# 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.

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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. –  Aaron May 30 '11 at 16:39
–  lhf May 30 '11 at 16:44

In a certain sense the answer is yes -- look at Hubbard and Douady's work concerning "external angles" and "Hubbard trees". Modulo a conjecture about local path-connectedness I believe they have a very explicit topological model of the Mandelbrot set which in some sense is derived from a "picture" of it.

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