Skip to main content
SpringerLink
Log in

Global Optimization in Geometry — Circle Packing into the Square

  • Chapter
Essays and Surveys in Global Optimization

Abstract

The present review paper summarizes the research work done mostly by the authors on packing equal circles in the unit square in the last years.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 129.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Alefeld, G. and Herzberger, J. (1983). Introduction to Interval Computations. Academic Press, New York.

    Google Scholar 

  • Althöfer, I. and Koschnick, K.U. (1991). On the convergence of threshold accepting. Applied Mathematics and Optimization, 24:183–195.

    Article  MathSciNet  Google Scholar 

  • Ament, P. and Blind, G.(2000). Packing equal circles in a square. Studia Scientiarum Mathematicarum Hungarica, 36:313–316.

    Article  MathSciNet  Google Scholar 

  • Boll, D.W., Donovan, J., Graham, R.L., and Lubachevsky, B.D. (2000). Improving dense packing of equal disks in a square. Electronic Journal of Combinatorics, 7:R46.

    MathSciNet  Google Scholar 

  • Bolyai. F. (1904). Tentamen Juventutem Studiosam in Elementa Matheseos Purae, Elementaris Ac Sublimioris, Methodo Intituitiva, Evidentiaque Huic Propria, Introducendi, Volume 2, Second edition, pp. 119–122.

    Google Scholar 

  • Casado, L.G., García, I., and Sergeyev, Ya.D. (2000). Interval branch and bound algorithm for finding the first-zero-crossing-point in one-dimensional functions. Reliable Computing, 6:179–191.

    Article  MathSciNet  Google Scholar 

  • Casado, L.G., García, I., Szabó, P.G., and Csendes, T. (2001) Packing equal circles in a square. II. New results for up to 100 circles using the TAMSASS-PECS stochastic algorithm. In: Optimization Theory: Recent Developments from Mátraháza, pp. 207–224. Kluwer, Dordrecht.

    Google Scholar 

  • Croft, H.T., Falconer, K.J., and Guy, R.K. (1991). Unsolved Problems in Geometry, pp. 108–110. Springer, New York.

    Google Scholar 

  • Csallner, A.E. Csendes, T., and Markót, M.Cs. (2000). Multisection in interval methods for global optimization. I. Theoretical results. Journal of Global Optimization, 16:371–392.

    Article  Google Scholar 

  • Csendes, T. (1988). Nonlinear parameter estimation by global optimization—Efficiency and reliability. Acta Cybernetica, 8:361–370.

    MATH  MathSciNet  Google Scholar 

  • Csendes, T. and Ratz, D. (1997). Subdivision direction selection in interval methods for global optimization, SIAM Journal on Numerical Analysis, 34:922–938.

    Article  MathSciNet  Google Scholar 

  • Du, D.Z. and Pardalos, P.M. (1995). Minimax and Applications. Kluwer, Dordrecht.

    Google Scholar 

  • Dueck, G. and Scheuer, T. (1990). Threshold accepting: A general purpose optimization algorithm appearing superior to simulated annealing. Journal of Computational Physics, 90:161–175.

    Article  MathSciNet  Google Scholar 

  • Fejes Tóth, G. (1997). Handbook of Discrete and Computational Geometry. CRC Press, Boca Raton.

    Google Scholar 

  • Fejes Tóth, L. (1972). Lagerungen in der Ebene, auf der Kugel und im Raum. Springer-Verlag, Berlin.

    Google Scholar 

  • Fodor, F. (1999). The densest packing of 19 congruent circles in a circle. Geometriae Dedicata 74:139–145.

    Article  MATH  MathSciNet  Google Scholar 

  • Folkman, J.H. and Graham, R.L. (1969). A packing inequality for compact convex subsets of the plane. Canadian Mathematical Bulletin, 12:745–752.

    MathSciNet  Google Scholar 

  • Fukagawa, H. and Pedoe, D. (1989). Japanese temple geometry problems. San gaku. Charles Babbage Research Centre, Winnipeg.

    Google Scholar 

  • Goldberg, M. (1970). The packing of equal circles in a square. Mathematics Magazine, 43:24–30.

    Article  MATH  Google Scholar 

  • Goldberg, M. (1971). Packing of 14, 16, 17 and 20 circles in a circles. Mathematics Magazine, 44:134–139.

    Article  MATH  Google Scholar 

  • Graham, R.L. and Lubachevsky, B.D. (1995). Dense packing of equal disks in an equilateral triangle from 22 to 34 and beyond. Electronic Journal of Combinatorics 2:A1.

    MathSciNet  Google Scholar 

  • Graham, R.L. and Lubachevsky, B.D. (1996). Repeated patterns of dense packing of equal circles in a square, Electronic Journal of Combinatorics, 3:R17.

    Google Scholar 

  • Graham, R.L., Lubachevsky, B.D., Nurmela, K.J., and Östergard, P.R.J. (1998). Dense packing of congruent circles in a circle. Discrete Mathematics 181:139–154.

    Article  MathSciNet  Google Scholar 

  • Grannell, M. (1990). An Even Better Packing of Ten Equal Circles in a Square. Manuscript.

    Google Scholar 

  • de Groot, C., Monagan, M., Peikert, R., and Würtz, D. (1992). Packing circles in a square: review and new results. In: System Modeling and Optimization, pp. 45–54. Lecture Notes in Control and Information Services, vol. 180.

    Google Scholar 

  • de Groot, C., Peikert, R. and Würtz, D. (1990). The Optimal Packing of Ten Equal Circles in a Square. IPS Research Report No. 90-12, Eidgenössiche Technische Hochschule, Zürich.

    Google Scholar 

  • Grünbaum, B. (1990). An Improved Packing of Ten Circles in a Square. Manuscript.

    Google Scholar 

  • Hadwiger, H. (1944). Über extremale Punktverteilungen in ebenen Gebieten. Mathematische Zeitschrift, 49:370–373.

    Article  MATH  MathSciNet  Google Scholar 

  • Hammer, R. Hocks, M., Kulisch, U., and Ratz, D. (1993). Numerical Toolbox for Verified Computing. I. Springer-Verlag, Berlin.

    Google Scholar 

  • Hansen, E. (1992). Global Optimization Using Interval Analysis. Marcel Dekker, New York.

    Google Scholar 

  • van Hentenryck, P., McAllester, D., and Kapur, D. (1997). Solving polynomial systems using a branch and prune approach, SIAM Journal on Numerical Analysis, 34:797–827, 1997.

    Article  MathSciNet  Google Scholar 

  • Horst R. and Thoai. N.V. (1999). D.C. programming: Overview, Journal of Optimization Theory and Applications, 103:1–43.

    Article  MathSciNet  Google Scholar 

  • Hujter, M. (1999). Some numerical problems in discrete geometry. Computers and Mathematics with Applications, 38:175–178.

    Article  MATH  MathSciNet  Google Scholar 

  • Karnop, D.C. (1963). Random search techniques for optimization problems. Automatica, 1:111–121.

    Article  Google Scholar 

  • Kearfott, R.B. (1996). Test results for an interval branch and bound algorithm for equality-constrained optimization. In: Computational Methods and Applications, pp. 181–200. Kluwer, Dordrecht.

    Google Scholar 

  • Kirchner, K. and Wengerodt, G. (1987). Die dichteste Packung von 36 Kreisen in einem Quadrat. Beiträge zur Algebra und Geometrie, 25:147–159, 1987.

    MathSciNet  Google Scholar 

  • Knüppel, O. (1993a). PROFIL—Programmer's Runtime Optimized Fast Interval Library. Bericht 93.4., Technische Universität Hamburg-Harburg.

    Google Scholar 

  • Knüppel, O. (1993b). A Multiple Precision' Arithmetic for PROFIL. Bericht 93.6, Technische Universität Hamburg-Harburg.

    Google Scholar 

  • Kravitz, S. (1967). Packing cylinders into cylindrical containers. Mathematics Magazine, 40:65–71.

    Article  MATH  Google Scholar 

  • Locatelli, M. and Raber, U. (1999). A Deterministic global optimization approach for solving the problem of packing equal circles in a square. In: International Workshop on Global Optimization (GO.99), Firenze.

    Google Scholar 

  • Locatelli, M. and Raber, U. (2002). Packing equal circles in a square: A deterministic global optimization approach. Discrete Applied Mathematics, 122:139–166.

    Article  MathSciNet  Google Scholar 

  • Lubachevsky, B.D. (1991). How to simulate billiards and similar systems. Journal of Computational Physics, 94:255–283.

    Article  MATH  MathSciNet  Google Scholar 

  • Lubachevsky, B.D. and Graham, R.L. (1997). Curved hexagonal packings of equal disks in a circle. Discrete and Computational Geometry, 18:179–194.

    Article  MathSciNet  Google Scholar 

  • Lubachevsky, B.D. Graham, R.L., and Stillinger, F.H. (1997). Patterns and structures in disk packing. Periodica Mathematica Hungarica, 34:123–142.

    Article  MathSciNet  Google Scholar 

  • Lubachevsky, B.D. and Stillinger, F.H. (1990). Geometric properties of random disk packing. Journal of Statistical Physics, 60:561–583.

    Article  MathSciNet  Google Scholar 

  • Maranas, C.D., Floudas, C.A., and Pardalos, P.M. (1998). New results in the packing of equal circles in a square. Discrete Mathematics, 128:187–193.

    Google Scholar 

  • Markót, M.Cs. (2000). An interval method to validate optimal solutions of the “packing circles in a unit square” problems. Central European Journal of Operational Research, 8:63–78.

    MATH  Google Scholar 

  • Markót, M.Cs. (2003a). Optimal packing of 28 equal circles in a unit square—The first reliable solution. Numerical Algorithms, 37:253–261.

    Article  Google Scholar 

  • Markót, M.Cs. (2003b). Reliable Global Optimization Methods for Constrained Problems and Their Application for Solving Circle Packing Problems (in Hungarian). Ph.D. dissertation. Szeged. Available at http://www.inf.u-szeged.hu/~markot/phdmm.ps.gz

    Google Scholar 

  • Markót, M.Cs. and Csendes, T. (2004). A New verified optimization technique for the “packing circles in a unit square” problems. Forthcoming in SIAM Journal on Optimization.

    Google Scholar 

  • Markót, M.Cs., Csendes, T., and Csallner, A.E. (2000). Multisection in interval methods for global optimization. II. Numerical tests. Journal of Global Optimization, 16:219–228.

    Article  Google Scholar 

  • Matyas, J. (1965). Random optimization. Automatization and Remote Control, 26:244–251.

    MATH  MathSciNet  Google Scholar 

  • McDonnell, J.R. and Waagen, D. (1994). Evolving recurrent perceptrons for time-series modeling. IEEE Transactions on Neural Networks, 5:24–38.

    Article  Google Scholar 

  • Melissen, J.B.M. (1993). Densest packing for congruent circles in an equilateral triangle. American Mathematical Monthly, 100:916–925.

    Article  MATH  MathSciNet  Google Scholar 

  • Melissen, J.B.M. (1994a). Densest packing of six equal circles in a square. Element der Mathematik, 49:27–31.

    MATH  MathSciNet  Google Scholar 

  • Melissen, J.B.M. (1994b). Densest packing of eleven congruent circles in a circle. Geometriae Dedicata, 50:15–25.

    Article  MATH  MathSciNet  Google Scholar 

  • Melissen, J.B.M. (1994c). Optimal packing of eleven equal circles in an equilateral triangle. Acta Mathematica Hungarica, 65:389–393.

    Article  MATH  MathSciNet  Google Scholar 

  • Melissen, J.B.M. and Schuur, P.C. (1995). Packing 16, 17 or 18 circles in an equilateral triangle. Discrete Mathematics, 145:333–342.

    Article  MathSciNet  Google Scholar 

  • Milano, R. (1987). Configurations optimales de desques dans un polygon régulier. Mémoire de licence, Université Libre de Bruxelles.

    Google Scholar 

  • Mollard, M. and Payan, C. (1990). Some progress in the packing of equal circles in a square. Discrete Mathematics, 84:303–307.

    Article  MathSciNet  Google Scholar 

  • Moore, R.E. (1966). Interval Analysis. Prentice-Hall, Englewood Cliffs.

    Google Scholar 

  • Moser, L. (1960). Problem 24 (corrected), Canadian Mathematical Bulletin, 8:78.

    Google Scholar 

  • Neumaier, A. (2001). Introduction to Numerical Analysis. Cambridge Univ. Press, Cambridge.

    Google Scholar 

  • Nurmela, K.J. (1993). Constructing Combinatorial Designs by Local Search. Series A: Research Reports 27, Digital Systems Laboratory, Helsinki University of Technology.

    Google Scholar 

  • Nurmela, K.J. and Österghrd, P.R.J. (1997). Packing up to 50 equal circles in a square. Discrete and Computational Geometry, 18:111–120.

    Article  MathSciNet  Google Scholar 

  • Nurmela, K.J. and Östergård, P.R.J. (1999a). More optimal packing of equal circles in a square. Discrete and Computational Geometry, 22:439–457.

    Article  MathSciNet  Google Scholar 

  • Nurmela, K.J. and Östergård, P.R.J. (1999b). Optimal packing of equal circles in a square. In: Y. Alavi, D.R. Lick, and A. Schwenk (eds.), Combinatorics, Graph Theory, and Algorithms, pp. 671–680.

    Google Scholar 

  • Nurmela, K.J., Östergård, P.R.J., and aus dem Spring, R. (1999). Asymptotic Behaviour of Optimal Circle Packings in a Square. Canadian Mathematical Bulletin, 42:380–385, 1999.

    MathSciNet  Google Scholar 

  • Oler, N. (1961a). An inequality in the geometry of numbers. Acta Mathematica, 105:19–48.

    MATH  MathSciNet  Google Scholar 

  • Oler, N. (1961b). A finite packing problem. Canadian Mathematical Bulletin. 4:153–155.

    MATH  MathSciNet  Google Scholar 

  • Peikert, R. (1994). Dichteste Packungen von gleichen Kreisen in einem Quadrat, Elemente der Mathematik, 49:16–26.

    MATH  MathSciNet  Google Scholar 

  • Peikert, R., Würtz, D., Monagan, M., and de Groot, C. (1992). Packing circles in a square: A review and new results. In: P. Kall (ed.), System Modelhng and Optimization, pp. 45–54. Lecture Notes in Control and Information Sciences, vol. 180. Springer-Verlag, Berlin.

    Google Scholar 

  • Petris, J. and Hungerbüler, N. (1990). Manuscript.

    Google Scholar 

  • Pirl, U. (1969). Der Mindestabstand von n in der Einheitskreisscheibe gelegenen Punkten. Mathematische Nachrichten, 40:111–124.

    MATH  MathSciNet  Google Scholar 

  • Raber, U. (1999). Nonconvex All-Quadratic Global Optimization Problems: Solution Methods, Application and Related Topics. Ph.D. thesis. University of Trier.

    Google Scholar 

  • Rao, S.S. (1978). Optimization Theory and Applications. John Willey and Sons, New York.

    Google Scholar 

  • Ratschek, H. and Rokne, J. (1988). New Computer Methods for Global Optimization. Ellis Horwood, Chichester.

    Google Scholar 

  • Reis, G.E. (1975). Dense packing of equal circles within a circle. Mathematics Magazine, 48:33–37.

    Article  MATH  MathSciNet  Google Scholar 

  • Ruda, M. (1969). Packing circles in a rectangle (in Hungarian). Magyar Tudományos Akadémia Matematikai és Fizikai Tudományok Osztályának Közieményei, 19:73–87.

    MATH  MathSciNet  Google Scholar 

  • Schaer, J. (1965). The densest packing of nine circles in a square, Canadian Mathematical Bulletin, 8:273–277.

    MATH  MathSciNet  Google Scholar 

  • Schaer, J. (1971). On the densest packing of ten equal circles in a square. Mathematics Magazine, 44:139–140.

    MATH  Google Scholar 

  • Schaer, J. and Meir, A. (1965). On a geometric extremum problem. Canadian Mathematical Bulletin, 8:21–27.

    MathSciNet  Google Scholar 

  • Schlüter, K. (1979). Kreispackung in Quadraten. Elemente der Mathematik, 34:12–14.

    MATH  Google Scholar 

  • Schwartz, B.L. (1970). Separating points in a square. Journal of Recreational Mathematics, 3:195–204.

    Google Scholar 

  • Solis, F.J. and Wets, J.B. (1981). Minimization by random search techniques. Mathematics of Operations Research, 6:19–50.

    Article  MathSciNet  Google Scholar 

  • E. Specht's packing web site. http://www.packomania.com

    Google Scholar 

  • Specht, E. and Szabó, P.G. (2004). Lattice and Near-Lattice Packings of Equal Circles in a Square. In preparation.

    Google Scholar 

  • Staar, Gy. (1990). The Lived Mathematics (in Hungarian). Gondolat, Budapest.

    Google Scholar 

  • Szabó, P.G. (2000a). Optimal packing of circles in a square (in Hungarian). Polygon, X:48–64.

    Google Scholar 

  • Szabó, P.G. (2000b). Some new structures for the “equal circles packing in a square” problem. Central European Journal of Operations Research, 8:79–91.

    MATH  Google Scholar 

  • P.G. Szabó (2001). Sangaku—Wooden boards of mathematics in Japanese temples (in Hungarian). KöMaL, 7:386–388.

    Google Scholar 

  • Szabó, P.G. (2004). Optimal substructures in optimal and approximate circle packing. Forthcoming in Beitrdge zur Algebra und Geometrie.

    Google Scholar 

  • Szabó, P.G. and Csendes, T. (2001). Dezső Lázár and the densest packing of equal circles in a square problem (in Hungarian). Magyar Tudomány, 8:984–985.

    Google Scholar 

  • Szabó, P.G. Csendes, T., Casado, L.G., and García, I. (2001). Packing equal circles in a square. I. Problem setting and bounds for optimal solutions. In: Optimization Theory: Recent Developments from Mátraháza, pp. 191–206. Kluwer, Dordrecht.

    Google Scholar 

  • Szabó, P.G. and Specht, E. (2005). Packing up to 200 Equal Circles in a Square. Submitted for publication.

    Google Scholar 

  • Tarnai, T. (1997). Packing of equal circles in a circle. Structural Morphology: Toward the New Millenium, pp. 217–224. The University of Nottingham, Nottingham.

    Google Scholar 

  • Tarnai, T. and Gáspár, Zs. (1995–96). Packing of equal circles in a square. Acta Technica Academiae Scientiarum Hungaricae, 107(1–2):123–135.

    Google Scholar 

  • Valette, G. (1989). A better packing of ten circles in a square. Discrete Mathematics, 76:57–59.

    Article  MATH  MathSciNet  Google Scholar 

  • Wengerodt, G (1983). Die dichteste Packung von 16 Kreisen in einem Quadrat. Beiträge zur Algebra und Geometrie, 16:173–190.

    MathSciNet  Google Scholar 

  • Wengerodt, G. (1987a). Die dichteste Packung von 14 Kreisen in einem Quadrat. Beiträge zur Algebra und Geometrie, 25:25–46.

    MATH  MathSciNet  Google Scholar 

  • Wengerodt, G. (1987b). Die dichteste Packung von 25 Kreisen in einem Quadrat. Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae. Sectio Mathematica, 30:3–15.

    MATH  MathSciNet  Google Scholar 

  • Würtz, D., Monagan, M., and Peikert, R. (1994). The history of packing circles in a square. Maple Technical Newsletter, 0:35–42.

    Google Scholar 

Download references

Authors
  1. Péter Gábor Szabó
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mihály Csaba Markót
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tibor Csendes
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. École Polytechnique de Montréal and GERAD, Montréal, Canada

    Charles Audet  & Gilles Savard  & 

  2. HEC Montréal & GERAD, Montréal, Canada

    Pierre Hansen

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Springer Science+Business Media, Inc.

About this chapter

Cite this chapter

Szabó, P.G., Markót, M.C., Csendes, T. (2005). Global Optimization in Geometry — Circle Packing into the Square. In: Audet, C., Hansen, P., Savard, G. (eds) Essays and Surveys in Global Optimization. Springer, Boston, MA. https://doi.org/10.1007/0-387-25570-2_9

Download citation

Publish with us

Policies and ethics

Search

Navigation