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Apollonian circle packings: Number theory
, 2003
"... 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 integral Apollonian circle packing. This paper st ..."
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Cited by 49 (3 self)
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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 integral Apollonian circle packing. This paper studies numbertheoretic 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 Descartes equation, which relates the radii of curvature of four mutually tangent circles: x2 þ y2 þ z2 þ w2 1 2ðx þ y þ z þ wÞ2: Each integral Apollonian circle packing is classified by a certain root quadruple of integers that satisfies the Descartes equation, and that corresponds to a particular quadruple of circles appearing in the packing. We express the number of root quadruples with fixed minimal element n as a class number, and give an exact formula for it. We study which integers occur in a given integer packing, and determine congruence restrictions which sometimes apply. 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 Corresponding author.
Apollonian Circle Packings: Geometry and Group Theory II. SuperApollonian Group and Integral Packings
, 2006
"... Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. Such packings can be described in terms of the Descartes configurations they contain, where a Descartes configuration is a set of four mutually tangent circles ..."
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Cited by 31 (4 self)
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Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. Such packings can be described in terms of the Descartes configurations they contain, where a Descartes configuration is a set of four mutually tangent circles in the Riemann sphere, having disjoint interiors. Part I showed there exists a discrete group, the Apollonian group, acting on a parameter space of (ordered, oriented) Descartes configurations, such that the Descartes configurations in a packing formed an orbit under the action of this group. It is observed there exist infinitely many types of integral Apollonian packings in which all circles had integer curvatures, with the integral structure being related to the integral nature of the Apollonian group. Here we consider the action of a larger discrete group, the superApollonian group, also having an integral structure, whose orbits describe the Descartes quadruples of a geometric object we call a superpacking. The circles in a superpacking never cross each other but are nested
Circle packing: a mathematical tale
 Notices Amer. Math. Soc
, 2003
"... The circle is arguably the most studied object in mathematics, yet I am here to tell the tale of circle packing, a topic which is likely to be new to most readers. These packings are configurations of circles satisfying preassigned patterns of tangency, and we will be concerned here with their creat ..."
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Cited by 27 (2 self)
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The circle is arguably the most studied object in mathematics, yet I am here to tell the tale of circle packing, a topic which is likely to be new to most readers. These packings are configurations of circles satisfying preassigned patterns of tangency, and we will be concerned here with their creation, manipulation, and interpretation. Lest we get off on the wrong foot, I should caution that this is NOT twodimensional “sphere ” packing: rather than being fixed in size, our circles must adjust their radii in tightly choreographed ways if they hope to fit together in a specified pattern. In posing this as a mathematical tale, I am asking the reader for some latitude. From a tale you expect truth without all the details; you know that the storyteller will be playing with the plot and timing; you let pictures carry part of the story. We all hope for deep insights, but perhaps sometimes a simple story with a few new twists is enough—may you enjoy this tale in that spirit. Readers who wish to dig into the details can consult the “Reader’s Guide ” at the end. Once Upon a Time … From wagon wheel to mythical symbol, predating history, perfect form to ancient geometers, companion to π, the circle is perhaps the most celebrated object in mathematics.
Beyond the Descartes Circle Theorem
 Amer. Math. Monthly
, 2002
"... The Descartes circle theorem states that if four circles are mutually tangent in the plane, with disjoint interiors, then their curvatures (or “bends”) bi = 1 ri satisfy the relation (b1 + b2 + b3 + b4) 2 = 2(b2 1 + b22 + b23 + b24). We show that similar relations hold involving the centers of the f ..."
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Cited by 21 (3 self)
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The Descartes circle theorem states that if four circles are mutually tangent in the plane, with disjoint interiors, then their curvatures (or “bends”) bi = 1 ri satisfy the relation (b1 + b2 + b3 + b4) 2 = 2(b2 1 + b22 + b23 + b24). We show that similar relations hold involving the centers of the four circles in such a configuration, coordinatized as complex numbers, yielding a complex Descartes Theorem. These relations have elegant matrix generalizations to the ndimensional case, in each of Euclidean, spherical, and hyperbolic geometries. These include analogues of the Descartes circle theorem for spherical and hyperbolic space.
Circle Packing And Discrete Analytic Function Theory
 R. Kühnau (Ed.), Handbook of Complex Analysis
"... . Circle packings  congurations of circles with specied patterns of tangency  came to prominence with analysts in 1985 when Thurston conjectured that maps between such congurations would approximate conformal maps. The proof by Rodin and Sullivan launched a topic which has grown steadily as ever m ..."
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Cited by 7 (1 self)
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. Circle packings  congurations of circles with specied patterns of tangency  came to prominence with analysts in 1985 when Thurston conjectured that maps between such congurations would approximate conformal maps. The proof by Rodin and Sullivan launched a topic which has grown steadily as ever more connections with analytic functions and conformal structures have emerged. Indeed, the core ideas have matured to the point that one can fairly claim that circle packing provides a discrete analytic function theory. There are two related but distinct aspects visavis the classical model  analogy and approximation. This survey concentrates on the analogies, with a largely pictorial tour intended for the reader familiar with classical conformal geometry. The companion survey [St99] treats approximation. Part I of the paper covers circle packing basics and introduces discrete analytic functions as maps between circle packings. Representatives of the standard classes of analytic functi...
A generalization of Apollonian packing of circles
 Journal of Combinatorics
"... Three circles touching one another at distinct points form two curvilinear triangles. Into one of these we can pack three new circles, touching each other, with each new circle touching two of the original circles. In such a sextuple of circles there are three pairs of circles, with each of the cir ..."
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Cited by 5 (1 self)
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Three circles touching one another at distinct points form two curvilinear triangles. Into one of these we can pack three new circles, touching each other, with each new circle touching two of the original circles. In such a sextuple of circles there are three pairs of circles, with each of the circles in a pair touching all four circles in the other two pairs. Repeating the construction in each curvilinear triangle that is formed results in a generalized Apollonian packing. We can invert the whole packing in every circle in it, getting a \generalized Apollonian superpacking". Many of the properties of the Descartes conguration and the standard Apollonian packing carry over to this case. In particular, there is an equation of degree 2 connecting the bends
Combinatorial excursions in moduli space
 Pacific J. Math
"... Given an abstract triangulation of a torus, there is a unique point in moduli space which supports a circle packing for this triangulation. We will describe combinatorial deformations analogous to the process of conformal welding. These combinatorial deformations allow us to travel in moduli space ..."
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Cited by 3 (1 self)
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Given an abstract triangulation of a torus, there is a unique point in moduli space which supports a circle packing for this triangulation. We will describe combinatorial deformations analogous to the process of conformal welding. These combinatorial deformations allow us to travel in moduli space from any packable torus to a point arbitrarily close to any other torus we choose. We also provide two proofs of Toki’s result that any torus can be transformed into any other by a conformal welding and compute the maps necessary to accomplish the welding. 1. Introduction. A circle packing is a configuration of circles with a prescribed pattern of tangencies. The interplay between the combinatorial “pattern ” and the geometry provided by the circles has generated intense interest in recent years. As a result, a discrete version of complex analysis based on maps between
On the ring lemma
, 2008
"... The sharp ring lemma states that if n >= 3 cyclically tangent discs with pairwise disjoint interiors are externally tangent to and surround the unit disc, then no disc has a radius below cn = (F 2 n−1 +F2 n−2 −1)−1 – where Fk denotes the k th Fibonacci number – and that the lower bound is attaine ..."
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Cited by 1 (0 self)
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The sharp ring lemma states that if n >= 3 cyclically tangent discs with pairwise disjoint interiors are externally tangent to and surround the unit disc, then no disc has a radius below cn = (F 2 n−1 +F2 n−2 −1)−1 – where Fk denotes the k th Fibonacci number – and that the lower bound is attained in essentially unique Apollonian configurations. Here we give a proof by transforming the problem to a class of strip configurations, after which we closely follow Aharonov’s and Stephenson’s method of proof [3]. Generalizations to three dimensions are discussed, a version of the ring lemma in three dimensions is proved, and a natural generalization of the extremal twodimensional configuration – thought to be extremal in three dimensions – is given. The sharp threedimensional ring lemma constant of order n is shown to be bounded from below by the twodimensional constant of order n − 1.
Apollonian circle packings:number theory
, 2000
"... 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 integral Apollonian circle packing. This paper st ..."
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Cited by 1 (0 self)
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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 integral Apollonian circle packing. This paper studies numbertheoretic 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 Descartes equation, which relates the radii of curvature of four mutually tangent circles: x2 þ y2 þ z2 þ w2 1 2ðx þ y þ z þ wÞ2: Each integral Apollonian circle packing is classified by a certain root quadruple of integers that satisfies the Descartes equation, and that corresponds to a particular quadruple of circles appearing in the packing. We express the number of root quadruples with fixed minimal element n as a class number, and give an exact formula for it. We study which integers occur in a given integer packing, and determine congruence restrictions which sometimes apply. 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 Corresponding author.