The Apollonian Gasket is a pretty well-known fractal that you get by taking four mutually tangent circles and then adding the circles tangent to any subset of three mutually tangent circles, recursively and ad infinitum. The union of these circles is also the limit set of a Kleinian Group, a discrete group of Möbius transformations.
In a way, the Apollonian Gasket is pretty similar to the simpler Sierpinski Gasket, but it's also more beautiful in the sense that it has a very natural group of exact symmetries, namely the Kleinian Group mentioned above.
However, one could be bothered by the fact that the "inside" and "outside" of any one of those circles are not symmetric with respect to one another. There's an obvious way to remedy that, leading to what I'm calling the "Super-Apollonian" fractal here. It's simply the closure of the Apollonian Gasket under its own symmetry group and one judiciously chosen additional transformation that swaps the inside and outside of one of the circles.
The image shown here is an attempt at a nice visualization of that concept.
This is a visualization of a particular two-generator Kleinian group. I'm working on a full description, but until that's done, maybe you can reverse-engineer my work from the image!
This is a quasi-periodic fractal made from wave interference patterns.