It's possible that what we're looking at is based on the map f(z) = 1/z applied to the cartesian coordinate grid.

This is old-school, but a long time ago I wrote a java applet to help explore this function on the complex plane:

http://www.math.nyu.edu/~neylon/applet1/

If you move around the red square, you see a lot of the circular shapes that look similar to those in the fractal.

I don't completely understand the algorithm being used, so I'm not sure how to gain more confidence on this guess.