I've lately been trying (without much success) to understand the combinatorics behind the integers that turn up as the reciprocals of the radii of circles in Apollonian gaskets. (See Jeffrey C. Lagarias, Colin L. Mallows, Allan R. Wilks, "Beyond the Descartes Circle Theorem", American Mathematical Monthly, vol. 109, pp. 338-361, available on the web as http://arxiv.org/abs/math.MG/0101066 .) With the somewhat perverse post-Polya attitude "if you can't solve a problem, replace it by a harder problem you also can't solve", I've thought about switching to even more confusing families of circles, which (if they haven't been named yet) OUGHT to be called "Dionysian space-fillers". To get one of this hairy objects, start with an Apollonian gasket and begin promiscuously inverting circles in the gasket with respect to other circles in the gasket. (The inverse of a point P relative to a circle with radius r centered at O is the point Q on the ray OP satisfying (OP)(OQ)= r^2.) Now invert new and old circles with respect to new and old circles. Etc. Taking this to the limit, we get a collection of circles with the property that for every point P in the plane, and any positive epsilon, there is a circle in the collection that is contained within an epsilon-neighborhood of P. Also, all the circles have radii that are reciprocals of integers (I think I know how roughly a proof of this last assertion might go, but I have not worked out the necessary details). Has anyone looked at collections of circles like this? A simplified version of this construction starts with the partial gasket consisting of just the circles of radius 1 centered on all the even integers (viewed as complex numbers). We're not using the "gasket swap" ("Given four mutually tangent circles, replace one of them by the other circle that's tangent to the other three"); we're just using inversion of circles with respect to circles (as in the book "Indra's Pearls", which I still mean to look at one of these days!). Are things known about this? It ought to be possible to give a complete description of the combinatorics of the set of circles, at least in this simplified case. To see one step in the construction of a Dionysian space-filler of a more genuinely two-dimensional kind, see Hal Canary's nice picture http://ups.physics.wisc.edu/~hal/SSL/apollonian.pdf Jim Propp