Abstract
In 1969 Pirl provided the densest packings ofn equal circles in a circle forn ≤ 10. We will prove the optimality for the packings that were conjectured forn=11. The proof is based on elementary combinatorial and analytical techniques.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berezin, A.: An unexpected result in classical electrostatics,Nature 315 (1985), 104.
Coxeter, H. S. M., Greening, M. G. and Graham, R. L.: Sets of points with given maximum separation (Problem E1921),Amer. Math. Monthly 75 (1968), 192–193.
Croft, H. T., Falconer, K. J. and Guy, R. K.:Unsolved Problems in Geometry, Springer Verlag, Berlin, 1991, pp. 107–111.
Fejes Tóth, L.:Lagerungen in der Ebene, auf der Kugel und im Raum, (1953) 2e Auflage, Springer-Verlag, Berlin, 1972.
Goldberg, M.: Packing of 14, 16, 17 and 20 circles in a circle,Math. Mag. 44 (1971), 134–139.
Graham, R. L.: Private communication.
de Groot, C., Peikert, R. and Würtz, D.: The optimal packing of ten equal circles in a square, ETH Zürich IPS Research Report No. 90–12 (1990). Submitted toDiscrete Math.
Janssen, A. J. E. M. and Melissen, J. B. M.: An unexpected result in classical electrostatics, Problem 92-16,SIAM Review 34 (1992), 648.
Kirchner, K. and Wengerodt, G.: Die dichteste Packung von 36 Kreisen in einem Quadrat,Beiträge Algebra Geom. 25 (1987), 147–159.
Kravitz, S.: Packing cylinders into cylindrical containers,Math. Mag. 40 (1967), 65–71.
Melissen, J. B. M.: Densest packings of congruent circles in an equilateral triangle,Amer. Math. Monthly 100 (1993), 916–925.
Melissen, J. B. M.: Densest packing of six equal circles in a square,Elem. Math. 49, (1994), 27–31.
Melissen, J. B. M.: Optimal packings of eleven equal circles in an equilateral triangle, to appear inActa Math. Hung. 65 (1994).
Melissen, J. B. M. and Schuur, P. C.: Packing 16, 17 and 18 circles in an equilateral triangle, to appear inDiscrete Math.
Moser, L.: Problem 24 (corrected),Canad. Math. Bull. 3 (1960), 78.
Moser, W. O. and Pach, J.:Research Problems in Discrete Geometry, Montreal, 1984.
Oler, N.: A finite packing problem,Canad. Math. Bull. 4 (1961), 153–155.
Peikert, R., Würtz, D., Monagan, M. and de Groot, C.: Packing circles in a square: a review and new resultsProceedings of the 15th IFIP Conference on System Modelling and Optimization, 1991, Springer Lecture Notes in Control and Information Sciences 180, Springer, New York, pp. 45–54.
Pirl, U.: Der Mindestabstand vonn in der Einheitskreisscheibe gelegenen Punkten,Math. Nachr. 40 (1969), 111–124.
Schaer, J. and Meir, A.: On a geometric extremum problem,Canad. Math. Bull. 8 (1965), 21–27.
Schaer, J.: The densest packing of 9 circles in a square,Canad. Math. Bull. 8 (1965), 273–277.
Schaer, J.: Unpublished manuscript (1964), private communication.
Wengerodt, G.: Die dichteste Packung von 16 Kreisen in einem Quadrat,Beiträge Algebra Geom. 16 (1983), 173–190.
Wengerodt, G.: Die dichteste Packung von 14 Kreisen in einem Quadrat,Beiträge Algebra Geom. 25 (1987), 25–46.
Wengerodt, G.: Die dichteste Packung von 25 Kreisen in einem Quadrat,Ann. Univ. Sci. Budapest Rolando Eötvös Sect. Math 30 (1987), 3–15.
Wille, L. T. and Vennik, J.: Electrostatic minimisation by simulated annealing,J. Phys. A: Math. Gen. 18 (1985), L1113–1117.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Melissen, H. Densest packings of eleven congruent circles in a circle. Geom Dedicata 50, 15–25 (1994). https://doi.org/10.1007/BF01263647
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01263647