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. Now, anytime you have four mutually tangent circles, you can replace any of the four by a different circle that is tangent to the other three. Repeat this sort of replacement operation ad infinitum, and you've got the kind of gasket I mean. Check out http://arxiv.org/abs/math.MG/0101066 for some pictures of this sort of gasket, and http://ups.physics.wisc.edu/~hal/SSL/apollonian.pdf for pictures that start down the road from gaskets to space-fillers. Jim