Apollonian Circle Packings: Geometry and Group Theory II. Super-Apollonian Group and Integral Packings

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 super-Apollonian group, also having an integral structure, whose orbits describe the Descartes quadruples of a geometric object we call a super-packing. The circles in a super-packing never cross each other but are nested to an arbitrary depth. Certain Apollonian packings and super-packings are strongly integral in the sense that the curvatures of all circles are integral and the curvature x centers of all circles are integral. We show that (up to scale) there are exactly eight different (geometric) strongly integral super-packings, and that each contains a copy of every integral Apollonian circle packing (also up to scale). We show that the super-Apollonian group has finite volume in the group of all automorphisms of the parameter space of Descartes configurations, which is isomorphic to the Lorentz group O(3, 1).     

Apollonian Circle Packings: Geometry and Group Theory I. The Apollonian Group

Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. We observe that there exist Apollonian packings which have strong integrality properties, in which all circles in the packing have integer curvatures and rational centers such that (curvature) $\times$ (center) is an integer vector. This series of papers explain such properties. A Descartes configuration is a set of four mutually tangent circles with disjoint interiors. An Apollonian circle packing can be described in terms of the Descartes configuration it contains. We describe the space of all ordered, oriented Descartes configurations using a coordinate system $M_ D$ consisting of those $4 \times 4$ real matrices $W$ with $W^T Q_{D} \bW = Q_{W}$ where $Q_D$ is the matrix of the Descartes quadratic form $Q_D= x_1^2 + x_2^2+ x_3^2 + x_4^2 - \frac{1}{2}(x_1 +x_2 +x_3 + x_4)^2$ and $Q_W$ of the quadratic form $Q_W = -8x_1x_2 + 2x_3^2 + 2x_4^2$. On the parameter space $M_ D$ the group $\mathop{\it Aut}(Q_D)$ acts on the left, and $\mathop{\it Aut}(Q_W)$ acts on the right, giving two different "geometric" actions. Both these groups are isomorphic to the Lorentz group $O(3, 1)$. The right action of $\mathop{\it Aut}(Q_W)$ (essentially) corresponds to Mobius transformations acting on the underlying Euclidean space $\rr^2$ while the left action of $\mathop{\it Aut}(Q_D)$ is defined only on the parameter space. We observe that the Descartes configurations in each Apollonian packing form an orbit of a single Descartes configuration under a certain finitely generated discrete subgroup of $\mathop{\it Aut}(Q_D)$, which we call the Apollonian group. This group consists of $4 \times 4$ integer matrices, and its integrality properties lead to the integrality properties observed in some Apollonian circle packings. We introduce two more related finitely generated groups in $\mathop{\it Aut}(Q_D)$, the dual Apollonian group produced from the Apollonian group by a "duality" conjugation, and the super-Apollonian group which is the group generated by the Apollonian and dual Apollonian groups together. These groups also consist of integer $4 \times 4$ matrices. We show these groups are hyperbolic Coxeter groups.     

Apollonian circle packings: number theory

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 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 Descartes equation, which relates the radii of curvature of four mutually tangent circles: . 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 congruence conditions. Finally, we discuss asymptotic properties of the set of curvatures obtained as the packing is recursively constructed from a root quadruple.     

On integral Apollonian circle packings

The curvatures of four mutually tangent circles with disjoint interiors form what is called a Descartes quadruple. The four least curvatures in an integral Apollonian circle packing form what is called a root Descartes quadruple and, if the curvatures are relatively prime, we say that it is a primitive root quadruple. We prove a conjecture of Mallows by giving a closed formula for the number of primitive root quadruples with minimum curvature −n. An Apollonian circle packing is called strongly integral if every circle has curvature times center a Gaussian integer. The set of all such circle packings for which the curvature plus curvature times center is congruent to 1 modulo 2 is called the “standard supergasket.” Those centers in the unit square are in one-to-one correspondence with the primitive root quadruples and exhibit certain symmetries first conjectured by Mallows. We prove these symmetries; in particular, the centers are symmetric around y=x if n is odd, around x=1/2 if n is an odd multiple of 2, and around y=1/2 if n is a multiple of 4.     

A generalized Malfatti problem

Malfatti?s problem, first published in 1803, is commonly understood to ask fitting three circles into a given triangle such that they are tangent to each other, externally, and such that each circle is tangent to a pair of the triangle?s sides. There are many solutions based on geometric constructions, as well as generalizations in which the triangle sides are assumed to be circle arcs. A generalization that asks to fit six circles into the triangle, tangent to each other and to the triangle sides, has been considered a good example of a problem that requires sophisticated numerical iteration to solve by computer. We analyze this problem and show how to solve it quickly.     

A topological characterization of the Sierpiński triangle

We present a topological characterization of the Sierpiński triangle. This answers question 58 from the Problem book of the Open Problem Seminar held at Charles University. In fact we give a characterization of the Apollonian gasket first. Consequently we show that any subcontinuum of the Apollonian gasket, whose boundary consists of three points, is homeomorphic to the Sierpiński triangle.     

Apollonian Circle Packings: Geometry and Group Theory III. Higher Dimensions

This paper gives $n$-dimensional analogues of the Apollonian circle packings in Parts I and II. Those papers considered circle packings described in terms of their Descartes configurations, which are sets of four mutually touching circles. They studied packings that had integrality properties in terms of the curvatures and centers of the circles. Here we consider collections of $n$-dimensional Descartes configurations, which consist of $n+2$ mutually touching spheres. We work in the space $M_D^n$ of all $n$-dimensional oriented Descartes configurations parametrized in a coordinate system, augmented curvature-center coordinates, as those $(n+2) \times (n+2)$ real matrices $W$ with $W^T Q_{D,n} W = Q_{W,n}$ where $Q_{D,n} = x_1^2 + \cdots + x_{n+2}^2 - ({1}/{n})(x_1 +\cdots + x_{n+2})^2$ is the $n$-dimensional Descartes quadratic form, $Q_{W,n} = -8x_1x_2 + 2x_3^2 + \cdots + 2x_{n+2}^2$, and $\bQ_{D,n}$ and $\bQ_{W,n}$ are their corresponding symmetric matrices. On the parameter space $M_D^n$ of augmented curvature-center matrices, the group ${\it Aut}(Q_{D,n})$ acts on the left and ${\it Aut}(Q_{W,n})$ acts on the right. Both these groups are isomorphic to the $(n+2)$-dimensional Lorentz group $O(n+1,1)$, and give two different "geometric" actions. The right action of ${\it Aut}(Q_{W,n})$ (essentially) corresponds to Mobius transformations acting on the underlying Euclidean space $\rr^n$ while the left action of ${\it Aut}(Q_{D,n})$ is defined only on the parameter space $M_D^n$. We introduce $n$-dimensional analogues of the Apollonian group, the dual Apollonian group and the super-Apollonian group. These are finitely generated groups in ${\it Aut}(Q_{D,n})$, with the following integrality properties: the dual Apollonian group consists of integral matrices in all dimensions, while the other two consist of rational matrices, with denominators having prime divisors drawn from a finite set $S$ depending on the dimension. We show that the Apollonian group and the dual Apollonian group are finitely presented, and are Coxeter groups. We define an Apollonian cluster ensemble to be any orbit under the Apollonian group, with similar notions for the other two groups. We determine in which dimensions there exist rational Apollonian cluster ensembles (all curvatures are rational) and strongly rational Apollonian sphere ensembles (all augmented curvature-center coordinates are rational).     

Not set in stone: nineteenth-century geometrical constructions and the Malfatti Problem

In 1803, Gian Francesco Malfatti posed the problem of constructing three parallel cylinders of maximal volume from a marble triangular prism. He then reduced the problem to that of inscribing three mutually tangent circles in a triangle. This apparently equivalent construction reappeared repeatedly through the nineteenth century. A decade after Malfatti, Joseph-Diaz Gergonne posed the problem to the readers in the first issue of his Annales. The construction gained a wider audience when Jakob Steiner offered a purely synthetic treatment in his 1826 article published in the first volume of Crelle's Journal. Many, including Gergonne, found Steiner's proof incomplete, and alternative approaches continued to emerge in France, Germany, and later Britain and the United States. The dispersion of the so-called Malfatti problem provides an illustrative thread through the evolution of nineteenth century geometry. From this perspective we observe efforts towards developing general theories to encompass the approaches for particular problems, the differentiation and competition of geometric methodologies, and the nationalization and internationalization of mathematical communities.     

The ring lemma in three dimensions

Suppose that n cyclically tangent discs with pairwise disjoint interiors are externally tangent to and surround the unit disc. The sharp ring lemma in two dimensions states that no disc has a radius below c _n (R ²) = (F _2n−3−1)⁻¹—where F _k denotes the kth Fibonacci number—and that the lower bound is attained in essentially unique Apollonian configurations. In this article, generalizations of the ring lemma to three dimensions are discussed, a version of the ring lemma in three dimensions is proved, and a natural generalization of the extremal two-dimensional configuration—thought to be extremal in three dimensions—is given. The sharp three-dimensional ring lemma constant of order n is shown to be bounded from below by the two-dimensional constant of order n − 1.     

Apollonian circle packings: Number theory II. Spherical and hyperbolic packings

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 of the group of integral automorphs of the indefinite quaternary quadratic form . This subgroup, called the Apollonian group, acts on integer solutions . This paper gives a reduction theory for orbits of acting on integer solutions to 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. 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.     

Director circles of conic sections

Given a conic section, the locus of a moving point in the plane of the conic section such that the two tangent lines drawn to the conic section from the moving point are all mutually perpendicular is a curve. In the case of an ellipse and hyperbola this curve is a circle referred to as the director circle. In the case of the parabola this curve coincides with the directrix of the parabola. The last section is devoted to the graphical illustrations of director circles for circles, parabolas, ellipses and hyperbolas using the built-in Maple V software of Scientific Work Place 3.0.     

度量方程应用于Krause定理的推广

本文用距离几何的方法证明了主要定理,对曲率为Ｋ的ｎ维常曲率空间,其内任意ｎ＋１个ｎ－１维球Ｓｉ（ｉ＝１,２,…,ｎ＋１）,它们中的任一个都与其它球不变,则与Ｓｉ交角为βｉ（ｉ＝１,２,…,ｎ＋１）的ｎ－１维一般有２＾ｎ＋１个,当ｎ为偶数时,它们的测地线曲率之交错和为零;当ｎ为奇数时,此结论不成立,该定理包括非欧情形,而当ｎ＝２,βｉ＝１（ｉ＝１,２,…,ｎ＋１）时,就是ｉｉｌｋｅｒＪＢ在「１」中所     

The asymptotic distribution of circles in the orbits of Kleinian groups

Let P\mathcal{P} be a locally finite circle packing in the plane ℂ invariant under a non-elementary Kleinian group Γ and with finitely many Γ-orbits. When Γ is geometrically finite, we construct an explicit Borel measure on ℂ which describes the asymptotic distribution of small circles in P\mathcal{P}, assuming that either the critical exponent of Γ is strictly bigger than 1 or P\mathcal{P} does not contain an infinite bouquet of tangent circles glued at a parabolic fixed point of Γ. Our construction also works for P\mathcal{P} invariant under a geometrically infinite group Γ, provided Γ admits a finite Bowen-Margulis-Sullivan measure and the Γ-skinning size of P\mathcal{P} is finite. Some concrete circle packings to which our result applies include Apollonian circle packings, Sierpinski curves, Schottky dances, etc.     

Classification of rotational motions for the baryon model “triangle”

A baryon model with three particles (quarks) pairwise connected by relativistic strings forming a curvilinear triangle is considered. The classical analytic solutions corresponding to a uniform plane rotation of the system with arbitrary quark masses m_i are found for this model. The sections of the related world surfaces by the plane t=const are curvilinear triangles composed of segments of a hypocycloid. A complete classification of the types of motion is suggested, based on differences in topological configuration and the presence and number of interior massless points on the string that move at the velocity of light. The classification results from investigating the limiting states as m_i»0. The calculated energy dependence of the angular momentum of the motions creates possibilities to model baryon states on Regge trajectories using these motions.     

The apollonian octets and an inversive form of Krause's theorem

Suppose we are given three disjoint circles in the Euclidean plane with the property that none of them contains the other two. Then there are eight distinct circles tangent to the given three, and R.M. Krause has shown that a certain alternating sum of the curvatures of these eight circles must vanish. We express this result in an inversively invariant way and determine the extent to which it generalizes to other configurations of three given circles.     

Spiral hexagonal circle packings in the plane

We discuss an intriguing geometric algorithm which generates infinite spiral patterns of packed circles in the plane. Using Kleinian group and covering theory, we construct a complex parametrization of all such patterns and characterize those whose circles have mutually disjoint interiors. We prove that these coherent spirals, along with the regular hexagonal packing, give all possible hexagonal circle packings in the plane. Several examples are illustrated.The last two authors gratefully acknowledge support of the National Science Foundation and the Tennessee Science Alliance.     

A generalization of the circumcircle theorem

The theorem relating the bisectors of the edges of a triangle and the corresponding circumscribing circle is established as a special case of a theorem for triangles with weighted vertices where the edges are partitioned with circular arcs in the proportions of the weights. The circular arcs are established as being uniquely determined by the weights and the triangle, and are given by three circles with collinear centres. These circles either intersect in zero, one or two real points, these latter points being the triple points.     

Effective Circle Count for Apollonian Packings and Closed Horospheres

The main result of this paper is an effective count for Apollonian circle packings that are either bounded or contain two parallel lines. We obtain this by proving an effective equidistribution of closed horospheres in the unit tangent bundle of a geometrically finite hyperbolic 3-manifold, whose fundamental group has critical exponent bigger than 1. We also discuss applications to affine sieves. Analogous results for surfaces are treated as well.     

3-Webs generated by confocal conics and circles

We consider families of confocal conics and two pencils of Apollonian circles having the same foci. We will show that these families of curves generate trivial 3-webs and find the exact formulas describing them.     

The construction of a special equilateral triangle

This article deals with the construction of an equilateral triangle that must satisfy the following special constraint conditions. If the equilateral triangle is denoted by ΔABC, then the radii of the inscribed circle, the three escribed circles of ΔABC, and the circumcircle of ΔABC all must have positive integral radii. The inscribed circle radius is required to be 1 unit. The three escribed circles that have equal radii must have 3 units each, and the circumcircle of the triangle must have 2 units. All these requirements may seem outlandish. The aim is to teach crucial Geometric principles that Geometric designs must take into account before the constructions are implemented. This article hopefully may be useful to students of College Geometry as well as teachers.