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Infinite Packings of Disks

Published online by Cambridge University Press:  20 November 2018

Z. A. Melzak
Z. A. Melzak*
Affiliation:
University of British Columbia
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Let U be the closed disk in the plane, centred at the origin, and of unit radius. By a solid packing, or briefly a packing, C of U we shall understand a sequence ﹛Dn﹜, n = 1, 2, … , of open proper disjoint subdisks of U, such that the plane Lebesgue measures of U and of are the same. If rn is the radius of Dn and the complex number cn represents its centre, then the conditions for C to be a packing are

It was proved by Mergelyan (3) that for any packing the sum of the radii diverges:

1

Mergelyan's demonstration of (1) is somewhat involved and leans heavily on the machinery of functions of a complex variable. An elegant direct proof of (1) is given by Wesler (5), who uses the technique of projecting the boundaries of the disks of the packing on a diameter I of U.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 18 , 1966 , pp. 838 - 852
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. Coxeter, H. S. M., Introduction to geometry (New York, 1961).Google Scholar
2. Gilbert, E. N., Randomly packed and solidly packed spheres, Can. J. Math., 16 (1964), 286–298.Google Scholar
3. Mergelyan, S. N., Uniform approximation to functions of a complex variable, Amer. Math. Soc. Transi., 101 (1954).Google Scholar
4. Soddy, F., Coxeter, cf. H. S. M., Introduction to geometry (New York, 1961). p. xx.Google Scholar
5. Wesler, O., Infinite packing theorem for spheres, Proc. Amer. Math. Soc, 11 (1960), 324–326.Google Scholar