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Jim: Since there are various versions of Apollonian gaskets, could you tell us exactly the one you are using to get started here?

Start with three mutually externally tangent circles, with a big circle that contains and is tangent to the other three. This gives a foursome of mutually tangent circles.

My original problem concerned situations with an additional constraint: all four circles should have radii that are reciprocals of integers. Then it can be shown that ALL of the infinitely many circles in the gasket have this property too, and I believe the same is true of the space-filling family of circles derived from the gasket by iterated inversion. I was trying to find combinatorial interpretations of the numbers that turn up, analogous to the fact that numerators and denominators of fractions in the Stern-Brocot tree count perfect matchings of "snake graphs" like o---o---o | | | o---o---o | | o---o---o---o---o | | | | | o---o---o---o---o These numbers are related to the Ford circles, which are an especially nice subset of the most fundamental of Apollonian gaskets, namely the one in which the four original "circles" are two parallel lines and two mutually tangent circles in between them. So, although my long-range goal is to understand this and other specific gaskets from a combinatorial point of view, I'd also like to find out what's known about the space-fillers derived from them from a purely geometrical point of view. Jim Propp