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Discrete Comput Geom 34:547–585 (2005) Discrete & Computational DOI: 10.1007/s00454-005-1196-9 Geometry © 2005 Springer Science+Business Media, Inc. Apollonian Circle Packings: Geometry and Group Theory I. The Apollonian Group∗ Ronald L. Graham,1 Jeffrey C. Lagarias,2 Colin L. Mallows,3 Allan R. Wilks,4 and Catherine H. Yan5 1Department of Computer Science and Engineering, University of California at San Diego, La Jolla, CA 92093-0114, USA [email protected] 2Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1109, USA [email protected] 3Avaya Labs, Basking Ridge, NJ 07920, USA [email protected] 4AT&T Labs, Florham Park, NJ 07932-0971, USA [email protected] 5Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA [email protected] Abstract. Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. We observe that there exist Apol- lonian packings which have strong integrality properties, in which all circles in the packing have integer curvatures and rational centers such that (curvature)×(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 T coordinate system MD consisting of those 4 × 4 real matrices W with W QDW = QW = 2 + 2 + 2 + 2 − 1 ( + where QD is the matrix of the Descartes quadratic form Q D x1 x2 x3 x4 2 x1 + + )2 =− + 2 + 2 x2 x3 x4 and QW of the quadratic form QW 8x1x2 2x3 2x4 . On the parameter ∗ Ronald L. Graham was partially supported by NSF Grant CCR-0310991. Catherine H. Yan was partially supported by NSF Grants DMS-0070574, DMS-0245526 and a Sloan Fellowship. She is also affiliated with Nankai University, China. 548 R. L. Graham, J. C. Lagarias, C. L. Mallows, A. R. Wilks, and C. H. Yan space MD the group Aut(Q D) acts on the left, and Aut(QW ) 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 Aut(QW ) (essentially) corresponds to M¨obius transformations 2 acting on the underlying Euclidean space R while the left action of 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 Aut(Q D), which we call the Apollonian group. This group consists of 4×4 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 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 × 4 matrices. We show these groups are hyperbolic Coxeter groups. 1. Introduction An Apollonian circle packing is a packing of circles arising by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. We call an initial arrangement of four mutually tangent circles with distinct tangents (necessarily six of them) a Descartes configuration. Starting from any Descartes configuration, we can recursively construct an infinite circle packing of the Euclidean plane, in which new circles are added which are tan- gent to three of the circles that have already been placed and have interiors disjoint from any of them. The infinite packing obtained in the limit of adding all possible such circles is called an Apollonian packing. The new circles added at each stage can be obtained using M¨obius transformations of Descartes configurations in the partial packing. An Apollonian packing is pictured in Fig. 1, in which each circle is labeled by its curvature, which is the inverse of its radius. The initial configuration consists of two 1 circles of radius 2 inscribed in a circle of radius 1, the latter being assigned negative curvature −1 by a convention given in Section 3. This particular Apollonian packing has the special property that all circles in the packing have integer curvatures. We call a packing with this property integral. More remarkably, if one regards the circle centers as complex numbers, then one can place the initial circles so that every circle in the packing has “curvature×center” a Gaussian integer (element of the ring Z[i].) This occurs for example when the center of the outer circle is placed at z = 0 and the two circles of 1 =−1 = 1 radius 2 have centers at z 2 and at z 2 . We call a packing with this extra property strongly integral. The object of this paper is to give a geometric explanation of the origin of these integrality properties involving both the curvatures and the circle centers. This is based on five facts: (1) An Apollonian packing can be described in terms of the Descartes configurations it contains. Apollonian Circle Packings, I 549 182 182 114 146 146 114 86 86 131 62 62 131 186 42 42 186 150 75 75 150 26 143 143 26 171 171 90 90 87 87 194 194 35 182 182 35 107 66 66 107 158 158 111 14 14 111 122 122 131 131 47 47 147 147 98 98 98 98 167 167 170 59 59 170 191 191 195 30 30 195 74 74 138 138 95 95 194 194 194 194 6 119 119 6 62 62 179 179 170 170 186 186 110 110 110 110 174 174 54 54 150 150 131 167 11 39 83 143 179 111 59 23 23 59 111 179 143 83 39 11 167 131 162 3 162 98 98 183 183 147 50 50 147 182 87 87 182 134 134 134 134 63 191 191 63 179 179 18 155 155 18 86 86 95 95 27 167 167 27 126 114 114 126 71 71 38 135 38 110 110 38 135 38 174 174 183 183 51 107 107 51 66 15 66 83 83 102 102 102 102 123 123 146 35 146 171 171 198 198 198 198 63 99 143 195 195 143 99 63 198 198 198 198 171 171 146 146 123 35 123 102 102 102 102 83 83 66 2 2 66 51 51 183 107 107 183 174 15 174 38 135 38 110 110 38 135 38 71 71 126 114 114 126 167 167 27 95 95 27 86 86 155 155 179 179 191 191 63 134 134 134 134 63 18 182 87 87 182 18 147 147 183 50 50 183 98 162 162 98 167 179 179 167 131 83 143 111 59 150 150 59 111 143 83 131 54 39 39 54 174 23 23 174 110 110 110 110 186 186 170 170 179 179 11 62 62 11 119 119 194 194 194 194 95 95 138 138 74 74 195 30 30 195 191 191 170 170 59 167 167 59 98 98 98 98 147 6 6 147 47 47 131 131 122 122 111 111 158 158 107 3 107 66 66 182 182 194 35 35 194 87 87 90 14 14 90 171 171 143 143 150 75 26 26 75 150 186 42 42 186 131 62 62 131 86 114 114 86 146 182 182 146 Fig. 1. An Apollonian packing, labeled with circle curvatures. (2) There is a coordinate representation of the set MD of all ordered, oriented Descartes configurations as a six-dimensional real-algebraic variety. It consists of the set of 4 × 4 real matrices W satisfying a system of quadratic equa- T tions, W QDW = QW , which state that W conjugates the Descartes quadratic = 2 + 2 + 2 + 2 − 1 ( + + + )2 form Q D x1 x2 x3 x4 2 x1 x2 x3 x4 to a quadratic form =− + 2 + 2 QW 8x1x2 2x3 2x4 that we call the Wilker quadratic form, after Wilker [42]. This coordinate system we call “augmented curvature-center coordinates,” as it encodes the curvatures and centers of the circles in the configuration. We term MD with these coordinates the parameter space of ordered, oriented Descartes configurations. (3) The variety MD is a principal homogeneous space for the Lorentz group O(3, 1), under both a left and a right action of this group, realized as Aut(Q D) and Aut(QW ), respectively. The right action corresponds to M¨obius transformations acting on the plane, while the left action acts only on the Descartes configuration space. (4) There is a discrete subgroup of the left action by Aut(Q D), the Apollonian group A, having the property that the (unordered, unoriented) Descartes configurations in any Apollonian packing are described by a single orbit of this group. If the Descartes configurations are regarded as ordered and oriented, then exactly 48 550 R. L. Graham, J. C. Lagarias, C. L. Mallows, A. R. Wilks, and C. H. Yan such orbits correspond to each packing, each one containing a copy of each unordered, unoriented Descartes configuration in the packing. (5) The Apollonian group A consists of integer 4 × 4 matrices. The last property explains the existence of Apollonian packings having integrality properties of both curvatures and centers.
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