Geodesic
Infinite Packings of Disks
Canadian journal of mathematics, Tome 18 (1966) no. 1, pp. 838-852

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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. 
Melzak, Z. A. Infinite Packings of Disks. Canadian journal of mathematics, Tome 18 (1966) no. 1, pp. 838-852. doi: 10.4153/CJM-1966-084-8
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[1] 1. Coxeter, H. S. M., Introduction to geometry (New York, 1961). Google Scholar

[2] 2. Gilbert, E. N., Randomly packed and solidly packed spheres, Can. J. Math., 16 (1964), 286–298. Google Scholar

[3] 3. Mergelyan, S. N., Uniform approximation to functions of a complex variable, Amer. Math. Soc. Transi., 101 (1954). Google Scholar

[4] 4. Soddy, F., Coxeter, cf. H. S. M., Introduction to geometry (New York, 1961). p. xx. Google Scholar

[5] 5. Wesler, O., Infinite packing theorem for spheres, Proc. Amer. Math. Soc, 11 (1960), 324–326. Google Scholar

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