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Thomas
@aviationdoctor.eth
Today I dove into how the Mandelbrot set is constructed, and I am awestruck by the simplicity of it. I knew that chaotic, nonlinear systems could be elegantly simple under the hood, but this takes the cake. The ability to zoom infinitely into this gorgeous fractal landscape using just one second-degree polynomial is mind-blowing
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John Camkiran
@johncamkiran
Not to detract from the beauty of this set, but I did want to note for the readership that such ‘closed-form’ solutions are quite rare in the study of chaotic phenomena. Most such systems are what Stephen Wolfram calls ‘computationally irreducible’, meaning that the only way to find out what happens at some time (in this case, zoom) is to sequentially compute all that which came before it.
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Thomas
@aviationdoctor.eth
Thanks John! I thought the M set was also computationally irreducible. Is there a way to figure out for which values of c the z^2 + c iterations will remain bounded, without actually doing the calculations? (within the trivial bound of 2 that is)
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Chaos 🎩
@multifractal.eth
The Julia set can be derived from the logistic nap with a simple transformation. c = r/2 - 1. So Julia set itself is indeed bounded within a finite region and exhibits folding & stretching properties.
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John Camkiran
@johncamkiran
There are no known general analytical solutions, but also no proof yet that it cannot be solved analytically (at least to the best of my knowledge). The latter is one thing I would demand before using the term, though a strong suspicion is enough for most people.
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