The sequence of radii of the Apollonian packing

Author:
David W. Boyd

Journal:
Math. Comp. **39** (1982), 249-254

MSC:
Primary 52A45

DOI:
https://doi.org/10.1090/S0025-5718-1982-0658230-7

MathSciNet review:
658230

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Abstract | References | Similar Articles | Additional Information

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 .

**[1]**David W. Boyd,*The disk-packing constant*, Aequationes Math.**7**(1971), 182–193. MR**0303420**, https://doi.org/10.1007/BF01818512**[2]**David W. Boyd,*Improved bounds for the disk-packing constant*, Aequationes Math.**9**(1973), 99–106. MR**0317180**, https://doi.org/10.1007/BF01838194**[3]**David W. Boyd,*The residual set dimension of the Apollonian packing*, Mathematika**20**(1973), 170–174. MR**0493763**, https://doi.org/10.1112/S0025579300004745**[4]**D. W. Boyd, "Solution to problem P. 276,"*Canad. Math. Bull.*, v. 23, 1980, pp. 251-253.**[5]**H. S. M. Coxeter, "Problem P. 276,"*Canad. Math. Bull.*, v. 22, 1979, p. 248.**[6]**Edward Kasner and Fred Supnick,*The Apollonian packing of circles*, Proc. Nat. Acad. Sci. U.S.A.**29**(1943), 378–384. MR**0009128****[7]**Z. A. Melzak,*Infinite packings of disks*, Canad. J. Math.**18**(1966), 838–852. MR**0203594**, https://doi.org/10.4153/CJM-1966-084-8**[8]**Z. A. Melzak,*On the solid-packing constant for circles*, Math. comp.**23**(1969), 169–172. MR**0244866**, https://doi.org/10.1090/S0025-5718-1969-0244866-8**[9]**J. B. Wilker,*Sizing up a solid packing*, Period. Math. Hungar.**8**(1977), no. 2, 117–134. MR**0641055**, https://doi.org/10.1007/BF02018498

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DOI:
https://doi.org/10.1090/S0025-5718-1982-0658230-7

Article copyright:
© Copyright 1982
American Mathematical Society