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Apollonian circle packings arise by repeatedly filling the intersticesbetween four mutually tangent circles with further tangent circles.We observe that there exist Apollonian packings which have strong integralityproperties, in which all circles in the packing have integer curvatures andrational 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 MDβ consisting of those 4times4 real matrices W with WTQDβbW=QWβ where QDβ is the matrix of the Descartes quadratic form QDβ=x12β+x22β+x32β+x42ββfrac12(x1β+x2β+x3β+x4β)2 and QWβ of the quadratic form QWβ=β8x1βx2β+2x32β+2x42β. On the parameter spaceMDβ the group mathopitAut(QDβ) acts on the left, and mathopitAut(QWβ) 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 mathopitAut(QWβ) (essentially) corresponds to Mobius transformations acting on the underlying Euclidean space rr2 while the left action of mathopitAut(QDβ) is defined only on the parameter space. We observe thatthe Descartes configurations in each Apollonian packing form an orbit of a single Descartes configuration under a certain finitely generated discrete subgroup of mathopitAut(QDβ), which we call the Apollonian group. This group consists of 4times4 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 mathopitAut(QDβ), 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 anddual Apollonian groups together. These groups also consist of integer 4times4 matrices. We show these groups are hyperbolic Coxeter groups.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41356/1/454_2005_Article_1196.pd
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