A very interesting cast of underdog characters appears in the article.<p>You’ve got one guy with a relentless spirit to continue with mathematics in is spare time after being blacklisted from mainstream academia because of antisemitism.<p>Another is a childhood prodigy, who set the record for the youngest American IMO team member, but got burned out as an adult and went into finance but found his way back through the mentorship of another mathematician.<p>And a third was a biology major. After graduating, he worked as a baker. But he wanted a career change so he entered a master’s program in math and proved an impressive result.
Decades ago two mathematicians argued about the area of the mandelbrot set. Both argued an asymtoically approaching different numbers. They started a distributed project to calculate the area.<p>I donated time on PA-risc workstations to the effort and was surprised to hear that the 2 machines contributed more to the final answer then 100s of other contributors. Something about how HP's compiler/chip preserved more accurate in the intermediate results than others. That surprised me since AFAIK the PA-risc is just a normal 64 bit floating point unit, which doesn't every have more precision for intermediate results. I believe PCs at the time often used the x86, which has 80 bits of precision for the intermediate results.<p>I believe the project was a success, but I don't remember the conclusion.
If you enjoyed learning about the Hashlife algorithm for Conway's Game of Life, the bilinear approximation algorithm for computing whether a point is in the mandelbrot set has the same je ne sais quoi. No one has done a really accessible writeup of it yet, but this blog post and the linked forum thread are a good start: <a href="https://mathr.co.uk/blog/2022-02-21_deep_zoom_theory_and_practice_again.html#a2022-02-21_deep_zoom_theory_and_practice_again_bilinear_approximation" rel="nofollow">https://mathr.co.uk/blog/2022-02-21_deep_zoom_theory_and_pra...</a>
Making a program to render a view of the Mandelbrot set is a fun exercise, I recommend it.<p>I found a viewer that works in the browser:<p>> Mandelbrot Viewer is a universal (desktop and mobile) vanilla JS implementation of a plain Mandelbrot set renderer – supporting mouse, touch and keyboard interaction.<p><a href="https://mandelbrot.silversky.dev/" rel="nofollow">https://mandelbrot.silversky.dev/</a>
This view on the Mandelbrot iteration seems much more interesting than the original set:<p><a href="https://en.wikipedia.org/wiki/Buddhabrot" rel="nofollow">https://en.wikipedia.org/wiki/Buddhabrot</a><p>It is the probability distribution (i.e. the most frequent locations visited) over the trajectory of the points that escape the plane.
'“We’ve got to try to train a neural network to zoom around the Mandelbrot set,” Kapiamba joked.'<p>This actually sounds to me to be fine goal. An AI that sounds out unexplored depths to reveal interesting sights that maybe resemble what we see in the world at our level or cool patterns that potentially are a delight to the eye would be pretty cool.
What is the conjectured topology of the Mandelbrot set if MLC is true?<p>My understanding is that there's a certain number of bulbs, each centred around a point which becomes periodic with period p after k steps. But how do they all stick together?
If you are interested in rendering the Mandelbrot set in a variety of ways, as well as its 3D extensions, look here: <a href="https://iquilezles.org/articles/" rel="nofollow">https://iquilezles.org/articles/</a><p>The last part is about fractals, especially the Mandelbrot set. With some theoretical and some practical articles.
Maybe Mandelbrot shape represents a state space or set of possible transformations, configurations or relationships of certain solvable or equilibrium dynamical systems, so maybe MLC is true if there's a certain structure-preserving relationship over sets of these dynamical systems.
Does anyone know of any good resources on Kolmogorov complexity and fractals such as the Mandelbrot set? Or even on information theory and fractals?<p>For some reason reading this article is making me wonder about the difference between the information required to generate something like a mandelbrot, knowing the underlying rule, and the information required to represent it as it is, without following the rule. Or e.g., the difference between the information of the generating rule and the information implicitly represented through the time or number of operations needed to generate it.<p>It seems like there's some analogy between potential and kinetic energy, and kolmogorov complexity and something else, that I'm having trouble putting my finger on. Even if you have a simple generating algorithm that might be small in a kolmogorov complexity sense, if that algorithm entails a repeating something over a large number of operations, the resulting object would be complex, so there's an implied total complexity as well as an "generating" one.<p>Maybe this is some basic computational complexity concept but if so I'm not recalling this, or am being dense. E.g., I'm used to discussions of "compressibility" but not of the "generating representation information cost" versus "execution cost".
My favourite way to think about this shape is <a href="https://en.m.wikipedia.org/wiki/File:Unrolled_main_cardioid_of_Mandelbrot_set_for_periods_8-14.png" rel="nofollow">https://en.m.wikipedia.org/wiki/File:Unrolled_main_cardioid_...</a><p>... It's all elephants. The -R spike at the main disc is period 2, the fork around +/-i is period 3, and so on to infinity at 0.25+0i.<p>MLC might just be one of those facts that is true but unprovable!
Since the Mandelbrot iteration happens on the complex plane, is there any recommended reading / research about its relation to scientific fields where the complex plane is applied, like mechanical oscillatory systems, quantum mechanics and electromagnetism?
For some fractals on the regular, Benoit Mandelbot on Mastodon - <a href="https://botsin.space/@benoitmandelbot" rel="nofollow">https://botsin.space/@benoitmandelbot</a>
I don't think I understand what 'locally connected' means. You can easily choose a rectanglular area that contains 2 areas of the set that are not joined.
i’ve often wondered if mandelbrot is what you get when you do a simple quadratic iterator in complex numbers, what are the comparable sets for quaternions and octonions??
>"When computers revealed all those smaller copies of the Mandelbrot set within itself, Douady and Hubbard wanted to explain their presence. They ended up turning to what’s known as <i>renormalization theory</i>, a technique that physicists use to tame infinities in the study of quantum field theories, and to connect different scales in the study of phase transitions."<p><a href="https://en.wikipedia.org/wiki/Renormalization" rel="nofollow">https://en.wikipedia.org/wiki/Renormalization</a><p>Rampant conjecture/speculation: In the future, perhaps some Mathematician might discover a link between <i>Renormalization Theory</i> -- and the <i>Digits Of Pi</i>... since they seem related...<p>More specifically, between Renormalization Theory -- and algorithms for the Digits of Pi.<p>Of which, one notable one is The Chudnovsky algorithm:<p><a href="https://en.wikipedia.org/wiki/Chudnovsky_algorithm" rel="nofollow">https://en.wikipedia.org/wiki/Chudnovsky_algorithm</a><p>Which leads to Binary Splitting:<p><a href="https://en.wikipedia.org/wiki/Binary_splitting" rel="nofollow">https://en.wikipedia.org/wiki/Binary_splitting</a><p>Which leads to Hypergeometric Series:<p><a href="https://en.wikipedia.org/wiki/Hypergeometric_function#The_hypergeometric_series" rel="nofollow">https://en.wikipedia.org/wiki/Hypergeometric_function#The_hy...</a><p>Which leads to Gauss' continued fraction:<p><a href="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d54114465d858bd4de8e8bb87818d19d9e2da38" rel="nofollow">https://wikimedia.org/api/rest_v1/media/math/render/svg/4d54...</a><p><a href="https://en.wikipedia.org/wiki/Hypergeometric_function#:~:text=Gauss%27%20continued%20fraction" rel="nofollow">https://en.wikipedia.org/wiki/Hypergeometric_function#:~:tex...</a><p><a href="https://en.wikipedia.org/wiki/Gauss%27s_continued_fraction" rel="nofollow">https://en.wikipedia.org/wiki/Gauss%27s_continued_fraction</a><p>Which leads to Analytic continuation of 3F2, 4F3 and higher functions:<p><a href="https://fredrikj.net/blog/2009/12/analytic-continuation-of-3f2-4f3-and-higher-functions/" rel="nofollow">https://fredrikj.net/blog/2009/12/analytic-continuation-of-3...</a><p>Which leads to my brain hurting ("Put down that Math book and step away from the Math!" <g>) -- because I can't handle all of this Math for now! :-) <g> :-)<p>But there is this very cool picture there:<p><a href="https://3.bp.blogspot.com/_rh0QblLk0C0/SzEG9q5FxaI/AAAAAAAAAL0/hfg6OZ2QHJA/s400/hcplot.png" rel="nofollow">https://3.bp.blogspot.com/_rh0QblLk0C0/SzEG9q5FxaI/AAAAAAAAA...</a>
The Mandelbrot set is quite well known.<p>Yet something I learned recently blew my mind. It's about the uncanny resemblance between the images generated by the Mandelbrot set, and among all things, the popular image of Buddha.<p>For example: <a href="https://en.wikipedia.org/wiki/Buddhabrot" rel="nofollow">https://en.wikipedia.org/wiki/Buddhabrot</a><p>Even when looking at the 2D Mandelbrot set renderings, I can't help but wonder whether the similarity of the "bulbs" to the rather unique Buddha "hairstyle" (of allegedly funny lumps of hair) was just a coincidence.<p>Also, the tower-like makuṭa headdress in some Buddhist traditions look exactly like the thin threads that connect the bulbs at around (-2, 0).<p>I'm not saying they mean anything, but just something uncanny, and once I learned about the resemblance, it's hard to unsee it...
> the word evoked the notion of a new kind of geometry — something fragmented, fractional or broken.<p>But that's not what "fractal" means; it means "fractional dimension". To say the word "fractal" evoked something is subjective - evoked it for whom?