Abstract
What is the densest packing of points in an infinite strip of widthw, where any two of the points must be separated by distance at least I? This question was raised by Fejes-Tóth a number of years ago. The answer is trivial for\(w \leqslant \sqrt 3 /2\) and, surprisingly, it is not difficult to prove [M2] for\(w = n\sqrt 3 /2\), wheren is a positive integer, that the regular triangular lattice gives the optimal packing. Kertész [K] solved the case\(w< \sqrt 2 \). Here we fill the first gap, i.e., the maximal density is determined for\(\sqrt 3 /2< w \leqslant \sqrt 3 \).
Article PDF
Similar content being viewed by others
References
[F1] G. Fejes-Tóth, New results in the theory of packing and covering, inConvexity and its Applications (P. Gruberet al., eds.) Birkhäuser, Basel, 1983, pp. 318–359. (Our problem is on p. 335.)
[F2] L. Fejes-Tóth, Parasites on the stem of a plant,Amer. Math. Monthly 78 (1971), 528–529.
[F3] L. Fejes-Tóth,Lagerungen in der Ebene auf der Kugel und in Raum, Springer-Verlag, Berlin, 1972.
[FG] J. H. Folkman and R. L. Graham, A packing inequality for compact discs,Canad. Math. Bull. 12 (1969), 745–752.
[G] R. L. Graham, Lecture at the 1986 AMS-IMS-SIAM Summer Research Conference on Discrete and Computational Geometry, Santa Cruz, California, July 1986.
[Gro] H. Groemer, Über dei Einlagerung von Kreisen in einen konvexen Bereich,Math. Z. 73 (1960), 285–294.
[HF] A. Heppes and L. Fejes-Tóth, Problem 1 of the M. Schweitzer mathematical competition 1966,Mat. Lapok 18 (1967), 108–109 (in Hungarian).
[H1] J. Horváth, The densest packing of ann-dimensional cylinder by unit spheres,Ann. Univ. Sci. Budapest, Eötvös Sect. Math. 15 (1972), 139–143 (in Russian);MR 49 #11391 (1973).
[H2] J. Horváth, Die Dichte einer Kugelpacking in einer 4-dimensionalen Schicht,Period. Math. Hungar. 5 (1974), 195–199.
[K] G. Kertész, On a problem of parasites, Dissertation, Budapest, 1982 (in Hungarian).
[M1] J. Molná, unpublished (see in [F1])
[M2] J. Molnár, Packing of congruent spheres in a strip,Acta Math. Hungar. 31 (1978), 173–183.
[MP] W. O. Moser and J. Pach, Research problems in discrete geometry, Problem 98, Montreal 1984 (mimeographed).
Author information
Authors and Affiliations
Additional information
This research was supported in part by the Hungarian National Science Foundation under Grant No. 1812.
Rights and permissions
About this article
Cite this article
Füredi, Z. The densest packing of equal circles into a parallel strip. Discrete Comput Geom 6, 95–106 (1991). https://doi.org/10.1007/BF02574677
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02574677