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Mathematics of Computation

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The sequence of radii of the Apollonian packing

Author: David W. Boyd
Journal: Math. Comp. 39 (1982), 249-254
MSC: Primary 52A45
MathSciNet review: 658230
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Abstract: We consider the distribution function $ N(x)$ of the curvatures of the disks in the Apollonian packing of a curvilinear triangle. That is, $ N(x)$ counts the number of disks in the packing whose curvatures do not exceed x. We show that $ \log N(x)/\log x$ 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 $ y = A{n^s}$ to the values of $ {N^ - }(1000n)$ for $ n = 1,2, \ldots ,6400$ produces the estimate $ S \approx 1.305636$ which is consistent with the known bounds $ 1.300197 < S < 1.314534$.

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Article copyright: © Copyright 1982 American Mathematical Society