The quest to decode the Mandelbrot set(quantamagazine.org)
quantamagazine.org
The quest to decode the Mandelbrot set
https://www.quantamagazine.org/the-quest-to-decode-the-mandelbrot-set-maths-famed-fractal-20240126/
90 comments
Sounds 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
Note 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).
> they couldn't move abroad to work in their field
Nobody could do that, Jews or not.
Nobody could do that, Jews or not.
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.
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.
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.
More 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
[...] 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
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: https://mathr.co.uk/blog/2022-02-21_deep_zoom_theory_and_pra...
I 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.
If you ever decide to put it out as-is, I'd love to read it!
I 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.
Making 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/
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/
If 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 . The math to make this possible gets pretty funky: I go over the tricks I used at https://www.hgreer.com/JavascriptMandelbrot/
Wow nice! Though I seem to get artifacts both in Brave and Firefox on Android:
https://imgur.com/a/vo2ZVOQ
https://mandeljs.hgreer.com/?;re=-0.751165720536567020384363...
https://mandeljs.hgreer.com/?;re=-0.751165720536567020384363...
That really is very fast!
(I don't know if this is a known compromise of your technique (I couldn't find it mentioned), but I occassionally get a ring of varying thickness centred in the render with an inverted coluring to the rest of the pixels, or sometimes a solid colour. It varies is size and comes and goes without an obvious correlation to zoom level.)
(I don't know if this is a known compromise of your technique (I couldn't find it mentioned), but I occassionally get a ring of varying thickness centred in the render with an inverted coluring to the rest of the pixels, or sometimes a solid colour. It varies is size and comes and goes without an obvious correlation to zoom level.)
Yeah, I've been chasing that for a while. I've traced it to subnormal float handling. Here's the scenario: if you multiply the 32 bit float float 1 x 2^-127 by 1/8, you can't get 1x2^-130 because 130 isn't representable in a 8 bit exponent. On the CPU or a modern GPU, this is special cased to result in a "subnormal" float: .125 x 2^-127. However, this adds a ton of extra complexitity to the FPU. Old GPUs just gave up and say "It's zero." WebGL standardized on the old GPU behavior, so a modern GPU running webgl sets a flag to behave like the old GPUs.
I'm representing complex numbers as (real_mantissa + i * imag_mantissa) * 2^exp where real_mantissa and imag_mantissa are themselves 32 bit floats, and I can reduce the frequency of the colored rings in my shader by keeping the mantissas around ~1000 instead of ~1 (and reducing the exponent to keep the value the same) so that they are less likely to go subnormal, but I can't seem to get rid of the colored rings entirely.
I don't actually know the pathway through the code from subnormal underflow -> colored rings, but if you run a CUDA renderer with the same algorithm, the rings appear when you set the flag for "round subnormal floats to zero."
I'm representing complex numbers as (real_mantissa + i * imag_mantissa) * 2^exp where real_mantissa and imag_mantissa are themselves 32 bit floats, and I can reduce the frequency of the colored rings in my shader by keeping the mantissas around ~1000 instead of ~1 (and reducing the exponent to keep the value the same) so that they are less likely to go subnormal, but I can't seem to get rid of the colored rings entirely.
I don't actually know the pathway through the code from subnormal underflow -> colored rings, but if you run a CUDA renderer with the same algorithm, the rings appear when you set the flag for "round subnormal floats to zero."
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Thanks for creating this, I had some fun exploring the set. What does the scale parameter stand for?
The color of each pixel is
fragColor = vec4(vec3(cos(c), cos(1.1214 * c) , cos(.8 * c)) / 2. + .5, 1.);
No scrolling with the scroll wheel though :(
The whole thing is a tower of spaghetti bit hacks in the name of depth and speed. Embarassingly, it actually can't actually render zoom levels that aren't powers of two and I'm not sure how to change that.
Can you make it so that the position you clicked stays at the same position after each zoom, rather than centering it on your cursor? That way you can click multiple times to zoom in on the same point.
I made quite a sophisticated viewer as a teen, but boy was it slow to run back then, back on a 486.
Now people write viewers that run lighting fast in a browser thanks to WebGPU! Like:
https://www.reddit.com/r/fractals/comments/o7l4bm/please_try...
Now people write viewers that run lighting fast in a browser thanks to WebGPU! Like:
https://www.reddit.com/r/fractals/comments/o7l4bm/please_try...
I remember as a teen I did mine in 68000 assembly on a Commodore Amiga to get it reasonable fast considering it was running on a 16 MHz CPU — i would say a few seconds to draw the canonical image — but IIRC it reached pretty fast the limit of math precision.
At that time I didn't know what was a complex number, but was fascinated by the whole concept of fractals and how complex structures could be created with a relative simple program.
At that time I didn't know what was a complex number, but was fascinated by the whole concept of fractals and how complex structures could be created with a relative simple program.
A few seconds? Wow! I did a science fair project on the Mandelbrot set using my Amiga 2000, and it took me a good 45+ minutes to generate a single 320x200 color image. IIRC, I wrote the generator in some variant of Pascal, and was so happy with the performance increase over Basic on my C=128...
I ran in to the precision limit pretty quickly, same as you. I didn't understand computers well enough to know that's what the problem was, and I remember spending hours pouring over my code, trying to figure out where the bug was. Good times. :D
I ran in to the precision limit pretty quickly, same as you. I didn't understand computers well enough to know that's what the problem was, and I remember spending hours pouring over my code, trying to figure out where the bug was. Good times. :D
If I'm not confusing it with something else, I seem to recall that when zooming on the set the calculation was nearly instantaneous. As you said, good times! ^__^
I still have that A500 but who knows where I put those floppies, I'm tempted to turn it on but I'm scared the PSU will blow itself...
I still have that A500 but who knows where I put those floppies, I'm tempted to turn it on but I'm scared the PSU will blow itself...
A500 PSUs (particularly the 2nd one) are the most reliable and powerful.
The A500 themselves are built like tanks. Chances are it just works.
But watch out for trapdoor expansion. Most likely has a varta barrel battery in it, which will eventually leak and damage the expansion, and possibly also the computer itself.
I would recommend opening that trapdoor and removing/inspecting anything installed there as soon as possible.
These barrel batteries are only used to keep RTC, and the board will be fine w/o.
The A500 themselves are built like tanks. Chances are it just works.
But watch out for trapdoor expansion. Most likely has a varta barrel battery in it, which will eventually leak and damage the expansion, and possibly also the computer itself.
I would recommend opening that trapdoor and removing/inspecting anything installed there as soon as possible.
These barrel batteries are only used to keep RTC, and the board will be fine w/o.
The difference is probably you used floats and he/she used fixed point (integers). Of course that way you do run out of precision very quickly.
Software emulated floats on Amiga 2000 were really, really slow.
Software emulated floats on Amiga 2000 were really, really slow.
I typed in a BASIC program for a C-64 from a computer magazine. That was a popular mechanism for software distribution back then.
I thought "this will never work", but I was amazed that it did in fact work. I remember letting it run overnight to find a complete 160x200 4-color image the next morning. Just clearing the screen using a loop to fill 8192 bytes with 0s took over a minute.
Everything about the computers at the time was so terrible, I don't feel any nostalgia for those days or that technology.
I thought "this will never work", but I was amazed that it did in fact work. I remember letting it run overnight to find a complete 160x200 4-color image the next morning. Just clearing the screen using a loop to fill 8192 bytes with 0s took over a minute.
Everything about the computers at the time was so terrible, I don't feel any nostalgia for those days or that technology.
The one I wrote in BASIC on my Apple IIe never finished, it hadn't even got the interesting bits of the set after a day or so and I needed to play games.
FRACTINT was great because I could use my 286 and it was fairly fast because used integer whenever possible.
FRACTINT was great because I could use my 286 and it was fairly fast because used integer whenever possible.
I did it in QuickBasic (also on a 486).
Now that's slow.
Now that's slow.
Here's my attempt done a long time ago, not very polished:
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.
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.
I'll add my julia attempt that animates and works with webgl:
http://julia.perses.games/
http://julia.perses.games/
Complexity 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.
Which implies something important, no doubt.
I did it in High School on my crappy TI programmable calculator. Took for ever to calculate but it worked.
This view on the Mandelbrot iteration seems much more interesting than the original set:
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.
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.
'“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.
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?
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?
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?
MLC 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.
Yes, 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?
The combinatorics of how the Mandelbrot set is put together is well-studied, and rather independent of MLC. The arrangement of the bulbs on the boundary of the "main cardiod" (which is where there is an attracting fixed point) is described here: https://en.wikipedia.org/wiki/Mandelbrot_set#Main_cardioid_a.... Generally, the patterns are given by something called Lavaur's Algorithm; see https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_com... for some explanation. Attachment points are always at the "root" of the Mandelbrot set, which is the cusp of the main cardioid.
A consequence of MLC is that the combinatorial picture given by Lavaur's algorithm and related analyses is "complete" -- all dynamical information is available from the combinatorial models.
A consequence of MLC is that the combinatorial picture given by Lavaur's algorithm and related analyses is "complete" -- all dynamical information is available from the combinatorial models.
Thank you, that's very helpful!
The entire set is connected iirc.
But what would be its homology, for instance?
It's known that it's connected and simply connected. So if it's locally connected then I think it has to be contractable.
If 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/
The last part is about fractals, especially the Mandelbrot set. With some theoretical and some practical articles.
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.
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Does 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".
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".
I 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. 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
Perhaps you’d be interested in 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
> 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://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
>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 fractal someone mentioned above.
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://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
>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 fractal someone mentioned above.
My favourite way to think about this shape is 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!
... 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!
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?
I 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?
For some fractals on the regular, Benoit Mandelbot on Mastodon - https://botsin.space/@benoitmandelbot
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.
Suppose you choose such a rectangle, let's say it incorporates the 'fringe' of the main cardioid and the fringe of the biggest circle. Within that rectangle, color green all the pieces which connect to the main cardioid and blue all the pieces which connect to the circle. Local connectedness means that there won't be any points in that rectangle which have both blue and green points arbitrarily close. So there are places where 'locally non-connected parts' of the set can be close together, but there must be a border between them, rather than them being hopelessly entangled.
Strictly speaking, you would do this coloring with all connected components of the intersection of your rectangle and M. (And the rectangle could be any region.)
The example messes this up, although it is a correct example, the square on the diagram showing the local piece containing non-connected parts is wrong. The comb has more and more teeth, infinitely many in a bounded space, on the left side. Only a rectangle which includes the left edge properly shows why the set isn't locally connected. The rectangle pictured includes finitely many teeth which have a separation between them. A rectangle overlapping the left edge of the comb would include separate components which get arbitrarily close to that left edge and so can't be separated by a border.
Strictly speaking, you would do this coloring with all connected components of the intersection of your rectangle and M. (And the rectangle could be any region.)
The example messes this up, although it is a correct example, the square on the diagram showing the local piece containing non-connected parts is wrong. The comb has more and more teeth, infinitely many in a bounded space, on the left side. Only a rectangle which includes the left edge properly shows why the set isn't locally connected. The rectangle pictured includes finitely many teeth which have a separation between them. A rectangle overlapping the left edge of the comb would include separate components which get arbitrarily close to that left edge and so can't be separated by a border.
Thanks for taking the time to explain that. So, in terms I find easier to understand, MLC would mean that if I:
-take any rectanglular section of the complex plain that includes part or all of the Mandelbrot set
-draw the Mandelbrot set in black
-pick an arbitary black point and colour it red
-recursively colour every black point touching a red point (flood fill)
Then every black point would be recoloured red. And this would work with a pixel based image of the mandelbrot if the image had a high enough resolution. Is that right?
-take any rectanglular section of the complex plain that includes part or all of the Mandelbrot set
-draw the Mandelbrot set in black
-pick an arbitary black point and colour it red
-recursively colour every black point touching a red point (flood fill)
Then every black point would be recoloured red. And this would work with a pixel based image of the mandelbrot if the image had a high enough resolution. Is that right?
No, that's not right.
Do those first four steps. You wouldn't (necessarily) cover every black point in your rectangle. Choose a remaining black point and flood fill from that, say green. Keep on doing this with different colours until you've covered every black point in your rectangle. You have a bunch of regions of different colours.
Now, if the different coloured regions are all nicely separate, then your set is locally connected. Because each point is either cleanly in one component or cleanly in the other.
If on the other hand your drawing looks like https://commons.m.wikimedia.org/wiki/File:Julia_set_for_the_... with mixed up boundaries where some points are infinitesimally close to more than one colour, then it's not locally connected.
The difficulty with intuition is that in our intuition, coloured regions always have reasonable boundaries (think countries in a map: the border can be wiggly but there's never infinitely many tiny bits of one country mixed up in the boundary of two others). In fractal geometry, things like the Newton fractal picture above are quite usual.
Do those first four steps. You wouldn't (necessarily) cover every black point in your rectangle. Choose a remaining black point and flood fill from that, say green. Keep on doing this with different colours until you've covered every black point in your rectangle. You have a bunch of regions of different colours.
Now, if the different coloured regions are all nicely separate, then your set is locally connected. Because each point is either cleanly in one component or cleanly in the other.
If on the other hand your drawing looks like https://commons.m.wikimedia.org/wiki/File:Julia_set_for_the_... with mixed up boundaries where some points are infinitesimally close to more than one colour, then it's not locally connected.
The difficulty with intuition is that in our intuition, coloured regions always have reasonable boundaries (think countries in a map: the border can be wiggly but there's never infinitely many tiny bits of one country mixed up in the boundary of two others). In fractal geometry, things like the Newton fractal picture above are quite usual.
I think I understand now. Much appreciated! 'Locally connected' seems like quite poor terminology.
You don't get to choose a rectangle, you choose a point.
The doesn't seem to fit with the comb analogy.
I still don't understand.
I still don't understand.
Non 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
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.
I think this part of the article is incorrect.
It'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.
It means you can choose two points inside the Mandelbrot set and always find a curve that connects the two without you ever needing to lift the pencil.
Isn't that "connected" rather than "locally connected"?
I'm aware that "locally connected" has a very specific meaning in math:
https://en.wikipedia.org/wiki/Locally_connected_space
Unfortunately I don't have the slightest idea what it actually means... that article does not have any ELI5 sentence within it.
https://en.wikipedia.org/wiki/Locally_connected_space
Unfortunately I don't have the slightest idea what it actually means... that article does not have any ELI5 sentence within it.
>Unfortunately I don't have the slightest idea what it actually means
Not just me then? ;0)
Not just me then? ;0)
There are some in hints in TFA:
"MLC posits that the Mandelbrot set isn’t just connected; it’s locally connected — no matter how much you zoom in on the Mandelbrot set, it will always look like one connected piece. For instance, a circle is locally connected. An extremely fine-toothed comb, on the other hand, is not. Though the entire shape is connected, if you skip over the shaft and instead zoom in on the tips of some of its teeth, you’ll just see a bunch of separate line segments."
"MLC posits that the Mandelbrot set isn’t just connected; it’s locally connected — no matter how much you zoom in on the Mandelbrot set, it will always look like one connected piece. For instance, a circle is locally connected. An extremely fine-toothed comb, on the other hand, is not. Though the entire shape is connected, if you skip over the shaft and instead zoom in on the tips of some of its teeth, you’ll just see a bunch of separate line segments."
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I’ve always wondered if you could make fractals into some sort of game.
There'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
Sadly, I can't run MMCE to test, since I'm using a PC and gfx card from 2009.
†: https://michaelmoroz.itch.io/mmce
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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??
https://en.m.wikibooks.org/wiki/Pictures_of_Julia_and_Mandel...
There's a whole world of this as well as iterating different functions.
There's a whole world of this as well as iterating different functions.
>"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
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
Which leads to Binary Splitting:
https://en.wikipedia.org/wiki/Binary_splitting
Which leads to 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/4d54...
https://en.wikipedia.org/wiki/Hypergeometric_function#:~:tex...
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-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/AAAAAAAAA...
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
Which leads to Binary Splitting:
https://en.wikipedia.org/wiki/Binary_splitting
Which leads to 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/4d54...
https://en.wikipedia.org/wiki/Hypergeometric_function#:~:tex...
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-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/AAAAAAAAA...
The 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
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...
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
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...
> 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?
Honest question: what else do you think it could be, if not a coincidence?
Honest 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...
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...
I 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".
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".
I've done a wide variety of hallucinogens. While I've seen things that could be described as fractal (in that they are nested and self similar), I've never seen a literal time escape fractal. In my experience, hallucinations look more like turbulent flow, like smoke. Deepdream isn't a terrible first approximation, but may give you the impression that the hallucinations are more dramatic or total than is typical.
I can't rule this hypothesis out entirely, but I'm very skeptical. Pareidolia seems more likely to me.
I can't rule this hypothesis out entirely, but I'm very skeptical. Pareidolia seems more likely to me.
There are a few others well-known fractals apart from the Buddhabrot that resemble real-world objects. My favourite is the burning ship fractal: https://en.wikipedia.org/wiki/Burning_Ship_fractal
I found these sorts of things really helpful in getting my kids interested in fractals. They love the idea that there are "things" they can find that are only viewable through math.
I found these sorts of things really helpful in getting my kids interested in fractals. They love the idea that there are "things" they can find that are only viewable through math.
The burning ship is still my favourite object in mathematics despite about twenty years of study since I first came across it. In an echo of the famous story about Tamm and the Taylor series error, I once had to explain the burning ship to border control in an impromptu test of my credentials (though I'm not sure if I'd have been shot if I got it wrong).
Also see the Mandelbrot monk (before you get too excited, it was a hoax)
https://abcnews.go.com/Technology/WhosCounting/story?id=9861...
https://abcnews.go.com/Technology/WhosCounting/story?id=9861...
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> 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?
But that's not what "fractal" means; it means "fractional dimension". To say the word "fractal" evoked something is subjective - evoked it for whom?
For 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.
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.
For 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.
I suppose the dimension is the something, to be charitable.
Compare to the etymology section here
https://en.wikipedia.org/wiki/Fractal
Compare to the etymology section here
https://en.wikipedia.org/wiki/Fractal
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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.