Abstract
We show that ifP is a convex polygon which has no parallel sides, then the densest packing of the plane with congruent copies ofP is not lattice-like. As a corollary we obtain that, in the sense of Baire categories, for most convex disks densest packing is not lattice-like.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
G. Fejes Tóth and T. Zamfirescu, For most convex discs thinnest covering is not lattice-like,Intuitive Geometry, Colloquia Mathematica Societates János Bolyai, vol. 63, North-Holland, Amsterdam, 1994, pp. 105–108.
L. Fejes Tóth, Some packing and covering theorems,Acta Sci. Math. (Szeged) 12A (1950), 62–67.
P. M. Gruber, Typical convex bodies have surprisingly few neighbours in densest lattice packings,Studia Sci. Math. Hungar. 21 (1986), 163–173.
P. M. Gruber, Baire categories in convexity,Handbook of Convex Geometry (P. M. Gruber and J. M. Wills, eds.), North-Holland, Amsterdam, 1993, pp. 1327–1346.
B. Grünbaum and G. C. Shephard,Tilings and Patterns, Freeman, San Francisco, 1987.
H. Minkowski, Dichteste gitterförmige Lagerung kongruenter Körper,Nachr. Ges. Wiss. Göttingen (1904), 311–355 =Gesammelte Abhandlungen, vol. 2, Teubner, Leipzig, 1911, pp. 3–42.
C. A. Rogers,Packing and Covering, Cambridge University Press, Cambridge, 1964.
P. Schmitt, Disks with special properties of densest packing.Discrete Comput. Geom. 6 (1991), 181–190.
T. Zamfirescu, Baire categories in convexity,Atti Semin. Mat. Fis. Univ. Modena 39 (1991), 139–164.
Author information
Authors and Affiliations
Additional information
This research was supported by the Hungarian National Foundation for Scientific Research (OTKA) under Grant Nos. 1907 and 14218.
Rights and permissions
About this article
Cite this article
Tóth, G.F. Densest packings of typical convex sets are not lattice-like. Discrete & Computational Geometry 14, 1–8 (1995). https://doi.org/10.1007/BF02570693
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02570693