Used to see these as demos oh how fast a processor wasFor me, fractals are the essence of the universe.
I like Benoît Mandelbrot.
I'm waiting for others who want to talk about fractals. For now, anything about them.
For things like Mandelbrot, the colors are assigned by the number of iterations it takes for iterated calculations to "escape" - i.e.: how long it takes for, given \(z_{i+1} = f(z_i)\) (initialized to some initial complex number in the plane), \(\left|z_i\right|^2 > r^2\) for some "radius" \(r\). If, after a certain number of maximum iterations it does not "escape," that point (the pixel corresponding to the starting point) is colored black. The more iterations you let a fractal run through, the sharper your edges will be. The points that do not "escape" and are colored black are considered part of the Mandelbrot set. The Julia sets are a subtype of the Mandelbrot set (indeed; you can form the Mandelbrot set from all the Julia sets).Used to see these as demos oh how fast a processor was
how do you convert a number to those lovely plots
is it just real and imaginary,
how do they assign colours ?
I just mowed some of those that had escaped into our lawn!
Well bear in mind there is no process/algorithm to determine that a point iteration will ever "escape", it's a bit like the halting problem, there's no way to prove that any pair of points when iterated will "escape" that's why they place some limit, if it has not escaped after say 1 million iterations then they assume it never will, but they don't know, they don't know whether it might escape if iterated just a few more times, this is the essence of chaos - unpredictability.I would be curious to see what the fractals would look like if you iterated them until the points "escaped," and instead of coloring based on iteration count, coloring them based off of the argument (angle) or the modulus (magnitude) of the "final" \(z_i\) after the last iteration.
Right. This, I already knew (I frequently use the Mandelbrot set as a kind of crude "benchmark" of processing power/speed, especially multi-threading). The thought/idea was less of a technical/mathematical question and more of an "artistic choice" question.Well bear in mind there is no process/algorithm to determine that a point iteration will ever "escape", it's a bit like the halting problem, there's no way to prove that any pair of points when iterated will "escape" that's why they place some limit, if it has not escaped after say 1 million iterations then they assume it never will, but they don't know, they don't know whether it might escape if iterated just a few more times, this is the essence of chaos - unpredictability.
You can do this with any software that draws the Mandelbrot set, set the iteration limit to 1,000,000 then run it again with the limit 1,000,001 and you find some additional points that now escape but did not escape with the limit of 1,000,000.
Note that the "set" part of Mandelbrot set is all those points that never "escape" but since we can never really prove if a point definitely will escape the members of the set are unknown.
fractals are deeply embedded into the makeup of nature. Just like your broccoli,fractals demonstrate how simple geometric shapes can form the incredibly
I read intently all your posts.Life is a fractal.
That's interesting, as is the title of the video with the word "analog".
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