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I think these fractals look really cool and would like to better understand how they are formally defined. I recently came across, optimized, and re-implemented someone's algorithm that renders an apollonian gasket by repeatedly inverting every pixel relative to each circle great based on which circle it was in until it escaped. As cool as my finished product is, I'm pretty sure it doesn't directly have anything to do with the definition of a gasket. The first thing I don't understand is the definition of a kleinian group, are they just groups generated by a finite number of mobius transformations? If so, how do the generators affect the nature of the group? Otherwise, I'd like that explained first. The second thing I'm unfamiliar with is the concept of a "limit set," a term I've only heard in the context of kleinian groups that I've never found defined. Is it like a Julia set? Finally, the algorithm that I used recently had the center of each of the three great circles and their radii as parameters; I've read that an Apollonian gasket  is defined by the three points of tangency of these circles. Could someone explain how the points of tangency relate to the associated kleinian group and outline and algorithm for rendering gaskets based on these points? Thank you

https://en.wikibooks.org/wiki/Fractals/Apollonian_fractals

HTH