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On Partitions of an Equilateral Triangle

Published online by Cambridge University Press:  20 November 2018

R. L. Graham
R. L. Graham*
Affiliation:
Bell Telephone Laboratories, Murray Hill, New Jersey
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Let T denote a closed unit equilateral triangle. For a fixed integer n, let dn denote the infimum of all those x for which it is possible to partition T into n subsets, each subset having a diameter not exceeding x. We recall that the diameter of a plane set A is given by

where ρ (a, b) is the Euclidean distance between a and b.

In this note we determined dn for some small values of n. Typical values of dn are given in Table I. These values were obtained by three methods. As would be expected, as the value of n increases, the complexity of the argument needed to obtain dn also increases. We begin with the simplest case.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 19 , 1967 , pp. 394 - 409
Copyright
Copyright © Canadian Mathematical Society 1967

References

1. Fejes Tóth, L., Lagerungen in der Ebene, auf der Kugel und im Raum (Berlin, 1953).Google Scholar
2. Debrunner, Hadwiger, and Klee, , Combinatorial geometry in the plane (New York, 1983).Google Scholar
3. Lenz, H., Zur Zerlegung von Punktmengen in solche kleineren Durchmessers, Arch. Math., 6 (1955), 413–416.Google Scholar
4. Lenz, H., Über die Bedeckung ebener Punktmengen durch solche kleineren Durchmessers, Arch. Math., 7 (1956), 34–40.Google Scholar
5.Problem E 1311, Amer. Math. Monthly, 65 (1958), 775.Google Scholar
6.Problem E 1374, Amer. Math. Monthly, 66 (1959), 513.Google Scholar