The sequence of radii of the Apollonian packing
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- by David W. Boyd PDF
- Math. Comp. 39 (1982), 249-254 Request permission
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 $1.300197 < S < 1.314534$.References
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Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Math. Comp. 39 (1982), 249-254
- MSC: Primary 52A45
- DOI: https://doi.org/10.1090/S0025-5718-1982-0658230-7
- MathSciNet review: 658230