Beyond the Mandelbrot set, an intro to holomorphic dynamics
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- Опубліковано 17 тра 2024
- An intro to holomorphic dynamics, the study of iterated complex functions.
Video on Newton's fractal: • From Newton’s method t...
Special thanks to these supporters: 3b1b.co/lessons/holomorphic-d...
Extra special thanks to Sergey Shemyakov, of Aix-Marseille University, for helpful conversations and for introducing me to this phenomenon.
Introduction to Fatou sets and Julia sets, including a discussion of Montel's theorem and its consequences:
www.math.stonybrook.edu/~scott...
Numberphile with Ben Sparks on the Mandelbrot set:
• What's so special abou...
Ben explains how he made the Geogebra files on his channel here:
• Julia Sets and Orbits ... (part 1)
• Mandelbrot Orbits and ... (part 2)
Excellent article on Acko.net, from the basics of building up complex numbers to Julia sets.
acko.net/blog/how-to-fold-a-j...
Bit of a side note, but if you want an exceedingly beautiful rendering of the quaternion-version of Julia fractals, take a look at this Inigo Quilez video:
• "Geodes" - 3D Julia se...
I first saw Fatou's theorem in this article:
projecteuclid.org/journals/co...
Moduli spaces of Newton maps:
arxiv.org/pdf/1512.05098.pdf
On Montel's theorem:
people.ucsc.edu/~fmonard/Sp17...
On Newton's Fractal:
core.ac.uk/download/pdf/16338...
Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
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github.com/ManimCommunity/manim/
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Music by Vincent Rubinetti.
www.vincentrubinetti.com/
Download the music on Bandcamp:
vincerubinetti.bandcamp.com/a...
Stream the music on Spotify:
open.spotify.com/album/1dVyjw...
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Timestamps:
0:00 - Intro
3:02 - Rational functions
4:15 - The Mandelbrot set
8:12 - Fixed points and stability
12:51 - Cycles
16:25 - Hidden Mandelbrot
21:17 - Fatou sets and Julia sets
26:24 - Final thoughts
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I enjoy how on twitter you asked recently whether we preferred two 17 minute videos or one 34 minute video. Instead you seem to have given two ~30 min videos:D Best of both worlds:D
Hi there !
WOAH WOAH WOAH!!! Let me get this perfectly straight: You comment something that is completely unrelated to the fact that I have two HAZARDOUSLY HANDSOME girlfriends? Considering that I am the unprettiest UA-camr worldwide, it is really incredible. Yet you did not mention it at all. I am VERY disappointed, dear dr
@@AxxLAfriku new profile picture
He interpolated
@@AxxLAfriku if they are so beautiful and you are so ugly, why would you cheat on them?
As a kid of the 80s who iterated Mandelbrot sets on an i386 and would wait patiently for hours to see patterns emerge, I have to draw attention to the computational miracle you’re looking at... Julia sets being near instantly populated with the waive of a mouse!
As someone who has programmed graphics engines before, these graphics are astounding and beautiful to me. Technology these days is amazing, being able to see things that past mathematicians never dreamed of seeing.
I used to program fractals on an 8088 running at 7.44 MHz. I would have to start it before going to bed and see what emerged some time the next day. So I had the same reaction as you.
@@JeffreyLWhitledge, I too programmed them on my 8088 and would fall apart in excitement at the meager results that emerged eons later. Three fewer pixels and it would have been radio.
Years ago I wrote a Mandelbrot program in C64 assembly. It took 25 hours to compute a 320 × 200 black-and-white image. (And that was with optimisation for the main cardioid, IIRC.)
@@germansnowman, exactly! A handful of pixels, at least a day to run, B&W only. But it was just so damn exciting!!
I recently got my Ph. D. in holomorphic dynamics. We often refer to the "stuff goes everywhere principle" as the "explosion property" of Julia sets. In fact, for the higher dimensional generalization of holomorphic dynamics (known as quasiregular dynamics), this explosion property is used as the definition of a Julia set.
If you’re still able to contact the school where you learned this, perhaps recommend the name “the shotgun property” for this effect, since birdshot shells scatter pellets all over and in a fairly random spread, and from what I’ve seen in the examples from this video, there’s usually one step where the points go from a relatively tight cluster to semi-triangular, fairly random spread.
@@Myne33 ....Do you think the school named that? These things aren't officially named, someone discovers something, calls it something in their paper and either it catches on or it doesn't. Someone else might call it something else in their paper, and then that becomes common parlance. Sure, you can call it something different in your paper but if a term is in common use there's little chance a new name will catch on. Clarity in what you're talking about is important
@@YaamFel So, @myne33, the best way to have your name stick is to write this decade's most important paper on holomorphic dynamics using that name :D
Could you help me out with this: given a neighborhood N of some point in the Julia set as initial values, each point in the plane corresponds to some iteration and initial value pair (k, x0) with x0 in N. It follows each point in Fatou set corresponds to some pair (k', x0'). However, as the recursive algorithm is memoryless, the process (k' + n, x0'), n in 0,1,... must be stable. Unless each point in the plain corresponds to some pair (k, x0) where value at (k - 1, x0) is in the Julia set, I fail to understand how the process can explode.
@@MrSuperkalamies Honestly, I can't come up with an intuitive explanation for the explosion property using your idea of individual iterated sequences. The actual proof of this property uses the idea of normal families, and it is remarkably simple once you wrap your head around normal families.
Maybe this idea helps. The Little Picard Theorem says that any map holomorphic on the entire complex plane (plus infinity) omits at most three points, otherwise it is constant. There is a similar theorem in the world of normal families, saying that any family of holomorphic maps on the 2-sphere that omit the same three points is a normal family. This is Montel's Theorem that Grant alluded to in the video.
The Julia set is defined as the set where the iterates of f do not form a normal family. So, if we look at the iterates of f on a neighborhood N of some point z_0 in the Julia set, if the iterates don't cover the whole plane except at most two points (and infinity), then the iterates are normal on N. This is a contradiction since z_0 is in the Julia set.
I highly recommend checking out Milnor's book Dynamics in One Complex Variable. It's a remarkably accessible (in my opinion as a mathematician, so grain of salt for non-mathematicians) introduction to holomorphic dynamics. If you skip partway into Section 3 and read through Section 4, it goes over what I just summarized in detail. From this perspective, the explosion property feels to me very natural and almost obvious, even though the sequence interpretation you give is entirely non-obvious.
@3:42 - "I think this distinguishes Julia as one of the greatest mathematicians of all time who had no nose."
Newton: Thank you for adding that critical qualifier at the end of your statement.
Finally, representation for hyper-intelligent Mermaid Man cosplayers
Tycho Brahe was thinking something similar, but for a different word in the sentence.
Gaston Julia is one of the "broken faces" ("gueules cassées") of World War 1
newton is overrated
@@tafazziReadChannelDescription 😂😂 yeah right, I have studied physics for 4 years and he is literally everywhere apart from stuff like quantum mechanics and electricity of course, all of mechanics is based on Newton works, optics is mostly Newton based of course leaving out stuff like YDSE, gave a very important thermodynamics law of cooling, which was the first significant law that described the physical relationship between heat and energy
Yeah, I used to think it was just recreational... then I started doin' it during the week... you know, simple stuff: differentiation, kinematics. Then I got into integration by parts... I started doin' it every night: path integrals, holomorphic functions. Now I'm on diophantine equations and sinking deeper into transfinite analysis. Don't let them tell you it's just recreational.
Fortunately, I can quit any time I want.
its not addictive i swear
Oxygen isn't "addictive" - either. -
But it is essential, -
"dependency"
[e.g. death without it.]
- - -
We find we are dependent upon "purpose":
Math is "sufficient"
I'm on the hard stuff man, Neron models, etale cohomology. There's no turning back once you decide to learn the proof of Fermat's last theorem.
Dang, I really wish I knew this Holomorphic Dynamics sooner. I came from strange attractor of 3D Nonlinear ODEs, going to Poincare Map and Bifurcation Diagram, just realizing that discrete dynamical systems is really amazing. Now, here I am, find out that my acceptance of discrete dynamics led to this beautiful stuff of complex analysis and hyperbolic geometry inside Holomorphic Dynamics. I feel really bad and late to this stuff.😢
@@muhammadismailyunus7950 how do you even get in that stuff tho? algebraic geoemtrey?
A second part! It was announced and actually came!
Edit: Thank you for all these wholesome videos. Waiting for them is always worth it, no matter what the topic is!
I like to think all sequel's I once promised will eventually come...it's just that the timeline sits somewhere between 1 week and 10 years.
@@3blue1brown lol
@@3blue1brown I can't wait for more awesome videos in the Probabilities of Probabilities series! 🙏🙏🙏
@@3blue1brown lmao
@@3blue1brown This has some kind of chaotic behaviour
I quite like this description:
“The Mandelbrot Set is a geography of iterative stability.”
I always wondered about the similarities between control theory and the Mandelbrot set but this description makes it blissfully obvious since control theory is all about feedback loops and iterative processes
@@Hoera290 yes, exactly. This simple description made a certain understanding “click” for me too!
Ben sparks!
@@whydontiknowthat …interest in the field.
Fractals such as the Mandelbrot and Julia sets are one of the things that, when I was in high school, convinced me that I would've done math at university. The others were chaos theory, non Euclidean geometries, and Simon Singh's book on Fermat's Last Theorem. Crucially, none of the books that got me interested in mathematics in high school were school books.
As someone doing mathematics at university, god i wish this is what we learned.
@@triciaf61 If you choose the courses correctly (assuming your university provides relevant courses), nothing should prevent you learning these things ;). Although Fermat's last theorem needs quite a lot of machinery that no single course would give sufficient knowledge to understand it's proof
@@triciaf61 what do you learn instead?
@@triciaf61 how far are you into your degree? AFAIK the cool complex stuff starts late
Ok
I personally will always be easy on 3blue1brown about his deadlines.
That's because..these videos are hard to make, and I mean at every single step.
It's hard to write a nonfiction narrative that's correct, then harder to write a narrative people can learn from and harder still to write a weaved story where listeners can come away feeling like they've seen something beautiful, which is of course what we want to communicate as artists: to convey our personal sense of beauty to someone we don't know.
Right now, I'm making an "explainer" with Manim because it looks incredible when it's done. But rendering and working and trying to make Manim work for me has been both fun and developmental because it's a test in both your fundamental programming, and your ability to articulate your math knowledge to a rigid computer. It's not harder than anything I've ever done. But it takes time, especially when you're caught up with other facets of life. Take it easy on yourself Grant!
have been toying with the idea of making a manim video. any thoughts on getting started?
@@hyperadapted any thoughts on what your topic would be?
@@nathanwycoff4627 time series econometrics or non-parametric statistics I think could be an addition
Making 30 minutes worth of Manim video is definitely an insanely large task.
Especially when some of the graphics are not in the library (i.e. how did he do the expanding circle shape in 23:32?)
not harder than anything you've ever done?
that would mean it's easier than breathing, or tying your shoes, etc.
think the correct wording is "not the hardest thing I've ever done"
@3:44 "I think this distinguishes Julia as easily being one of the greatest mathematicians of all time who had no nose." Actually, his nose was just in a different plane, so he was perfectly capable of detecting complex smells.
If the photographer had waited long enough, he would have seen the nose appearing while the back of the head disappeared
Honestly, why did 3B1B even bother to mention that? That utterly random distinction is unnecessary.
Obviously that would be Tycho Brahe
@@deleetiusproductions3497it's called a joke
When I was taught Newton Raphson many year ago, I was told: "Make sure your initial guess for the root is good, otherwise it doesn't always work." *Who knew "doesn't always work" was code for all this incredible beauty?* Thank you very much for being our guide.
*"Mathematics is like a very good detective novel. At first everything is shrouded in mystery and nothing is clear. But as you dive deeper to understand more, the plot gets crystal clear."*
Mathematics is honestly, truly amd genuinely very beautiful and I've fallen in love with this channel.
does it get clear tho?
@@sergey1519 there is always a cliffhanger. Like in a good detective novel series.
@@sergey1519 It is only as clear as we can make it... we are asymptotically approaching knowing all of mathematics, we will never reach the end, but we can learn as much as we can while we're here...
@@Anonymous-ow6jz "... we are asymptotically approaching knowing all of mathematics..." this quote is beautiful! where did you get this?
@@mayabartolabac Thank you! I didn't get it from anyone... I have been saying that for years...
I almost started writing my homework essay, thanks for showing me cool fractals instead
underrated
Priorities
I'm at a loss for words. What a fantastically coherent, clear, beautiful and exciting video. And, by the way, I really loved the exercises on this one. Never thought I would get to understand why the Mandelbrot set, of all things, has the shape it does. I felt like I was starting to _actually_ understand the topic, so don't ever feel like you're assigning "too many" exercises, please.
Thank you for your work, Grant.
This an outstanding and brilliant exposition. I am absolutely gobsmacked at the painstaking effort and talent it must have taken to produce a video of this quality on this subject.
Wanted to say something like that but I see you did it already .
Absolutely
This channel is an attractor point of math knowledge and beauty.
He really did the topic he promised this time :o
And gave us homework.
@@moonik665 a small price to pay
12:26 I did those three exercises, and the Cardioid fell out like magic. I swear, you are the best math teacher on the internet. I never did this kind of stuff when I got my degree, but these kind of recreational applications are by far the most fun and rewarding parts of pure math that I've encountered.
Could you help me with that for a second please? I'm a bit confused. z^2+c=z, if the initial z value is equal to c then the only fixed point is z=c=0, is it not?
If c is constant instead then the fixed points are z=(+/-sqrt(1-4c)+1)/2 and the derivative at those points is +/-sqrt(1-4c)+1
@@randomname7918 the solution to the equation z²+c=z depends on the quadratic formula. But to use to the quadratic formula correctly, it helps to write the equation as z²-z+c=0. Now use the quadratic formula, and think about what a, b, and c should be. Also, note that there will be solutions with a plus and solutions with a minus. You should think, maybe one of those gives attractive fixed points, and one might only give repulsive ones.
@@jmcsquared18 if we just use the quadratic equation, we get points that vary depending on c. I can't wrap my head around it. I mean, after all we are trying to find the attrating points, but what does it mean if they vary with c?
I'm sort of confused, sorry
@@kevinarturourrutiaalvarez2613 go back to 4:15 and watch carefully how the Mandelbrot set is defined. We are iterating the function f(z)=z²+c. Notice that it's a function of z only. That is, c is understood to be a fixed parameter when we do the iterations. However, the Mandelbrot set is defined to be what happens when we iterate the function f(z) specifically for z=0 for different choices of the parameter c.
So, we think of the iteration function as depending only on z, while we think of the fixed points as functions of c. In other words, we first choose a c beforehand, and then f(z) is an iteration map which acts only on z (starting with z=0). But when we ask what kinds of fixed points z we could get, those explicitly depend on which choice of c we make beforehand. So, it helps to think of the fixed points z(c) as functions of c.
@@PiercingSight I don't mind at all! I am a teacher, it's sort of my calling card, after all.
I think there are infinitely many fixed points z(c) which depend smoothly on the complex number c. Use the quadratic formula on the fixed point equation f(z)=z²+c=z (make sure to rewrite it in standard form first). Should be a square root function of c somewhere in the answer.
Then try to figure out which branch (plus or minus) to choose in order to make them attractive fixed points (the derivative f'(z) must be less than 1 in magnitude in order for the fixed point to be attractive).
I think the coolest thing here is that even topics like the Mandelbrot set which seem so abstracted from useful math, have connections to some of the most practical algorithms used in engineering like Newton's method. If this isn't a justification for simply doing math for the sake of doing math, idk what is.
You're still one of, if not the most, amazing math youtuber out here. Seriously dude, your video are high quality, and you are a very good teacher. Thanks for this two parter, and for the rest of your channel too.
@@JonathanTrevatt lol
And he's not condescending and anti-christianity like mathologer
Grant pushing out videos faster than most of us could understand.
Collabs help the channel grow. This channel should
do some with other S-Channels!
Anyway: And theres many Science-Channel who's Fan's dont know each other's channels.
So here comes my plan into account: I drop random comments about 'Hey, want
some recommendations about something? Anything?',
get called a bot sometimes, but who cares,
and sometimes people say 'Thanks, i take a look',
which makes my Day!
Its arguably a hobby and arguably not an Obsession.
XD
Yesterday I was literally trying to figure out why we use Stochastic Gradiend Descent instead of Newton-Raphson method in ML, after watching the video, and I ended up deep diving into it and stopped at complex analysis and holomorphic functions. It's so great to see a new video on the same topic today itself 😁
Just wanted to know if you have found the answer to your question about SGD yet? My guess would be that we can't use Newton's method instead of SGD because we're not finding a root of a function, we are finding a minima, and we don't even know if a root would exist. Also you would end up computing the derivative of a high dimensional function if you use Newton's method anyway. But still would like to hear about your research into this matter.
@@kumarkartikay and maybe with newton's method you can be trapped in those cycles, what would be terrible for an AI, but just a guess too
@@kumarkartikay I think that's the answer. SGD is simpler because it needs less setup, but otherwise they're quite similar in their approach to solving.
That also applies to the stability too
Newton and quasi-Newton methods are entirely applicable for general minimization and therefore ML. Their main drawback is needing second order information (the Hessian of the function or approximations thereof) which doesn't scale well: for n dimensions, O(n^2) storage and computation are needed. (Reducing this cost is a main feature of quasi-Newton methods.) Another drawback is having to know what even the second derivatives are, or otherwise having to numerically approximate them.
In any case, any proper implementation of an optimization procedure will have checks for instability and (usually) ways to recover so as to avoid the issues mentioned.
I am currenty doing my bachelors in mathematics and my thesis is on the topic of quasiconformal maps and their use in constructing Herman rings from Siegel disks.
That being said, I had to learn a lot of complex dynamics to even apreciate the topic of my thesis. And then I find these two videos from 3B1B in my recomendation feed. It was a surreal feeling.
This video summes up my past month and a half of self study and I am happy to say, that I understand the topic on a deeper level. I to fell victim to te confusion between Julia sets and filled Julia sets, and the fact that Grant had the same problems, makes me feel more confident in my future work.
So on the off chance Grant is reading this, thank you for all your work and the inspiration you instilled, not only in me, but in all up and coming mathematicians.
This might be one of the best channels on the entire UA-cam platform, basically the apex of what real, free, and beautifully taught knowledge is. A tear in my eye here.
This can't be happening! Two great uploads in the span of less than a week!
Well, the upload schedule is stated in percentages after all, so videos with short time between them are bound to happen given enough time.
Right? He's loving these fractals, as am I!
I'm just loving what's happening nowadays. For your new initiative on submitting quality math videos, my UA-cam recommendation is blessed!
As a kid - it was thinking about what the math I was doing looked like - that got me into enjoying visualization and graphics. I love this topic. Thank you.
Every student deserves a teacher like him.. M soo glad that we are living in an era where we can understand and share knowledge so easily.... Ooh and as for all your videos.... They are always a visual treat to watch!
This content is fundamentally beautiful and awe inspiring. There is something naturally beautiful about math in general, but videos like this border on a kind of spiritual awakening - for me at least. The hidden relationships between seemingly disparate mathematical functions and the patterns that emerge when taken to great lengths and infinity are something no person throughout the entirety of humanity has experienced until relatively very recently. I feel humbled and lucky that someone like yourself is providing the means and knowledge to see these beautiful aspects of reality.
What I find fascinating about the Mandelbrot set is how well embedded it is in the cross-section of math and art. So often when a mathematical function or concept gets embraced by artists, there's a very consistent inverse relation between how aesthetically interesting it is vs how mathematically interesting it is (ie the principle that cool visuals normally means unexciting math, and exciting math normally means there's nothing much to see). Yet the Mandelbrot set and its associated Julia sets manage to be endlessly interesting in *both* aesthetics and mathematics.
I really love the depth you are going here! Much appreciated for not just assuming people don't get it! You have a real gift here, making something like this still understandable is far from trivial!
You're incredibly talented to be both a great mathematician and a great communicator. It's likely you're single handedly inspiring young people to investigate mathematics. Good work.
This was my favorite topic in mathematics as a high school student. I didn’t really understand it until much later, but the patterns were so beautiful and captivating I would spend hours torturing my parents home computer with fractint. Thanks for making such a clear explanation and introduction to the subject.
Another one in just a few days! Great work, Grant.
Grant I have to say this. I remember some of these concepts from classes and I was sometimes unable to understand _why_ something worked the way it works. But the way you explain it, suddenly I remember those things and I go "ooooh that's why that thing worked that I once did in class". Without a doubt, you are the best educator I've seen.
Your animations are so amazing, every time I watch them, I just lost myself in the fascinating patterns. Numbers are really magical. The glory of nature lies in the numbers themselves.
Just finished coding a program poorly generating fractals after watching your last video, so glad to see this
That’s awesome! Generating them poorly is the first step to generating them optimally, never stop!
@@BobWidlefish thanks man! Ima work on that
OMG yes! Thank you for this upload schedule!
I think this might be my favourite video of yours by far, it does such a good job at tickling your curiosity and showing just how much there is to explore beyond what is presented. It doesn't feel like an explainer video like most of your other work (which i obviously still love in its own right), it feels so open ended in a sort of new way. I already know how much I like learning about maths and other things, but this way of presenting things felt really special to me in a way I hadn't really seen done to this extent before. Thank you, I hope to see more things like this in the future both from you and others!
It's obvious that a LOT of work went into this video, and I want to appreciate that. Thanks.
That was a lot to digest. I usually don't have to rewatch your videos, but I definitely have to go over this one a few times. Fascinating stuff.
I'm in awe how you use manim to make all these beautiful fractals and even do all the zooming.
ABSOLUTELY mind bending, and as a side note, you made it clear to me how the complex plane represents the simultaneous processes of rotation and expansion, in such a simple way that I looked back and realized it was there all along.
As always, WELL DONE.
This is the kind of math that gives me chills and I love it. Thanks for a great demonstration.
Method for solving Exercise 3 at 19:10:
Let r, t, y denote roots of polynomial. Then we can write it as (x-r)(x-t)(x-y). Then multiply it all out and take second derivative.
Do you know what the "B." in Beniot B. Mandelbrot stands for?
for Beniot B. Mandelbrot.
*harddrive noises intensify*
I LOVE your channel 3b1b.
Not only is the content ultra fascinating (I learn so much), your graphics are 2nd to none.
I don't think everyone puts enough admiration to your visualisations.
I have no idea how you do it, but the are absolutely INCREDIBLE. So much effort must have gone into creating the capability. In constant awe 🙏🏻
Captures the true essence of mathematical exploration. So, beautifully done. This is going to be my top video of the year for all of UA-cam.
"Gaston Julia is one of the greatest mathematicians of all time who had no nose"
*Sad Tycho Brahe noises*
Tycho Brahe noses? Oops, I misread. But he was for sure the greatest noseless astronomer ever.
I didn't know Tycho that didn't have a nose. Thanks for the useful fact.
@@capitaopacoca8454 He lost it in a sword duel. Replaced it with a brass one.
@@JBOboe720 I read it on wikipedia, thanks. Living in the past was certainly more interesting than we think.
@@JBOboe720 I suppose he'll have to have a second sword duel - this time against Julia, in order to determine the greatest noseless mathematician.
Grant, please do more dynamic systems, all the tech applied to vision and robotics. the demand on learning those is surging and will continue.
These videos are really mind-blowing, they're really expanding my understanding of what analysis can be achieved
This is one of the best maths videos I have ever seen (if not the best). You didn’t just say “hey here’s the Mandelbrot set, and here’s a pretty fractal!”, you actually went into the intricate detail and even provided exercises for the viewers to do! This is the pinnacle of mathematics education videos. I’ve seen how much better your videos have become in the past 3 years, and I can’t wait to see what’s to come for you.
My favourite part was when you were drawing the two complicated diagrams at 19:40, yet playing extremely calming music 😁
I get excited when he finally uploads. Reminds me of How I used to feel about a Vsauce video suddenly coming out.
Some years ago, I tried iterating different functions (don't remember exactly which ones) and coloring it like the Mandelbrot set. Most gave either noise or a non-fractal shape, but when there was a fractal, it often contained a distorted Mandelbrot set somewhere.
These videos are so humbling. It is impressive what mathematicians of the past have found and created and what the human brain is capable of. I am also impressed with the comprehension, editing and story telling by 3B1B. I always have to watch these videos 3 or 4 times to understand and appreciate it completely, and I have a relatively strong math background.
I wish I saw your video during my years learning holormophic functions. It helps so much to see the dynamic of the function in full display, which you can't reproduce with a pen and paper. You did an outstanding job, Thank you!
Great vid, there's an incredible animation on youtube called "Hidden Structures of the Mandelbrot set" which sweeps through both and reveals hidden 3D shapes.
ok so the REASON that Newton’s fractals must have boundaries touching all n regions for polynomial degree n is because the boundary points are, by their nature, a julia set. Julia sets, by THEIR nature, repel points across the entire space. Then, you necessarily approach all n roots if you have points across the entirety of your space.
ok cool
It’s really amazing how even if I can’t understand like 35% of the video because it’s too much for me to process, Grant can make us understand the essence, the intuition of these complex (pun intended) concepts
Especially because those roots are attractive, so you only need to get somewhat near them
@@AndresFirte I seriously wouldn't have gotten the pun if it wasn't mentioned specifically
This is actually a really cool way to sum it up, thank you
absolutely love this content.
been writing my own fractal renderer in c++ for about a month in my free time and this is yet another great example I've never seen anywhere else with high arbitrarity. definitely going to have to add this
I think this is easily one of the best Videos about the Mandelbrot set, eventhough it wasn't the main topic.
*About Montel's theorem:* You don't need the full strength. It's enough to know that the union of the images of a non-normal function family is dense.
So you don't need the universal covering (j-function). And for the Newton fractal you only need to know this property for rational functions for which Montel's theorem can be deduced easier (you don't need the Cauchy integral formula for holomorphic functions).
It's interesting that points on a Julia set seem to jump from one point to another point _on_ the Julia set perfectly. Sure, that's probably part of the definition, but it's really cool watching that happen. And, for why pointd within an arbitrarily small circle around a point on the Julia set explode to EVERYWHERE (except possibly two points?) Must have something to do with the fact that there is a subset of that circle of points that are in each color area.
So, what happens at the boundary of the boundary, you could ask. At one spot, we bounce perfectly from one chaotic point to the next, never approaching any attracting root, and at another spot arbitrarily close, we eventually settle on a root. In between those two points (or, to cover my ass, within a circle centered on the point in the Julia set, with a radius of the distance between our two points) is an infinite set of points which must hit every possibility of end result (of one iteration) between approaching one root and bouncing within the Julia set infinitely. Somewhere in there, every point must be hit. Except for two points you said? I am interested in knowing which up to two points are excluded from this property.
If we ignore the fact that I couldn't remember my identity and why I clicked on this in the first place, really nice video, I will watch it ten more times after I received my PhD in physics (considering I've just started studying for the undergraduate degree and it would be two weeks from the start of the first class tomorrow). In my eyes, the effort you put in these two recent videos is more worthy than all the money that is on earth right now!
These last two videos you've posted on the emergence of fractals are some of the best - if not the best - mathematical explanations I've ever seen.
And now I see how people spend entire careers on this stuff. In answering a couple question this immediately opens dozens of others, every one of which feels "natural" and not just like math for the sake of math.
But he doesn't answer it. He just says that the shape appears to be something more general that relates to parameter spaces of processes like this. Kindof a let down based on the title of the video.
Well, Montels theorem is just the kind of complex-function analogon of Bolzano-Weierstraß' theoream: For BOUNDED complex number sequences you get a converging sub-sequence. Montel now says: For "bounded" holomorphic function sequences you get a converging sub-sequence. Here "bounded" means that your holomorphic function at least does not hit two different complex numbers. As far as i know, you dont need the j-function for the proof, just some easy sub-sequence constructions and cantors diagonal sequence method for getting it to convergen.
Nice videos btw ^^
Links?
Admittedly in the source I was looking at, the j-function was not playing a primary role, just a convenient function to precompose with so that proving it for a specific two points can extend to the general statement. I like the analogy with Bolzano-Weierstrass, it's a good way to put it, but for me, at the moment it feels like the details are fiddly.
Also, for the sake of bringing it up in this video, it would have required a least a little discussion of normal families and uniform convergence, and if you don't want to assume people have taken analysis that can take a little time.
@@3blue1brown I sadly cant post a link to my handwritten lecture notes from university ^^
Maybe you could just mention the analogy without the mathy details. The "bounded-ness" with not hitting two points also sounds just a bit stronger than the outcome of picards great theorem, hitting any point except one. There could lie an analogy as well, allthough Im a bit less sure about that. Just my intuition..
The world is not only stranger than we know, it is stranger than we can ever know.
But that makes it no less beautiful, and no less worthy of striving to understand.
Thank you for helping me understand, and showing me this beauty.
Just like your video about the Fourrier transform... When I learnt about Newton's method at the university the professor talked about the Mandelbrot set in one of the lectures, we viewed fractional dimensions and the reason why Benoit Mandelbrot came with that shape. Unfortunately, the explanations were so confusing and misleading that it remained misunderstood and mysterious to almost everybody present at that moment. In order to get more into it, I even went as far as writing my own renderer in C back then consulting text books about it, drawing the iteration path, doing the derivative etc... To me it remained mysterious.
Here, you manage to explain the content of these 12 hours of lectures in roughly 1 hour, and now I understand is the link, FINALLY!... you're a mind blowing teacher.
"Holomorphic dynamics" might be more _fancy_ than "Where Newton meets Mandelbrot," but it's not more _fanciful._ The idea of an abstract meeting of minds across centuries is more fanciful than a field of analysis.
I was just rewatching the Monster group video, and the image of the Klein's j function seemed familiar, so I looked it up, and yes, the function mentioned at 25:56 is seemingly related to the monstrous moonshine, which leads me to wonder if there is a connection between these fractals and group theory...
Consulting wikipedia, the importance of the j function for the proof seems to come from the fact that the j function gives a homomorphic universal covering of the complex plane without two points(?). So probably not.
@@zairaner1489 Fair enough. I just watched that video, so that the same function appears there seemed like a weird coincidence, so I was just wondering.
I think this may be one of the most fascinating videos I've ever seen in my life.
What a pleasant surprise! I was satisfied until December for a new 3b1b video. Thank you!
_Don't worry Julia, of course we'll all keep seeing each other after high school!_
This might only be me but in my opinion, mathematics is actually pretty similar to art.
it's no coincidence that the only fields where true "child geniuses" get born are music and mathematics.
Edit: more precisely, I generally meant field where pattern recognition is the main requirement.
Art is similar to Maths*
Yes, because they both are human products.
Maybe not a product, optimization problems are more like the force driving natural selection more than the product of it.
"This might be only me" yeah sure buddy, you're the first person to make that comparison
Well depends on what "art" means in this context. If it's the philosophical art then mathematics is definitely a subset of it.
it's amazing the work you put in to teach about things that are so theoretical and amazing. i told my students that they must look into your lineal algebra series if they want to really understand the subject. and now here i am, finally really understanding Mandelbrot set. thank you so much.
I LOVE your choices of colours for this video. The blue to white to yellow to black gradient for the Julia and Mandelbrot sets was absolutely gorgeous
I’ll admit that was blatantly pattern matched off of some of my favorite renderings of the Mandelbrot set, which I also thought were especially nice looking.
I don't know exactly wether this is true, but I have a strong feeling that if you weaken the "Stuff goes everywhere" principle to "hits a dense subset of C" instead of the unnecessary strong "hits every point in C except two", it becomes exponentially easier to prove and easier to understand while still proving it to be as chaotic as you could possible need
Idk because the infinite n-cycles that are in the Fatou set are right next to the Julia set and they go will go nearly all over as well but without going actually everywhere.
Hello.
When using the Newton-Raphson algorithm to find a root of a function (root you do not know), how do you know exactly how close you are to the root ? How can you quantify ?
You run the algorithm for some iterations. Let's say there are 10 roots. You start at some initial point z = a + ib, and after some iterations, z = a' + ib' (some new point). You pick whichever root this new point is closest and color the initial point a + ib accordingly. The limit is when iterations = infinity - that's when you get the whole fractal.
Amazing video about maths that I didn't even knew existed and would've stayed that way if it were not for your videos. I absolutely love this channel.
This channel is totally amazing! I've learnt sooo much about calculus and topology! Thank you... and amazing video! Also, THE SECOND PART ACTUALLY CAME!!!
I have a question about the boundary of the image of Newton's method for a complex polynomial P(z). I noticed that the recursive method z_n+1 = z_n - P(z_n)/P'(z_n) produces an undefined value for z_n+1 where P'(z_n) = 0. This makes sense to me, because we should expect P'(z_n) = 0 to be true when z_n is a root of P(z), and thus there is no z_n+1 which exists "closer" to the root of our chosen z_0. Additionally, at local minimums or maximums of P(z), we should expect points with P'(z_n) = 0, such that any choice of P(z) with such local minima/maxima ought to necessitate the existence of some set of complex numbers u (for which P(u) != 0), where z_0 = u, and as n goes from 0 to infinity, P'(z_n) = 0.
My question is this: does it make sense to consider the "size" of this set u? By this, I mean, can we make statements about which function P_1(z) or P_2(z) has a greater boundary length?
If P(z) is a polynomial with degree d, then P'(z) is a polynomial of degree (d-1) so there will be (d-1) critical points (local minima/ maxima) where P'(u) = 0 for some u. It seems your question is about how many points will eventually land at one of these undefined points. We can work backwards to say that one such point u = w - P(w)/P'(w) for some other w which will land on u in one step. Solving this yields 0 = w*P'(w) - u*P'(w) - P(w) which is another polynomial meaning it will have some other number of solutions, say D solutions.
Repeating this backwards we can see that after n steps we will only have finitely many "undefined points." Taking this to infinity we will only have countably many undefined points, so the 'measure' of such points will be zero compared to the entire real line/ complex plane. Maybe you could ask how fast this set grows, but I think it will mostly depend on the degree of the original polynomial. Maybe you could also look at when P'(z) is not zero but is very, very small (this will be bad for the same reasons, numerically you will be left with an extremely large number which is far from the roots/ zeroes.)
rubbing my hands together like a greedy little mouse going hehehehe after realizing this was the follow up to the last video and it came so soon
3b1b and SoME1 is the reason why i was inspired to start my own channel. Keep up the great work.
You're a blessing to the math community.
Is there any relation between lagrange points and the black regions of attracting cycles?
My intuition for why the one-or-all colors on the boundary rule exists goes something like this:
When you've zoomed infinitely close to a boundary (as when following the the definition of a boundary; using "infinite" as a short-hand for a limit), the colors beside it, if any, must go somewhere. But, given that you're infinitely zoomed in, you can't really choose to which point of the infinitely-far roots to go towards, as they all look the same "infinity" from the zoomed in view. But, given that you're already going towards at least two (the definition of a boundary), you can only choose to go towards all of them.
Similar to how dividing numbers gives you only one result (6÷3 is only 2), or infinitely many (0÷0 can give you any result you want, depending on what "zeroes" you have).
Does this make any sense? I assume you'd have to go a lot more precise on the interaction between the infinite zoom and non-infinite end-points to have any actual working definition (besides this seemingly saying that all boundaries must be like this, not only ones for rational functions).
for your division by zero example it makes more sense to use 0÷0. 6÷0 can never be anything other than infinity, but 0÷0 could be any nunmber depending on the context.
@@Tumbolisu oh yep, of course 0÷0 is the example to use.. edited
@@Tumbolisu Division by zero is undefined, not infinity. It will take an infinite amount of time to compute division by zero, therefore the answer cannot be defined.
@@davidmcgill1000 If anyone ever says something equals infinity, the obvious and only way to interpret it is as a short-hand for a limit. My comment even explicitly says to assume implicit limits, and "6÷0 is infinity" is perfectly clear in meaning anyways
We are 3blue1brown viewers. We all know that 6÷0 is undefined. From the context of the original comment, it should be clear that we aren't talking about rigorous facts here, but intuition.
Brilliant, the connection between what seems to be unrelated topics in mathematics is astonishing 😲
Two part video which shows the process of thinking and discovering in mathematics. Thanks a lot of making such videos.
It would interesting to explore how this effects A.I behavior because it just so happens that you are fundamentally discussing A.I model training.
Loose ends being “What the $!?@ is going on here”
That quote will go down in history.
3b1b is the best UA-camr period, the quality of his content is unmatched.
I have a few questions, but first I just want to say how, yet again, I am seed by the beauty of deep math and the amazing skill of Grant's in demonstrating it.
Lol
Lol
Dear person that's reading this, we may not know each other but i wish you all the best in life.Stop blaming yourself, accept things and go forward. Your smile is precious😊. All the keys of happiness is in your hands, so open it up.❤❤💗💗
S H U S H . . . . .
I just watched spectacular "The Colours of Infinity" on Netflix. The breathtaking documentary on Mandelbrot set narrated by sir Arthur C. Clarke himself and featuring Stephen Hawking and Benua Mandelbrot. And then opened UA-cam to see this in recommendations. What a day, what a day!
Brings back good memories from the 1990's of playing with Fractint, before I knew what a complex number was - things like that make you want to learn more.
Damn, didn’t realize physics were homophobic. Smh my head
Just when I thought the Mandelbrot set couldn't get any more iconic, it turns out to be a universal aspect of recursive functions. Mind-blowing.
I've been playing around with Manim for the first time, and am struck by just how beautiful your presentation is, compared to what I have managed to do. Gorgeous work, with awesome mathematics and explanations backing it all!
Finally you involved fractals. I've been waiting for you to cover this topic for years!!!