Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-13T07:11:50.079Z Has data issue: false hasContentIssue false
  • English
  • Français

Packing and covering with convex discs

Published online by Cambridge University Press:  26 February 2010

L. Fejes Tóth  and
A. Florian
L. Fejes Tóth
Affiliation:
Mathematical Institute, Hungarian Academy of Sciences, Reáltanoda ut. 13–15, H-1053 Budapest, Hungary
A. Florian
Affiliation:
Institut für Mathematik, Universität Salzburg, PetersbrunnstraBe 19, A-5020 Salzburg, Austria
Get access

Extract

Before turning to the questions to be considered in this paper, we recall two other problems. Let C(a, p) be the class of all convex discs of area not less than a given constant a and perimeter not greater than a given constant p. What is the densest packing and what is the most economical covering of the Euclidean plane with discs from C(a, p)?

Both problems are interesting only if p2/a < 8√3, i.e. if p is less than the perimeter of a regular hexagon of area a. In this case, the densest packing arises from a regular hexagonal tiling by rounding off the corners of the tiles by equal circular arcs so as to obtain smooth hexagons of area a and perimeter p.

Type
Research Article
Information
Mathematika , Volume 29 , Issue 2 , December 1982 , pp. 181 - 193
Copyright
Copyright © University College London 1982

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Bambah, R. P. and Rogers, C. A.Covering the plane with convex sets. J. London Math. Soc, 27 (1952), 304–314.CrossRefGoogle Scholar
2.Blaschke, W.. Vorlesungen über Differentialgeometrie II (Chelsea Publishing Comp., New York, 1967).Google Scholar
3.Tóth, G. Fejes Covering with convex discs of equal area and perimeter. Manuscript.Google Scholar
4.Tóth, L. FejesSome packing and covering theorems. Acta Univ. Szeged, Acta Sci. Math., 12/A (1950), 62–67.Google Scholar
5.Tóth, L. FejesÜber den Affinumfang. Math. Nachrichten., 6/A (1951), 51–64CrossRefGoogle Scholar
6.Tóth, L. FejesFilling of a domain by isoperimetric discs. Publ. Math. Debrecen., 5 (1957), 119–127CrossRefGoogle Scholar
7.Tóth, L. FejesRegular Figures (Pergamon Press, Oxford-London, 1964)Google Scholar
8Tóth, L. FejesLagerungen in der Ebene, aufder Kugel und im Raum (Springer, Berlin-Heidelberg-New York, second edition, 1972).CrossRefGoogle Scholar
9.Tóth, L. FejesApproximation of convex domains by polygons. Studia Sci. Math. Hungar, 15 (1980).Google Scholar