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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



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
MathSciNet review: 0278193
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Abstract: An osculatory packing of a disk, $U$, is an infinite sequence of disjoint disks, $\{ {U_n}\}$, contained in $U$, chosen so that, for $n \geqq 2$, ${U_n}$ has the largest possible radius, ${r_n}$, of all disks fitting in $U\backslash ({U_1} \cup \cdots \cup {U_{n - 1}})$. The exponent of the packing, $S$, is the least upper bound of numbers, $t$, such that $\sum {r_n^t} = \infty$. Here, we present a number of methods for obtaining lower bounds for $S$, based on obtaining weighted averages of the curvatures of the ${U_n}$. We are able to prove that $S > 1.28467$. We use a number of well-known results from the analytic theory of matrices.

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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