Lower bounds for the disk packing constant
Author:
David W. Boyd
Journal:
Math. Comp. 24 (1970), 697704
MSC:
Primary 52.45; Secondary 40.00
MathSciNet review:
0278193
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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 wellknown results from the analytic theory of matrices.
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 H. S. M. Coxeter, "Loxodromic sequences of tangent spheres," Aequationes Math., v. 1, 1968, pp. 104121. MR 38 #3765. MR 0235456 (38:3765)
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 G. H. Hardy, J. E. Littlewood & G. Pólya, Inequalities, Cambridge Univ. Press, New York, 1934.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197002781938
PII:
S 00255718(1970)02781938
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
