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The residual set dimension of the Apollonian packing

Published online by Cambridge University Press:  26 February 2010

David W. Boyd
David W. Boyd
Affiliation:
Department of Mathematics, The University of British Columbia, Vancouver, Canada
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In this paper we show that, for the Apollonian or osculatory packing C0 of a curvilinear triangle T, the dimension d(C0, T) of the residual set is equal to the exponent of the packing e(Co, T) = S. Since we have [5, 6] exhibited constructible sequences λ(K) and μ(K) such that λ(K) < S < μ(K), and μ(K)–λ(K) → 0 as κ → 0, we have thus effectively determined d(C0, T). In practical terms it is thus now known that 1·300197 < d(C0, T) < 1·314534.

Type
Research Article
Information
Mathematika , Volume 20 , Issue 2 , December 1973 , pp. 170 - 174
Copyright
Copyright © University College London 1973

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References

1. Artin, E., The gamma function (Holt, Rinehart and Winston, 1964).Google Scholar
2. Boyd, D. W., “Oscillatory packings by spheres”, Can. Math. Bull., 13 (1970), 59–64.CrossRefGoogle Scholar
3. Boyd, D. W., “Lower bounds for the disk packing constant”, Math. Comp., 24 (1970), 697–704.CrossRefGoogle Scholar
4. Boyd, D. W., “Disk packings which have non-extreme exponents”, Can. Math. Bull., 15 (1972), 341–344.CrossRefGoogle Scholar
5. Boyd, D. W., “The disk packing constant”, Aeq. Math., 7 (1972), 182–193.CrossRefGoogle Scholar
6. Boyd, D. W., “Improved bounds for the disk packing constant”, Aeq. Math., 9 (1973), 99–106.CrossRefGoogle Scholar
7. Eggleston, H. G., “On closest packing by equilateral triangles”, Proc. Camb. Phil. Soc, 49 (1953), 26–30.CrossRefGoogle Scholar
8. Eggleston, H. G., Problems in Euclidean space, pp. 160–165 (Pergamon Press, 1957).Google Scholar
9. Furstenberg, H. and Kesten, H., “Products of random matrices”, Ann. Math. Stat., 31 (1960) 457–469.CrossRefGoogle Scholar
10. Hirst, K. E., “The Apollonian packing of circles”, Lond. Math. Soc, 42 (1967), 281–291.CrossRefGoogle Scholar
11. Householder, A. S., The theory of matrices in numerical analysis (Blaisdell, Waltham, Mass., 1964).Google Scholar
12. Larman, D. G., “On the exponent of convergence of a packing of spheres”, Mathematika, 13 (1966), 57–59.CrossRefGoogle Scholar
13. Larman, D. G., “On the Besicovitch dimension of the residual set of arbitrarily packed disks in the plane”, J. Lond. Math. Soc., 42 (1967), 292–302.CrossRefGoogle Scholar
14. Melzak, Z. A., “Infinite packings of disks”, Canad. J. Math., 18 (1966), 838–852.CrossRefGoogle Scholar
15. Pólya, G. and Szegö, G., Aufgaben und Lehrsätze aus der Analysis, 4. Auflage (Springer-Verlag, Berlin, 1970).CrossRefGoogle Scholar
16. Schmeling, H. H. K.-B. von and Tschoegl, N. W., “Osculatory packing of finite areas with circles”, Nature, 225 (1970), 1119–1122.Google Scholar
17. Wilker, J. B., “Open disk packings of a disk”, Can. Math. Bull., 10 (1967), 395–415.CrossRefGoogle Scholar
18. Wilker, J. B., Almost perfect packings, thesis (Toronto, 1968).Google Scholar