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)Γ(center) is an integer vector. This series
of papers explain such properties. A {\em Descartes configuration} is a set of
four mutually tangent circles with disjoint interiors. We describe the space of
all Descartes configurations using a coordinate system \sM_\DD consisting of
those 4Γ4 real matrices \bW with \bW^T \bQ_{D} \bW = \bQ_{W} where
\bQ_D is the matrix of the Descartes quadratic form QDβ=x12β+x22β+x32β+x42ββ1/2(x1β+x2β+x3β+x4β)2 and \bQ_W of the quadratic form
QWβ=β8x1βx2β+2x32β+2x42β. There are natural group actions on the
parameter space \sM_\DD. We observe that the Descartes configurations in each
Apollonian packing form an orbit under a certain finitely generated discrete
group, the {\em Apollonian group}. This group consists of 4Γ4 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, the dual Apollonian group and the super-Apollonian
group, which have nice geometrically interpretations. We show these groups are
hyperbolic Coxeter groups.Comment: 42 pages, 11 figures. Extensively revised version on June 14, 2004.
Revised Appendix B and a few changes on July, 2004. Slight revision on March
10, 200