Compressing the Mandelbrot set

This question may not have a definitive answer. However, if someone is able to illuminate the topic for me, I would be very grateful.

The Mandelbrot set is the set obtained from the quadratic recurrence equation{1}:

$$$$z_{n+1}=z_n^2 + c$$$$

I'm sure most of you know what the graphical representation of the Mandelbrot set looks like, so I won't post a picture of it here.

Question

Have there been any attempts to derive the Mandelbrot set equation purely from it's graphical representation?

I would imagine that this would involve some sort of machine learning process which searches through program space trying to find a correct program with the smallest Kolmogorov complexity{2}.

What branch of mathematics works on solving this type of problem?

Thank you.

• As far as I am aware, there is no computationally precise way of encoding the Mandlebrot set except for the definition of the Mandlebrot set (although there are slightly different theoretical descriptions which are easily equivalent). You can't compress a collection of data if you don't at least have some finite yet inefficient way of representing it. The "graphical representation" of the Mandlebrot set is not actually such a representation, it is just a series of approximations. Commented May 30, 2011 at 16:39
• – lhf
Commented May 30, 2011 at 16:44
• While not a direct answer to your question I found learning about the bifurcation diagram - the "third dimension" of the Mandlebrot set very useful - this video very helpful to understand what the Mandlebrot set "means" mathematically - it might be helpful - particularly the understanding that the Mandlebrot set is another way to represent the bifurcation diagram Bifurcation Diagram youtube.com/watch?v=ovJcsL7vyrk Commented May 10, 2022 at 4:57