Apollonian circle packings: Number theory

Journal of Number Theory

Volumn 100, Issue 1, 2003, Pages 1-45

  Graham, Ronald L ^a,c   Lagarias, Jeffrey C ^a   Mallows, Colin L ^a,d   Wilks, Allan R ^a   Yan, Catherine H ^b  

^a AT AND T LABS RESEARCH   (United States)

^b TEXAS A AND M UNIVERSITY   (United States)

^c UNIVERSITY OF CALIFORNIA   (United States)

^d AVAYA LABS RESEARCH   (United States)

Author keywords

Apollonian circles; Circle packings; Diophantine equations

Indexed keywords

EID: 0037691992     PISSN: 0022314X     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0022-314X(03)00015-5     Document Type: Article

References (44)

1. D. Anharonov, K. Stephenson, Geometric sequences of dics in the Apollonian packing, Algebra 1 Analiz 9 (3) (9) 104-140 [English version: St. Petersburg Math. J. 9 (1998) 509-545]
2. A.N. Andrianov, Dirichlet series that correspond to representations of zero by indefinite quadratic forms, Algebra i Analiz 1 (30) (1989) 71-82 [English Version: St. Petersburg Math. J. 1 (1990) 635-646].
3. D. Boyd, The disk-packing constant, Aequationes Math. 7 (1971) 182-193.
4. D. Boyd, Improved bounds for the disk-packing constant, Aequationes Math. 9 (1973) 99-106.
5. D. Boyd, The osculatory packing of a three-dimensional sphere, Canad. J. Math. 25 (1973) 303-322.
6. D. Boyd, The residual set dimenesion of the Apollonian packing, Mathematika 20 (1973) 170-174.
7. D. Boyd, The sequence of radii of the Apollonian packing, Math. Comp. 39 (1982) 249-254.
8. H.S.M. Coxeter, The problem of Apollonius, Amer. Math. Monthly 75 (1968) 5-15.
9. T.W. Cusick, Continuants with bounded digits, Mathematika 24 (1977) 166-172.
10. I. Daubechies, J.C. Lagarias, Sets of matrices all infinite products of which converge, Linear Algebra Appl. 161 (1992) 227-263. Corrigendum and Addendum, Linear Algebra Appl. 327 (2001) 69-83.
11. H. Davenport, Multiplicative Number Theory, 3rd Edition, Revised and with and introduction by H. L. Montgomery, Springer, New York, 2000.
12. 2, unpublished manuscript, February, 1993.
13. K.J. Falconer, The Geometry of Fractal Sets, in: Cambridge Tracts in Mathematics, Vol. 85, Cambridge University Press, Cambridge, 1986.
14. L.R. Ford, Fractions, Amer. Math. Monthly 45 (1938) 586-601.
15. C.F. Gauss, Disquisitiones Arithmeticae, Leipzig 1801. (Reprinted in: Werke.) English translation: Springer, Berlin.
16. R.L. Graham, J.C. Lagarias, C.L. Mallows, A. Wilks, C. Yan, Apollonian packings: geometry and group theory, I. Apollonian group, eprint: arXiv math. MG/0010298.
17. R.L. Graham, J.C. Lagarias, C.L. Mallows, A. Wilks, C. Yan, Apollonian packings: geometry and group theory, II. Super-Apollonian group and integral packings, eprint: arXiv math. MG/0010302.
18. R.L. Graham, J.C. Lagarias, C.L. Mallows, A. Wilks, C. Yan, Apollonian packings: geometry and group theory, III. Higher dimensions, eprint: arXiv math. MG/0010324.
19. G.H. Hardy, E.M. Wright, Introduction to the Theory of Numbers, 4th Edition, Oxford University Press, Oxford, 1960.
20. K.E. Hirst, The Apollonian packing of circles, J. Lond. Math. Soc. 42 (1967) 281-291.
21. A. Hurwitz, Solution to Problem 3084, Vol. 13, 1906, p. 164. In: Mathematische Werke, Vol II, Birkhäuser, Basel, 1934, p. 751.
22. E. Kasner, F. Supnick, The Apollonian packing of circles, Proc. Nat. Acad. Sci. USA 29 (1943) 378-384.
23. J.C. Lagarias, C.L. Mallows, A. Wilks, Beyond the Descartes circle theorem, Amer. Math. Monthly 109 (2002) 338-361 eprint: arXiv math.MG/0101066.
24. E. Landau, Vorlesungen über Zahlentheorie, Vol. I, Teubner, Leipzig, 1927. Translation: Elementary Number Theory, Chelsea, New York.
25. D.G. Larman, On the exponent of convergence of a packing of spheres, Mathematika 13 (1966) 57-59.
26. P.D. Lax, R.S. Phillips, The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces, J. Funct. Anal. 46 (1982) 280-350.
27. B.B. Mandelbrot, The Fractal Geometry of Nature, Freeman, New York, 1982.
28. G.B. Mathews, Theory of Numbers, 2nd Edition, Chelsea, New York, 1961 (First edition: Cambridge, 1892.).
29. J.G. Mauldon, Sets of equally inclined spheres, Canad. J. Math. 14 (1962) 509-516.
30. Z.A. Melzak, Infinite packings of disks, Canad. J. Math. 18 (1966) 838-853.
31. P.J. Nicholls, Diophantine approximation via the modular group, J. London Math. Soc. 17 (1978) 11-17.
32. S. Northshield, On Apollonian circle packings, preprint.
33. H. Rademacher, Lectures on Elementary Number Theory, Blaisdell, New York, 1964.
34. H. Rademacher, Topics in Analytic Number Theory, Springer, Berlin, 1973.
35. J.G. Ratcliffe, S.T. Tschantz, On the representation of integers by the Lorentzian quadratic form, J. Funct. Anal. 150 (1997) 498-525.
36. H.F. Sandham, A square as the sum of seven squares, Quart. J. Math. (Oxford) 4 (1953) 230-236.
37. F. Soddy, The bowl of integers and the Hexlet, Nature 139 (1937) 77-79.
38. B. Söderberg, Apollonian tiling, the Lorentz group, and regular trees, Phys. Rev. A 46 (4) (1992) 1859-1866.
39. P.B. Thomas, D. Dhar, The Hausdorff dimension of the Apollonian packing of circles, J. Phys. A 27 (1994) 2257-2268.
40. C. Tricot, A new proof for the residual set dimension of the Apollonian packing, Math. Proc. Cambridge Philos. Soc. 96 (1984) 413-423.
41. A.I. Weiss, On isoclinal sequences of spheres, Proc. Amer. Math. Soc. 88 (1983) 665-671.
42. J.B. Wilker, Open disk packings of a disk, Canad. Math. Bull. 10 (1967) 395-415.
43. J.B. Wilker, Inversive geometry, in: C. Davis, B. Grünbaum, F.A. Sherk (Eds.), The Geometric Vein, Springer, New York, 1981, pp. 379-442.
44. S.K. Zaremba, La methode des "bonnes treillis" pour le calcul des integralesfc multiples, in: S.K. Zaremba (Ed.), Applications of Number Theory to Numerical Analysis, Montreal, 1971, Academic Press, New York, 1972, pp. 39-119.