Apollonian Circle Packings: Number Theory

By R. Graham, Jeffrey C. Lagarias, Colin L. Mallows, A. Wilks, Catherine H Yan. 2000

125 citations recorded by the source corpus.

Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. It is possible for every circle in such a packing to have integer radius of curvature, and we call such a packing an {\em integral Apollonian circle packing.} This paper studies number-theoretic properties of the set of integer curvatures appearing in such packings. Each Descartes quadruple of four tangent circles in the packing gives an integer solution to the {\em Descartes equation}, which relates the radii of curvature of four mutually tangent circles: $2 (x^2 + y^2 + z^2 + w^2) - (x + y + z + w)^2 = 0.$ Each integral Apollonian circle packing is classified by a certain {\em root quadruple} of integers that satisfies the Descartes equation, and that corresponds to a particular quadruple of circles appearing in the packing. We determine asymptotics for the number of root quadruples of size below $T$. We study which integers occur in a given integer packing, and determine congruence restrictions which sometimes apply. Finally, we present evidence suggesting that the set of integer radii of curvatures that appear in an integral Apollonian circle packing has positive density, and in fact represents all sufficiently large integers not excluded by congruence conditions. In a series of companion papersr ``Apollonian Circle Packings: Geometry and Group Theory,'' we investigate a variety of group-theoretic properties of these configurations, as well as various extensions to higher dimensions and other spaces, such as hyperbolic space.

Subfields: Computational Theory and Mathematics, Materials Chemistry

View Paper: https://www.semanticscholar.org/paper/7e87c83b70652e66afa1ada254ea397dd8c54955

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Citations in the corpus

Slowly decaying waves on spherically symmetric spacetimes and ultracompact neutron stars (2014, methodology): “In packed swiss cheese cosmology models, a configuration of such spheres is chosen so that they are tangent to each other and arranged into a higher-dimensional version of the Apollonian packing of circles [29].”
A proof of the positive density conjecture for integer Apollonian circle packings (2010, methodology): “in Figure 2 with the packing generated by starting with circles of curvatures 1;2;2; and 3. There are various problems associated to the diophantine properties of integer ACP’s which are addressed in [GLMWY] by the ﬁve authors Graham, Lagarias, Mallows, Wilks, and Yan. They make considerable progres”
Notes on Thin Matrix Groups (2012, background): “The expected local-to-global conjecture [Graham et al. 2003; Fuchs and Sanden 2011] is proven for all but a zero density set of integers (the conjecture asserts that there are only a finite number of exceptions).”
Equidistribution and counting for orbits of geometrically finite hyperbolic groups (2010, background): “we continue to repeatedly ll the interstices between mutually tangent circles with further tangent circles and obtain an innite circle packing, called an Apollonian circle packing, say, P (cf. [16], [15], [38], [19]). A natural question is to compute the asymptotic number of circles in P of radius a”
Pointwise tube formulas for fractal sprays and self-similar tilings with arbitrary generators (2010, background): “Given the radii of four mutually tangent circles (one of which contains the other three), it is possible (but nontrivial) to determine the radii of the other circles in the packing (and also to obtain asymptotics for their rate of decay; see [11,10]).”
Expander graphs in pure and applied mathematics (2011, background): “N in (w.r.t. fS ig4 i=1 ). Are there innitely many twin prime? i.e., are there innitely many kissing pairs of circles with prime curvatures. A rich theory has started to emerge in recent years (cf. [GLMWY], [S5], [S8], [KO1], [Fu], [BF], [FS] and the reference therein). This is a fascinating crossro”
Affine linear sieve, expanders, and sum-product (2010, background): “5) For details concerning this and the related basic facts that we record below see [25].”