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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
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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.

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

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