The Mandelbrot set isn't just deterministic. It is "deterministic chaos". That may seem contradictory, because chaos is usually understood to imply unpredictability, whereas determinism is understood to imply complete and perfect predictability.

The reason why this is not a contradiction is this: "prediction" is expected to be a shortcut of sorts. The purpose of prediction is to gain knowledge with less effort than actually letting events unfold. In our case, the purpose of prediction would be to obtain a deeply zoomed Mandelbrot image, but with less effort than actually computing iterations for all pixels. Instead, we want to predict the colors of those pixels.

*CLARIFICATION: I am not talking about numerical optimizations here. The perturbation method, for example, is not a prediction. Under the right conditions (which are currently not fully understood, but close enough in practice), the results of the perturbation method enjoy a mathematical guarantee to be the same as brute force calculation. Speed is gained because the perturbation method allows using the computer's highly optimized circuits for floating point arithmetic, while brute force calculation of deep zooms requires software subroutines for a less efficient "BigNum" data format.*This is the heart of "deterministic chaos". Everything is fully determined, but there are no shortcuts. If you want the true result, you have to do all the math, with no shortcuts. And "all the math" is not merely the formula "z*z + c"; it is a potentially infinite number of iterations of that formula, which potentially never approaches a periodic cycle. It is determined, yes. But a table of all values would still be infinitely big, with no shortcuts, no repetitions.

Alternatively, I have another argument for you which is a little less formal, but still abstract.

That 2nd argument is based on an informal idea of "information". The amount of information you gain does not strictly correlate with the amount of data you receive. For example if instead of the former sentence, I had written a long string "aaaaa...." with the same number of letters, that would not have been as informative (at least I should hope so

).

In an informal sense, you gain the more information, the more you are surprised by the incoming data stream. When things are too predictable, they quickly become boring (this is true not just for the scripts of Hollywood movies).

That sets up my punchline: the only reason you are still interested in seeing ever more ever deeper zooms into the Mandelbrot set is because you are

**not** bored. You continue being

**informed** by those images. There are yet more

**surprises** for you there. And that is despite your claim that you already know your way down there in the infinite details of the set's boundary. That is despite you using the most complex and most capable neural network known to us: a human brain.

Is it possible to build a specialized neural network, one that can beat humans at learning deep zoom Mandelbrot shapes? Sure, eventually, or even today already. But those will be finite, too. Those will eventually zoom deeper than their own "experience" and then continue to be surprised.

Chaos is not so easily defeated by puny mortals. Or by unnecessarily luxurious computing machinery, for that matter.