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Mathematics of Computation

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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

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