  We extend fundamental results concerning Apollonian packings, which
constitute a major object of study in number theory, to certain homogeneous
sets that arise naturally in complex dynamics and geometric group theory. In
particular, we give an analogue of D. W. Boyd's theorem (relating the curvature
distribution function of an Apollonian packing to its exponent and the
Hausdorff dimension of the residual set) for Sierpi\'nski carpets that are
Julia sets of hyperbolic rational maps.
