  • János D. Pintér
  • Frank J. Kampas
  • Ignacio Castillo
Original Paper


In this work we discuss the following general packing problem: given a finite collection of d-dimensional spheres with (in principle) arbitrarily chosen radii, find the smallest sphere in \(\mathbb {R}^{d}\) that contains the given d-spheres in a non-overlapping arrangement. Analytical (closed-form) solutions cannot be expected for this very general problem-type: therefore we propose a suitable combination of constrained nonlinear optimization methodology with specifically designed heuristic search strategies, in order to find high-quality numerical solutions in an efficient manner. We present optimized sphere configurations with up to \(n = 50\) spheres in dimensions \(d = 2, 3, 4, 5\). Our numerical results are on average within 1% of the entire set of best known results for a well-studied model-instance in \(\mathbb {R}^{2}\), with new (conjectured) packings for previously unexplored generalizations of the same model-class in \(\mathbb {R}^{d}\) with \(d= 3, 4, 5.\) Our results also enable the estimation of the optimized container sphere radii and of the packing fraction as functions of the model instance parameters n and 1 / n, respectively. These findings provide a general framework to define challenging packing problem-classes with conjectured numerical solution estimates.


General finite sphere packings in \(\mathbb {R}^{d}\) LGO solver suite for global-local constrained optimization Hybrid solution strategy Numerical results Regression analysis based optimum estimates 


  1. 1.
    Black, K., Chakrapani, C., Castillo, I.: Business Statistics for Contemporary Decision Making, 2nd Canadian edn. Wiley, Toronto (2014)Google Scholar
  2. 2.
    Castillo, I., Kampas, F.J., Pintér, J.D.: Solving circle packing problems by global optimization: numerical results and industrial applications. Eur. J. Oper. Res. 191, 786–802 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Castillo, I., Sim, T.: A spring-embedding approach for the facility layout problem. J. Oper. Res. Soc. 55, 73–81 (2004)CrossRefzbMATHGoogle Scholar
  4. 4.
    Chaikin, P.M.: Thermodynamics and hydrodynamics of hard spheres: the role of gravity. In: Cates, M.E., Evans, M.R. (eds.) Soft and Fragile Matter: Nonequilibrium Dynamics, Metastability and Flow, vol. 53, pp. 315–348. Institute of Physics Publishing, Bristol (2000)CrossRefGoogle Scholar
  5. 5.
    Chaikin, P.M., Lubensky, T.C.: Principles of Condensed Matter Physics. Cambridge University Press, New York (2000)Google Scholar
  6. 6.
    Cheng, Z.D., Russell, W.B., Chaikin, P.M.: Controlled growth of hard-sphere colloidal crystals. Nature 401, 893–895 (1999)CrossRefGoogle Scholar
  7. 7.
    Cohn, H.: Order and disorder in energy minimization. In: Proceedings of the International Congress of Mathematicians, Hyderabad, India, pp. 2416–2443. Hindustan Book Agency, New Delhi (2010)Google Scholar
  8. 8.
    Cohn, H., Kumar, A., Miller, S.D., Radchenko, D., Viazovska, M.S.: The sphere packing problem in dimension 24. (2016) arXiv:1603.06518v1
  9. 9.
    Conway, J.H.: Sphere packings, lattices, codes, and greed. In: Proceedings of the International Congress of Mathematicians, Zürich, Switzerland 1994, pp. 45–55. Birkhäuser Verlag, Basel (1995)Google Scholar
  10. 10.
    Conway, J.H., Sloane, N.J.A.: Sphere Packings, Lattices and Groups, 3rd edn. Springer Science + Business Media, New York (1998)zbMATHGoogle Scholar
  11. 11.
    de Gennes, P.G.: Granular matter: a tentative view. Rev. Mod. Phys. 71, S374–S382 (1999)CrossRefGoogle Scholar
  12. 12.
    Fasano, G.: Solving Non-standard Packing Problems by Global Optimization and Heuristics. Springer, New York (2014)CrossRefzbMATHGoogle Scholar
  13. 13.
    Fasano, G., Pintér, J.D. (eds.): Optimized Packings with Applications. Springer, New York (2015)zbMATHGoogle Scholar
  14. 14.
    Fejes Tóth, L.: Regular Figures. Pergamon Press, Macmillan, New York (1964)zbMATHGoogle Scholar
  15. 15.
    Friedman, E.: Erich’s Packing Center. (2017).
  16. 16.
    GNU Project: The GNU Compiler Collection (GCC). (2015).
  17. 17.
    Griess, R.L.: Positive definite lattices of rank at most 8. J. Number Theory 103, 77–84 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Griess, R.L.: An Introduction to Groups and Lattices: Finite Groups and Positive Definite Rational Lattices. International Press, Somerville, MA (2011)zbMATHGoogle Scholar
  19. 19.
    Hales, T.C.: A proof of the Kepler conjecture. Ann. Math. 162, 1065–1185 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hifi, M., M’Hallah, R.: A literature review on circle and sphere packing problems: models and methodologies. In: Advances in Operations Research (2009). doi: 10.1155/2009/150624
  21. 21.
    Jadrich, R., Schweizer, K.S.: Equilibrium theory of the hard sphere fluid and glasses in the metastable regime up to jamming. I. Thermodynamics. J. Chem. Phys. (2013a). doi: 10.1063/1.4816275
  22. 22.
    Jadrich, R., Schweizer, K.S.: Equilibrium theory of the hard sphere fluid and glasses in the metastable regime up to jamming. II. Structure and application to hopping dynamics. J. Chem. Phys. (2013b). doi: 10.1063/1.4816276
  23. 23.
    Kampas, F.J., Pintér, J.D.: Configuration analysis and design by using optimization tools in Mathematica. Math. J. 10, 128–154 (2006)Google Scholar
  24. 24.
    Kepler, J.: The Six-Cornered Snowflake. Oxford Classic Texts in the Physical Sciences (Illustrated reprint). Oxford University Press, Oxford (2014)Google Scholar
  25. 25.
    Leech, J.: Notes on sphere packings. Can. J. Math. 19, 251–267 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Melissen, J.B.M.: Packing and Covering with Circles. Ph.D. Dissertation, Universiteit Utrecht (1997)Google Scholar
  27. 27.
    Nesterenko, V.F.: Dynamics of Heterogeneous Materials. Springer, New York (2001)CrossRefGoogle Scholar
  28. 28.
    Olmos, L., Martin, C.L., Bouvard, D.: Sintering of mixtures of powders: experiments and modelling. Powder Technol. 190, 134–140 (2009)CrossRefGoogle Scholar
  29. 29.
    Pintér, J.D.: Global Optimization in Action. Kluwer, Dordrecht (1996)CrossRefzbMATHGoogle Scholar
  30. 30.
    Pintér, J.D.: LGO—a program system for continuous and Lipschitz global optimization. In: Bomze, I., Csendes, T., Horst, R., Pardalos, P.M. (eds.) Developments in Global Optimization, pp. 183–197. Kluwer, Dordrecht (1997)CrossRefGoogle Scholar
  31. 31.
    Pintér, J.D.: Globally optimized spherical point arrangements: model variants and illustrative results. Ann. Oper. Res. 104, 213–230 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Pintér, J.D.: LGO—A Model Development and Solver System for Global–Local Nonlinear Optimization, User’s Guide, Current edn. Published and distributed by Pintér Consulting Services Inc, Halifax, NS (2016)Google Scholar
  33. 33.
    Pintér, J.D.: How difficult is nonlinear optimization? A practical solver tuning approach, with illustrative results. Ann. Oper. Res. (2017). doi: 10.1007/s10479-017-2518-z
  34. 34.
    Pintér, J.D., Kampas, F.J.: Nonlinear optimization in Mathematica with MathOptimizer Professional. Math. Educ. Res. 10(1), 1–18 (2005)Google Scholar
  35. 35.
    Pintér, J.D., Kampas, F.J.: MathOptimizer Professional: key features and illustrative applications. In: Liberti, L., Maculan, N. (eds.) Global Optimization: From Theory to Implementation, pp. 263–280. Springer, New York (2006)CrossRefGoogle Scholar
  36. 36.
    Pintér, J.D., Kampas, F.J.: Benchmarking nonlinear optimization software in technical computing environments. I. Global optimization in Mathematica with MathOptimizer Professional. TOP 21, 133–162 (2013)CrossRefzbMATHGoogle Scholar
  37. 37.
    Pintér, J.D., Kampas, F.J.: Getting Started with MathOptimizer Professional. Published and distributed by Pintér Consulting Services Inc, Halifax, NS (2015)Google Scholar
  38. 38.
    Riskin, M.D., Bessette, K.C., Castillo, I.: A logarithmic barrier approach to solving the dashboard planning problem. INFOR 41, 245–257 (2003)Google Scholar
  39. 39.
    Sahimi, M.: Heterogeneous Materials I: Linear Transport and Optical Properties. Springer, New York (2003a)zbMATHGoogle Scholar
  40. 40.
    Sahimi, M.: Heterogeneous Materials II: Nonlinear and Breakdown Properties and Atomistic Modeling. Springer, New York (2003b)zbMATHGoogle Scholar
  41. 41.
    Sloane, N.J.A.: The sphere-packing problem. (2002) arXiv:math/0207256
  42. 42.
    Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423, 623–656 (1948)Google Scholar
  43. 43.
    Specht, E.: (2017).
  44. 44.
    Stortelder, W.J.H., de Swart, J.J.B., Pintér, J.D.: Finding elliptic Fekete point sets: two numerical solution approaches. J. Comput. Appl. Math. 130, 205–216 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Steinby, M., Thomas, W.: Trees and term rewriting in 1910: On a paper by Axel Thue. Bull. Eur. Assoc. Theor. Comput. Sci. 72, 256–269 (2000)MathSciNetGoogle Scholar
  46. 46.
    Szabó, P.G., Markót, M.Cs, Csendes, T., Specht, E., Casado, L.G., García, I.: New Approaches to Circle Packing in a Square With Program Codes. Springer, New York (2007)zbMATHGoogle Scholar
  47. 47.
    Szpiro, G.G.: Kepler’s Conjecture. Wiley, New York (2003)zbMATHGoogle Scholar
  48. 48.
    Thue, A.: Om nogle geometrisk taltheoretiske theoremer. Forhdl. Skand. Naturforsk. 14, 352–353 (1892)zbMATHGoogle Scholar
  49. 49.
    Thue, A.: Über die dichteste Zusammenstellung von kongruenten Kreisen in der Ebene. Christ. Vid. Selsk. Skr. 1, 3–9 (1910)zbMATHGoogle Scholar
  50. 50.
    Torquato, S.: Random Heterogeneous Materials: Microstructure and Macroscopic Properties. Springer, New York (2002)CrossRefzbMATHGoogle Scholar
  51. 51.
    Viazovska, M.S.: The sphere packing problem in dimension 8. (2016). arXiv:1603.04246
  52. 52.
    Wolfram Research: Mathematica (Release 11, December 2016). Wolfram Research Inc, Champaign, IL (2016)Google Scholar
  53. 53.
    Zallen, R.: The Physics of Amorphous Solids. Wiley, New York (1983)CrossRefGoogle Scholar
  54. 54.
    Zohdi, T.I.: Variational bounds for thermal fields in media with heterogeneous microstructure. Math. Mech. Solids 19, 434–439 (2014a)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Zohdi, T.I.: Additive particle deposition and selective laser processing: a computational manufacturing framework. Comput. Mech. 54, 171–191 (2014b)CrossRefzbMATHGoogle Scholar
  56. 56.
    Zong, C.: Sphere Packings (edited by Talbot, J.) Springer, New York (1999)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of Industrial and Systems EngineeringLehigh UniversityBethlehemUSA
  2. 2.Pintér Consulting Services, Inc.HalifaxCanada
  3. 3.Physicist at Large Consulting LLCBryn MawrUSA
  4. 4.Lazaridis School of Business and EconomicsWilfrid Laurier UniversityWaterlooCanada

Personalised recommendations