JSL, attached below, creates a set of still frames in a directory on the desktop that can be combined to make this video. There are a lot of sites describing Soddy Circles, Kissing Circles, Apollonian Gaskets, and Descartes' Theorem. The first link has a poem too, which includes all the math. The short story: in two dimensions, four circles can all touch each other, and there is a simple relationship between their sizes. And, for any three of the four, there is another circle that will make different set of four.
To make this video, I used a zero-to-two-PI sin and cos function to vary the sizes of the seed circles, and to rotate the image. The final frame is the same as the first. The circle at 3-o'clock in the initial frame is the one to observe if you are trying to figure out what is going on.
Update 4 Feb 2017: repair formatting for new community.
