Fractals and Kleinian groups - Rendering the limit set

Question

I am talking about this:

I recently read the book "The fractal geometry of nature" by Benoit B. Mandelbrot. There was one particular fractal I found very beautiful: A limit set of some group of ... well, it is "Homographien" in German, I don't exactly know the correct terminology in English. (I have the German translation​, there it is picture 190)

Now the problem: I don't understand it, but I really want to! I did quite some research, but I found nothing that was helpful to me. (Neither I found stuff that I understand)

I know it is somehow related to IFS (right?), but again: I have no clue. But I know that the formula $f(z) = \frac{az + b}{cz + d}$ with $ad - bc = 1$, which is called a möbius-transformation, I think. But I don't know if $z$ is complex and the same goes for the parameters.

My goal is to a) understand what is going on in that picture and b) use that knowledge to write a programm that can draw it.

If I messed up the equations (or anything else), please correct me. Thanks!

Answer 1

The book Indra's Pearls covers this stuff. Quoting Wikipedia:

Indra's Pearls: The Vision of Felix Klein is a geometry book written by David Mumford, Caroline Series and David Wright, and published by Cambridge University Press in 2002.

The book explores the patterns created by iterating conformal maps of the complex plane called Möbius transformations, and their connections with symmetry and self-similarity. These patterns were glimpsed by German mathematician Felix Klein, but modern computer graphics allows them to be fully visualised and explored in detail.

Also, Jos Leys has recently developed an escape time algorithm for Kleinian group limit sets, which is faster and more effective for certain cases. Figure 16 in the article is quite similar to your image.