Lower bounds for the disk packing constant

Author:
David W. Boyd

Journal:
Math. Comp. **24** (1970), 697-704

MSC:
Primary 52.45; Secondary 40.00

DOI:
https://doi.org/10.1090/S0025-5718-1970-0278193-8

MathSciNet review:
0278193

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Abstract | References | Similar Articles | Additional Information

Abstract: An osculatory packing of a disk,, is an infinite sequence of disjoint disks, , contained in , chosen so that, for , has the largest possible radius, , of all disks fitting in . The exponent of the packing, , is the least upper bound of numbers, , such that . Here, we present a number of methods for obtaining lower bounds for , based on obtaining weighted averages of the curvatures of the . We are able to prove that . We use a number of well-known results from the analytic theory of matrices.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1970-0278193-8

Keywords:
Packing of disks,
exponent of packing,
nonnegative matrix,
numerical computation of eigenvalues,
Descartes's formula,
Soddy's formula,
osculatory packing,
Apollonian packing

Article copyright:
© Copyright 1970
American Mathematical Society