On the solid-packing constant for circles

Author:
Z. A. Melzak

Journal:
Math. Comp. **23** (1969), 169-172

MSC:
Primary 52.45

DOI:
https://doi.org/10.1090/S0025-5718-1969-0244866-8

MathSciNet review:
0244866

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Abstract: A solid packing of a circular disk is a sequence of disjoint open circular subdisks whose total area equals that of . The MergelyanWesler theorem asserts that the sum of radii diverges; here numerical evidence is presented that the sum of ath powers of the radii diverges for every . This is based on inscribing a particular sequence of 19660 disks, fitting a power law for the radii, and relating the exponent of the power law to the above constant.

**[1]**S. N. Mergelyan,*Uniform approximations to functions of a complex variable*, Amer. Math. Soc. Translation**1954**(1954), no. 101, 99. MR**0060015****[2]**Oscar Wesler,*An infinite packing theorem for spheres*, Proc. Amer. Math. Soc.**11**(1960), 324–326. MR**0112078**, https://doi.org/10.1090/S0002-9939-1960-0112078-8**[3]**Z. A. Melzak,*Infinite packings of disks*, Canad. J. Math.**18**(1966), 838–852. MR**0203594**, https://doi.org/10.4153/CJM-1966-084-8**[4]**E. N. Gilbert,*Randomly packed and solidly packed spheres*, Canad. J. Math.**16**(1964), 286–298. MR**0162183**, https://doi.org/10.4153/CJM-1964-028-8**[5]**H. S. M. Coxeter,*Introduction to geometry*, John Wiley & Sons, Inc., New York-London, 1961. MR**0123930****[6]**S. Saks,*Theory of the Integral*, 2nd ed., Hafner, New York, 1958.

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DOI:
https://doi.org/10.1090/S0025-5718-1969-0244866-8

Article copyright:
© Copyright 1969
American Mathematical Society