Abstract
Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. In Euclidean space it is possible for every circle in such a packing to have integer radius of curvature, and we call such a packing an integral Apollonian circle packing. There are infinitely many different integral packings; these were studied in Part I (J. Number Theory 100, 1–45, 2003). Integral circle packings also exist in spherical and hyperbolic space, provided a suitable definition of curvature is used and again there are an infinite number of different integral packings. This paper studies number-theoretic properties of such packings. This amounts to studying the orbits of a particular subgroup \({\mathcal{A}}\) of the group of integral automorphs of the indefinite quaternary quadratic form \(Q_{{\mathcal{D}}}(w,x,y,z)=2(w^{2}+x^{2}+y^{2}+z^{2})-(w+x+y+z)^{2}\) . This subgroup, called the Apollonian group, acts on integer solutions \(Q_{{\mathcal{D}}}(w,x,y,z)=k\) . This paper gives a reduction theory for orbits of \({\mathcal{A}}\) acting on integer solutions to \(Q_{{\mathcal{D}}}(w,x,y,z)=k\) valid for all integer k. It also classifies orbits for all k≡0 (mod 4) in terms of an extra parameter n and an auxiliary class group (depending on n and k), and studies congruence conditions on integers in a given orbit.
Similar content being viewed by others
References
Aharonov, D., Stephenson, K.: Geometric sequences of discs in the Apollonian packing. Algebra Anal. 9(3), 104–140 (1997) [English version: St. Petersburg Math. J. 9, 509–545 (1998)]
Buell, D.: Binary Quadratic Forms. Classical Theory and Modern Computations. Springer, New York (1989)
Coxeter, H.S.M.: Introduction to Geometry, 2nd edn. Wiley, New York (1969)
Davenport, H.: Multiplicative Number Theory, 2nd edn. Springer, New York (1980). Revised by H. Montgomery
Graham, R.L., Lagarias, J.C., Mallows, C.L., Wilks, A., Yan, C.: Apollonian circle packings: geometry and group theory, I. The Apollonian group. Discrete Comput. Geom. 34, 547–585 (2005)
Graham, R.L., Lagarias, J.C., Mallows, C.L., Wilks, A., Yan, C.: Apollonian circle packings: geometry and group theory, II. Super-Apollonian group. Discrete Comput. Geom. 35, 1–36 (2006)
Graham, R.L., Lagarias, J.C., Mallows, C.L., Wilks, A., Yan, C.: Apollonian circle packings: geometry and group theory, III. Higher dimensions. Discrete Comput. Geom. 35, 37–72 (2006)
Graham, R.L., Lagarias, J.C., Mallows, C.L., Wilks, A., Yan, C.: Apollonian circle packings: number theory. J. Number Theory 100, 1–45 (2003)
Kasner, E., Supnick, F.: The Apollonian packing of circles. Proc. Natl. Acad. Sci. U.S.A. 29, 378–384 (1943)
Lagarias, J.C., Mallows, C.L., Wilks, A.: Beyond the Descartes circle theorem. Am. Math. Mon. 109, 338–361 (2002)
Lang, S.: Algebraic Number Theory, 2nd edn. Springer, New York (1994)
Mandelbrot, B.B.: The Fractal Geometry of Nature. Freeman, New York (1982)
Mathews, G.A.: Theory of Numbers, 2nd edn. Cambridge (1892). Reprint: Chelsea, New York (1962)
Maxwell, G.: Sphere packings and hyperbolic reflection groups. J. Algebra 79, 78–97 (1982)
Northshield, S.: On integral Apollonian circle packings. J. Number Theory 119, 171–193 (2006)
Ramaré, O.: Approximate formulae for L(1,χ). Acta Arith. 100, 245–266 (2001)
Ratcliffe, J.G., Tschantz, S.T.: On the representation of integers by the Lorentzian quadratic form. J. Funct. Anal. 150, 498–525 (1997)
Siegel, C.L.: Über die analytische Theorie der quadratischen Formen. Ann. Math. 36, 527–606 (1935)
Söderberg, B.: Apollonian tiling, the Lorentz group, and regular trees. Phys. Rev. A 46(4), 1859–1866 (1992)
Wilker, J.B.: Inversive geometry. In: Davis, C., Grünbaum, B., Sherk, F.A. (eds.) The Geometric Vein, pp. 379–442. Springer, New York (1981)
Author information
Authors and Affiliations
Corresponding author
Additional information
Much of this work was done while the authors were at AT&T Labs-Research, whom the authors thank for support. N. Eriksson was also supported by an NDSEG fellowship and J.C. Lagarias by NSF grant DMS-0500555.
Rights and permissions
About this article
Cite this article
Eriksson, N., Lagarias, J.C. Apollonian circle packings: Number theory II. Spherical and hyperbolic packings. Ramanujan J 14, 437–469 (2007). https://doi.org/10.1007/s11139-007-9052-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-007-9052-6