The sequence of radii of the Apollonian packing
Author:
David W. Boyd
Journal:
Math. Comp. 39 (1982), 249254
MSC:
Primary 52A45
MathSciNet review:
658230
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Abstract: We consider the distribution function of the curvatures of the disks in the Apollonian packing of a curvilinear triangle. That is, counts the number of disks in the packing whose curvatures do not exceed x. We show that approaches the limit S as x tends to infinity, where S is the exponent of the packing. A numerical fit of a curve of the form to the values of for produces the estimate which is consistent with the known bounds .
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 D. W. Boyd, "The disk packing constant," Aequationes Math., v. 1, 1972, pp. 182193. MR 0303420 (46:2557)
 [2]
 D. W. Boyd, "Improved bounds for the disk packing constant," Aequationes Math., v. 9, 1973, pp. 99106. MR 0317180 (47:5728)
 [3]
 D. W. Boyd, "The residual set dimension of the Apollonian packing," Mathematika, v. 20, 1973, pp. 170174. MR 0493763 (58:12732)
 [4]
 D. W. Boyd, "Solution to problem P. 276," Canad. Math. Bull., v. 23, 1980, pp. 251253.
 [5]
 H. S. M. Coxeter, "Problem P. 276," Canad. Math. Bull., v. 22, 1979, p. 248.
 [6]
 E. Kasner & F. Supnick, "The Apollonian packing of circles," Proc. Nat. Acad. Sci. U.S.A., v. 29, 1943, pp. 378384. MR 0009128 (5:106e)
 [7]
 Z. A. Melzak, "Infinite packings of disks," Canad. J. Math., v. 18, 1966, pp. 838852. MR 0203594 (34:3443)
 [8]
 Z. A. Melzak, "On the solidpacking constant for circles," Math. Comp., v. 23, 1969, pp. 169172. MR 0244866 (39:6179)
 [9]
 J. B. Wilker, "Sizing up a solid packing," Period. Math. Hungar. (2), v. 8, 1977, pp. 117134. MR 0641055 (58:30759)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198206582307
PII:
S 00255718(1982)06582307
Article copyright:
© Copyright 1982
American Mathematical Society
