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The quest to decode the Mandelbrot set
- QuadmasterXLII 3y agoIf 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: 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 https://mathr.co.uk/blog/2022-02-21_deep_zoom_theory_and_pra...
- RBerenguel 3y agoI did some research on this on the side for my dissertation, but never published it. The fact centers approximate the boundary generalises to almost any point in the plane as a consequence of normality of some sequences, and generalises to most families of complex iteration under very mild conditions. I’ve had a preprint that I never felt like finishing for something like 15 years lying around.
- QuadmasterXLII 3y agoIf you ever decide to put it out as-is, I'd love to read it!
- RBerenguel 3y agoI should probably brush it up and just upload it to Arxiv. But then I need ok from my advisor, and maybe she’d rather not (since she’d be a coauthor of that..). I’ll try again, thanks for the encouragement.
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- keepamovin 3y agoMaybe 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.
- deleted 3y ago[deleted]
- sliken 3y agoDecades 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. 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. I believe the project was a success, but I don't remember the conclusion.
- shrx 3y agoMore context here (alt.fractals discussion from February 1991) [0]: [...] by computing the area of the M-set using lots of terms in a series (Laurent Series?), the upper bound of the area seems to converge about at 1.72 (the graph gets quite flat, and seems to have an asymptote there), and by counting pixals more and more accurately, you seem to get a lower bound of very close to 1.52. Both these bounds are close to the values the methods would produce in the limit - that is, it is NOT the case that these numbers would get closer if a finer grid were used, or more terms were taken in the series. So, why the difference of 10% or so? No one knows. [0] https://ics.uci.edu/~eppstein/junkyard/mand-area.html https://ics.uci.edu/~eppstein/junkyard/mand-area.html
- denton-scratch 3y ago> the word evoked the notion of a new kind of geometry — something fragmented, fractional or broken. But that's not what "fractal" means; it means "fractional dimension". To say the word "fractal" evoked something is subjective - evoked it for whom?
- pohl 3y agoI suppose the dimension is the something, to be charitable. Compare to the etymology section here https://en.wikipedia.org/wiki/Fractal https://en.wikipedia.org/wiki/Fractal
- pvg 3y agoFor people who know similar-sounding words. Words can evoke things well outside their etymology or denotation, one of the many reasons people like and use words.
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- TheOtherHobbes 3y agoFor people who don't know what a fractional dimension is. And mostly don't care. Fractal art of all kinds was part of a certain trend in 80s/90s culture, which also influenced the look of the early Internet. And electronic dance music, clubs, and raves. It was maximalist, psychedelic, colourful, busy, inclusive, recursive, and complex. Whatever the math was doing, it was a very popular signifier of certain kinds of experience. I suspect it's not a surprise that if faded into the background when the Internet began to commercialise and blandify in the later 90s.
- infogulch 3y agoMaking a program to render a view of the Mandelbrot set is a fun exercise, I recommend it. I found a viewer that works in the browser: > Mandelbrot Viewer is a universal (desktop and mobile) vanilla JS implementation of a plain Mandelbrot set renderer – supporting mouse, touch and keyboard interaction. https://mandelbrot.silversky.dev/ https://mandelbrot.silversky.dev/
- brazzy 3y agoHere's my attempt done a long time ago, not very polished: https://brazzy.de/en/Mandelbrot.php https://brazzy.de/en/Mandelbrot.php Even for that, the actual Mandelbrot calculations were the smaller part. It's really amazing how such a trivially simple formula spawns such endless complexity.
- rnentjes 3y agoI'll add my julia attempt that animates and works with webgl: http://julia.perses.games/ http://julia.perses.games/
- swayvil 3y agoComplexity is trivial. That's the lesson here. It's human perception that's special. It's special in that it has an uncommonly low bar on what's considered impressively complex. Which implies something important, no doubt.
- QuadmasterXLII 3y agoIf you want to keep zooming in the browser and have snappy gpu accelerated performance while basically never hitting “reached limit of numerical precision,” check out https://mandeljs.hgreer.com https://mandeljs.hgreer.com . The math to make this possible gets pretty funky: I go over the tricks I used at https://www.hgreer.com/JavascriptMandelbrot/ https://www.hgreer.com/JavascriptMandelbrot/
- sapiogram 3y agoNo scrolling with the scroll wheel though :(
- martyvis 3y ago'“We’ve got to try to train a neural network to zoom around the Mandelbrot set,” Kapiamba joked.' 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.
- hnfong 3y agoThe Mandelbrot set is quite well known. 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. For example: https://en.wikipedia.org/wiki/Buddhabrot https://en.wikipedia.org/wiki/Buddhabrot 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. Also, the tower-like makuṭa headdress in some Buddhist traditions look exactly like the thin threads that connect the bulbs at around (-2, 0). I'm not saying they mean anything, but just something uncanny, and once I learned about the resemblance, it's hard to unsee it...
- flohofwoe 3y agoAlso see the Mandelbrot monk (before you get too excited, it was a hoax) https://abcnews.go.com/Technology/WhosCounting/story?id=98615&page=1 https://abcnews.go.com/Technology/WhosCounting/story?id=9861...
- ggambetta 3y ago> 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. Honest question: what else do you think it could be, if not a coincidence?
- hnfong 3y agoHonest answer: some early Buddhist followers taking some form of acid/mushrooms and saw geometric visions of the Mandelbrot set and thought it was a divine manifestation of Buddha? I mean, I don't think this is likely, but that's the best I got. I hear the brain likes to go into geometry mode when hallucinogens are ingested, and I suppose the brain is theoretically powerful enough to compute the Mandelbrot sets...
- armchairdweller 3y agoI have had similar thoughts. The prominent modes and paths of this 2D probability distribution also show some resemblance to the kabbalistic tree of life, which is its own, but fairly related topic of study. DMT use within a connected strand of this "inner science" has been suspected. Drawing more of these far-fetching connections: The complex plane is related to several areas of physics, which might somehow find expression in electromagnetic brain dynamics. In any case, the buddhabrot distribution seems quite understudied both from a scientific / mathematical PoV, and from the perspectives of the occluded study of the "inner realms".
- thanatos519 3y agoMy favourite way to think about this shape is https://en.m.wikipedia.org/wiki/File:Unrolled_main_cardioid_of_Mandelbrot_set_for_periods_8-14.png https://en.m.wikipedia.org/wiki/File:Unrolled_main_cardioid_... ... 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. MLC might just be one of those facts that is true but unprovable!
- pronoiac 3y agoFor some fractals on the regular, Benoit Mandelbot on Mastodon - https://botsin.space/@benoitmandelbot https://botsin.space/@benoitmandelbot
- DanielleMolloy 3y agoThis view on the Mandelbrot iteration seems much more interesting than the original set: https://en.wikipedia.org/wiki/Buddhabrot https://en.wikipedia.org/wiki/Buddhabrot It is the probability distribution (i.e. the most frequent locations visited) over the trajectory of the points that escape the plane.
- DanielleMolloy 3y agoSince 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?
- asplake 3y agoI had a couple of related questions. To what extent does complex dynamics map to physical phenomena? And in the opposite direction, how is renormalisation used outside of quantum physics?
- markisus 3y agoA very interesting cast of underdog characters appears in the article. 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. 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. 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.
- dylan604 3y agoSounds like a couple of people decided to take an extended gap year, and then came back more focused and in a better place for the number crunching
- fweimer 3y agoNote that the article refers to Soviet antisemitism: Jews were denied academic jobs, and they couldn't move abroad to work in their field, either. Not really related to mathematics as such (and definitely not about the mathematical community rejecting an antisemite).
- eps 3y ago> they couldn't move abroad to work in their field Nobody could do that, Jews or not.
- OscarCunningham 3y agoWhat is the conjectured topology of the Mandelbrot set if MLC is true? 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?
- bongodongobob 3y agoThe entire set is connected iirc.
- qazxcvbnm 3y agoBut what would be its homology, for instance?
- OscarCunningham 3y agoIt's known that it's connected and simply connected. So if it's locally connected then I think it has to be contractable.
- clintonc 3y agoMLC stands for "Mandelbrot Locally Connected". It's not obvious, but this is equivalent to the bulbs of the Mandelbrot set (the domains of parameters where almost all points get attracted toward periodic orbits) are dense in the Mandelbrot set. Everyone believes it to be true.
- OscarCunningham 3y agoYes, but how exactly are the bulbs arranged? Wikipedia says 'Not every hyperbolic component can be reached by a sequence of direct bifurcations from the main cardioid of the Mandelbrot set. Such a component can be reached by a sequence of direct bifurcations from the main cardioid of a little Mandelbrot copy'. Which sequences of bulbs have little copies at the end of them? And how do the little copies attach?
- derbOac 3y agoDoes anyone know of any good resources on Kolmogorov complexity and fractals such as the Mandelbrot set? Or even on information theory and fractals? 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. 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. 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".
- dcow 3y agoI think you’re forgetting that there’s no definitive way to represent something, compressed or raw. So while it’s interesting to acknowledge that we can compress data and sometimes rather efficiently, I’m not sure it maps to anything physical beyond the fact that decompressing data creates entropy. Perhaps you’d be interested in https://en.wikipedia.org/wiki/Landauer%27s_principle https://en.wikipedia.org/wiki/Landauer%27s_principle. Turns out there may be a minimum energy required to decrease entropy. Jade has a really good overview https://youtu.be/XY-mbr-aAZE?si=7DvSs2DMudsh6gk8 https://youtu.be/XY-mbr-aAZE?si=7DvSs2DMudsh6gk8
- kjqgqkejbfefn 3y ago> so there's an implied total complexity as well as an "generating" one. Dessalles's algorithmic simplicity theory of (cognitive) relevance is formulated in these terms. >Situations are relevant to human beings when they appear simpler to describe than to generate The discrepancy between generation complexity >the complexity (minimal description) of all parameters that have to be set for the situation s to exist in the "world" i.e, the "pixels" and description complexity > the length of the shortest available description of s (that makes s unique) i.e. the mandelbrot formula is named Unexpectedness in this framework. https://telecom-paris.hal.science/hal-03814119/document https://telecom-paris.hal.science/hal-03814119/document https://simplicitytheory.telecom-paris.fr/ https://simplicitytheory.telecom-paris.fr/ Dessalles published a paper in 2022, Unexpectedness and Bayes’ Rule https://cifma.github.io/Papers-2021/CIFMA_2021_paper_13.pdf https://cifma.github.io/Papers-2021/CIFMA_2021_paper_13.pdf >A great number of methods and of accounts of rationality consider at their foundations some form of Bayesian inference. Yet, Bayes’ rule, because it relies upon probability theory, requires specific axioms to hold (e.g. a measurable space of events). This short document hypothesizes that Bayes’ rule can be seen as a specific instance of a more general inferential template, that can be expressed also in terms of algorithmic complexities, namely through the measure of unexpectedness proposed by Simplicity Theory. Maybe there is a way to plug this into the https://en.wikipedia.org/wiki/Buddhabrot https://en.wikipedia.org/wiki/Buddhabrot fractal someone mentioned above.
- GuB-42 3y agoIf you are interested in rendering the Mandelbrot set in a variety of ways, as well as its 3D extensions, look here: https://iquilezles.org/articles/ https://iquilezles.org/articles/ The last part is about fractals, especially the Mandelbrot set. With some theoretical and some practical articles.
- auroralimon 3y agoi’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??
- bunabhucan 3y agohttps://en.m.wikibooks.org/wiki/Pictures_of_Julia_and_Mandelbrot_Sets/Quaternions https://en.m.wikibooks.org/wiki/Pictures_of_Julia_and_Mandel... There's a whole world of this as well as iterating different functions.
- hermitcrab 3y agoI 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.
- nonsensikal 3y agoYou don't get to choose a rectangle, you choose a point.
- hermitcrab 3y agoThe doesn't seem to fit with the comb analogy. I still don't understand.
- returningfory2 3y agoI think this part of the article is incorrect.
- matt-noonan 3y agoIt's almost correct, but misses the point in an annoying way that kind of ruins the example. What does work is something like the subset of the plane given by { (x, y) | x real, y rational } U { (0, y) | y real }. This is connected, because you can walk from any point (x,y) to any other point (x',y') by traveling horizontally to the Y axis at (0,y), vertically to (0,y'), then horizontally to (x',y'). But it isn't locally connected away from the Y axis because for a tiny enough open set S around a point (x,y), there are other points in S that you can't get to from (x,y) without leaving S.
- nhatcher 3y agoNon locally connected spaces are a bit pathological. Means that given a point there is always a neighborhood of the point (might be very small) that is connected. An example of a connected but not locally connected is: https://en.m.wikipedia.org/wiki/Topologist%27s_sine_curve https://en.m.wikipedia.org/wiki/Topologist%27s_sine_curve From (0, O) any neighborhood, no matter how small contains points that belong to the curve but cannot reach (0, 0) and stay in the neighborhood.
- bilsbie 3y agoI’ve always wondered if you could make fractals into some sort of game.
- netmare 3y agoThere's MMCE†, the spiritual successor of Marble Marcher. I've only played the original some years ago and it was pretty awesome. You simply have to move a ball across a 3D fractal surface that's constantly evolving in real-time! Sadly, I can't run MMCE to test, since I'm using a PC and gfx card from 2009. †: https://michaelmoroz.itch.io/mmce https://michaelmoroz.itch.io/mmce
- deleted 3y ago[deleted]
- peter_d_sherman 3y ago>"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 renormalization theory, 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." https://en.wikipedia.org/wiki/Renormalization https://en.wikipedia.org/wiki/Renormalization Rampant conjecture/speculation: In the future, perhaps some Mathematician might discover a link between Renormalization Theory -- and the Digits Of Pi... since they seem related... More specifically, between Renormalization Theory -- and algorithms for the Digits of Pi. Of which, one notable one is The Chudnovsky algorithm: https://en.wikipedia.org/wiki/Chudnovsky_algorithm https://en.wikipedia.org/wiki/Chudnovsky_algorithm Which leads to Binary Splitting: https://en.wikipedia.org/wiki/Binary_splitting https://en.wikipedia.org/wiki/Binary_splitting Which leads to Hypergeometric Series: https://en.wikipedia.org/wiki/Hypergeometric_function#The_hypergeometric_series https://en.wikipedia.org/wiki/Hypergeometric_function#The_hy... Which leads to Gauss' continued fraction: https://wikimedia.org/api/rest_v1/media/math/render/svg/4d54114465d858bd4de8e8bb87818d19d9e2da38 https://wikimedia.org/api/rest_v1/media/math/render/svg/4d54... https://en.wikipedia.org/wiki/Hypergeometric_function#:~:text=Gauss%27%20continued%20fraction https://en.wikipedia.org/wiki/Hypergeometric_function#:~:tex... https://en.wikipedia.org/wiki/Gauss%27s_continued_fraction https://en.wikipedia.org/wiki/Gauss%27s_continued_fraction Which leads to Analytic continuation of 3F2, 4F3 and higher functions: https://fredrikj.net/blog/2009/12/analytic-continuation-of-3f2-4f3-and-higher-functions/ https://fredrikj.net/blog/2009/12/analytic-continuation-of-3... 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> :-) But there is this very cool picture there: https://3.bp.blogspot.com/_rh0QblLk0C0/SzEG9q5FxaI/AAAAAAAAAL0/hfg6OZ2QHJA/s400/hcplot.png https://3.bp.blogspot.com/_rh0QblLk0C0/SzEG9q5FxaI/AAAAAAAAA...