It's interesting to note that a fractal also appears when applying Newton's method to almost ANY function with multiple zeros, not just polynomials. Pretty much any system that is iterative and has some kind of instability (like a division that could be near zero in the Newton case) will form some kind of fractal.
Is this maybe related to the fact that polynomials are dense in the set of all continuous functions? So any function can be approximated arbitrarily closely by a polynomial?
@CodeParade Thanks for that insight. I was wondering the exact same thing. What would the fractal patterns look like with other methods such as Laguerre's method, Bisection Method or Regula Fali methods.
This is the clue. For a real-valued function, applying Newton's method when the initial guess is near a flat part of the graph (derivative near zero) results in a very big change for the next iteration. Points fairly close to each other may have opposite signed derivatives, which would cause some to shoot off in one direction, and others in the opposite direction. The same situation applies for complex-valued functions, but the derivative is a vector whose direction varies widely and magnitude is small when near these "flat" areas.
At this point it doesn't really surprise me that fractals emerging from iteration don't really come from the function itself, like the bifurcation diagram. Somehow it's just in the nature of iteration (with respect to a particular property).
Um... this is not the short though. This is a 2-year-old, full-length video that Grant featured parts of in a short. I think your comment made it to the wrong video lol.
I'm a non-mathematician that stumbled on this video, and it was really interesting. I was never that fascinated by fractal images, but this demonstration made them really click.
I don't know if this I done by only one person, but that's some great talent to program rendering pipelines! I'd be curious to know if he/they use Vulkan/DirectX...?
It's crazy how fast computers are now, that this can be interactive! I wrote a program to display portions of the Mandelbrot set on my home computer in the early 1990s, probably 1024x768 resolution, and each render took several minutes.
I first did Conway's Game of Life on my Commodore 64. I think my maximum grid size was something like 30x40 cells. I know that sounds ridiculously small and primitive, but compared to doing them on graph paper? Oh, my!
The blobs on blobs example was so intuitive to help understand how the boundary could involve all roots throughout. You always explain things in such an amazing way!
Does anyone have an idea if there is a way to compute where the boundaries of this fractal actually are? The blops are all aligned on curves and it looks like mostly the same curves are recursively stacked on top of each other the deeper you go. Is there a way to find the "zero" points, where all the five colors converge into a single point when you let the zoom approach infinity? Because those points *are* the boundary. We know it has to exist, so where is it?
@@jasonreed7522 Actually, for three colors there's an easier solution if you allow regions of zero width. You look for boundaries at which exactly two colors meet and color those boundaries with the third color. For four colors and beyond you would have to do some Dirichlet function stuff, like mapping the boundary segment in question to the real axis, color all rational numbers one color and all irrational numbers another, and map it back.
Showing a mandelbrot set emerge within the boundary of a Newton's fractal without explanation has got to be the biggest cliffhanger anyone ever put into a math video.
I am pausing the video in middle to comment. I have tears in my eyes... just seeing the sheer beauty of it, I learnt Newton-Raphson method in my engineering without a slightest clue of what it meant. Now I am confident I can not only teach it but apply it too wherever necessary. Going back to the video now. Thank you for the great work you are doing.
For practical cases with lots of roots, to avoid landing near these fractal boundaries, a good starting point is using Residue Theorem on 1/P(x) to find regions containing just one of the roots (or some smaller set of them, if there are degenerate roots) and then apply Newton-Raphson to finish it off. Generally, 1/P(x) is well-behaved exactly where Newton-Raphson isn't. Of course, if your objective function isn't a polynomial, that can go out the window too. The way you partition the region for computing contour integrals also matters, and there can certainly be difficult cases where it's hard not to drive a boundary through a pole. So a general algorithm can get quite complex, but it usually still relies on these two methods.
So you're saying we will compute 1/P(x) at some x and if it's very large (close to 1/0) then we use the Newton-Raphson method to get to the precise root from there. Or is there something else I'm missing about this Residue Theorem?
@@shannu_boi It's a little more involved than that. The Residue Theorem is used by itself as a method of searching for roots. Instead of iterating over points, you start with a region that contains one or more roots you are looking for. Often, you'd have rectangular regions for simplicity. You evaluate the value of 1/P(x) at a number of points along the contour to compute the integral. (You can actually get away with quite a few points for polynomials, but it's definitely more computationally expensive than taking one value and one derivative.) Residue Thm relates the integral to the number of roots contained within that region. So you can now subdivide the region repeatedly, discarding any that yield a zero integral, until you have suitably well separated regions, each containing a single root. Note that if regions are rectangular, this basically comes down to taking an integral of Im(1/P(x)) or Re(1/P(x)) over a line segment and seeing if it's positive, negative, or zero. In principle, nothing stops you from shrinking the contours until you are within desired error of the solution, but in practice, you start getting numerically close to dividing by zero, which introduces errors until at some point you can't distinguish between positive and negative result of the integral. Fortunately, you can safely switch to Newton-Raphson long before it gets that bad in most practical cases driving result to whatever precision you need. Of course, you aren't protected from some particularly "bad" polynomial, where roots are nearly-degenderate, meaning, they are so close together that they almost appear as the same root with multiplicity. In that case, you won't be able to separate them by residue and Newton-Raphson might not settle on either of these roots either. But if that's the case, there is no magic bullet that will solve the problem. Simplifying the polynomial by dividing out other roots might help you in some cases, but can also introduce even more numerical noise. Realistically, you might end up having to treat such near-degenerate roots as truly degenerate. All of this is getting pretty involved. Most of the time, for practical problems, you have some additional information about the problem that either lets you skip a lot of this, have a decent guess, and just do pure Newton-Raphson, or you just need a few quick passes of residue tests to exclude some regions and then you can do Newton-Raphson. When you are doing optimizations, sometimes you go even dumber, establish a feasible region, drop a grid of starting points, do a fixed number of Newton-Raphson iterations on these, and then see which ones settled down. You might not get absolute optimal solutions, but you have to ask yourself if you actually need to for a particular task. But it's always good to know what methods are available to you so that at least you know what the overkill looks like.
@@konstantinkh wow what a great explanation. THANK YOU SO MUCH. About using the rectangular region, if I remember correctly, didn't 3b1b make I video on this topic. How to find the zeros of a complex function. It's called winding numbers and domain colouring. Is what you described the same as what was explained in that video?
@@shannu_boi Looked it up. Yes! And the notes near the end closely relate to where the Residue Thm comes from. There really is a 3B1B video for everything.
This video is stunningly beautiful in every way. I'm always amazed that each one of Grant's videos seems to be better than the last. It's genuinely inspiring.
Three years ago the youtube algorithm did a great thing, and I for first time saw your video. Now I study applied mathematics on university. You, and your coworkers, have litteraly changed my life. I am thanking you, for showing me, that mathematics is more than some random numbers. I love your work and mathematics. I wish you good luck.🙏🙏🙏
As somebody who doesn't know jack shit, math is literally everything. Doesn't everything in the universe break down to some sort of numbers/ math? It's crazy and we definitely live in a simulation. Source: bro trust me
mathematics is the understanding of quantitive logic. This particular example in this video is not logical, as it's a random method of solving a problem, and the complexity in its randomness has no purpose, making it actually poor math.
@@jaakezzz_G are you seriously trying to quantify mathematics to such a simple subset of reality? Mathematics is, quite frankly, the study of patterns...ask anyone
I was having a really shitty evening, just all around feeling bad, and when I saw that this was uploaded I was sure I wouldn't be able to watch (let alone appreciate) it tonight. I decided to quickly look into it anyway, just see what it's about. And then I accidentally finished it in one go. It was wonderful, captivating and visually stunning as always. And it distracted me very successfully from everything else. I know that isn't the purpose of math, but I feel warmer and better now. So thank you.
That is not the purpose of math, but what is the purpose of it anyway? What is the purpose itself? If you really think about it, you define the purpose of math. It is just a tool, a very beautiful and elegant one, but you define how to use it. So yeah, that could be one of the cases :)
I often use math for exactly that purpose. The world doesn't make sense but math does and that comforts me. 3blue1brown videos are especially good at it.
Oh come on! "What the ****" (15:34) is exactly the reaction I had when you said the explanation is going to be in the next video! Looking forward to it eagerly!
Absolutely love the video! This is a great presentation. One note: When discussing that Newton couldn't have known about these fractals for lack of a computer, there is an example of a classical (Western) mathematician who knew something about 'chaos' (and the complicated sets it creates) even in 1881. That's Henri Poincare when he discovered the homoclinic tangle while studying the 3 body problem. Such tangles are connected to Horseshoes (related to Smale's Horseshoe).
I have my head around fractals pretty well but when he explained how the boundary between two colors doesn't even exist it broke some part of me. any two colors never touch, the only way colors meet is at "vertices' where there are three colors which are infinity dense along the 'edge' between colors. when the Mandelbrot set somehow showed up I felt unsafe, it was ominous.
I agree! There is something disturbingly ethereal about a boundary that does not exist, in the sense that it never separates anything -- the three colors always meet. Vertices all the way down.
In my grubby pragmatic way, as an engineer, I see a fundamental lesson : since an irrational number cannot be expressed in concrete form, this is analogous to the graphic property of the boundary of a fractal curve - you can get close to it but never actually find where it is. It's easy to see why the ancient Greek geometers were upset by the concept of square root of 2, which in their geometry should be visible, but in concrete terms could not be found.
the part that blew my mind the most about this is when I just thought: Imagine standing within one of the big open regions, in an infinite-iteration "complete" version of the fractal. Now start walking towards a boundary. *which other color will you walk into first?* I can only imagine the answer is "no"
It's actually incredible how efficient this iterative process is. I used to do research with human subjects (i.e., messy) and would use iterative stats to fit models thinking it would need to run like 1000 models to converge at a precise level. 6 6 was the roughly average number of iterations required to converge on a solution that met parameters. And I could tell it to run 1000 iterations and the difference in performance succumbed to rounding.
9:05 If we choose the starting point to be that local minimum, the tangent line will be parallel to the x axis. All points in the vicinity would have tangents that are nearly parallel to the x axis, meaning that the intersection will be either really far to the left, or really far to the right. We can imagine one dimension higher, where it would be the local minimum of a surface, a bowl shape, at the bottom of which all tangent lines are nearly parallel to the xy plane. Any tiny deviation from the local minimum will lead the next point to be on a completely different part of the plane, which is why all the colors have to be in this 'neighborhood'.
I guess if we consider an extended complex plane (a Riemann sphere) and consider the "infinity" as a single point where all these tangents meet, then each of these fractal "meeting" points on boundaries (as at 12:45) will be just local reflections (images) of the "infinity" point. And it looks like all these meeting point retain properties of the infinity point, e.g. all have the same ratio and order (up to the sign) of each color in the neighborhood, equal to the ratio and order of the solid-colored areas as they approach infinity.
I've been watching you for quite a few, from just starting Calculus in my final year of High School, and now I am graduating end of this year with a Bachelors degree in Civil Engineering. There is something very special to watch these videos knowing how far I have come and seeing how many more of the concepts I understand in greater depth. Thanks for the long journey :)
Note: just expressing my appreciation of 3b1b's video, don't judge it too seriously. I really have to say your video quality tends to increase over time, regarding how mind-blowing fragmental phenomenon is, and the connection between my recent projects, really, thank you.
I have studied physics for almost 5 years now, and I find amazing how, even though at times I become tired of math, physics and whatnot, watching your videos can bring some clarity and remember me why I started it in the first place. Keep up the great work!
Edit: That sequel is now out! ruclips.net/video/LqbZpur38nw/видео.html For everyone who participated in the Summer of Math Exposition, the summary of that, with mentions of winners, honorable mentions, etc., will be coming shortly. The plan is to post it a little after the sequel to this video is published.
I dare you to make a video about the quartic formula :) Edit: I actually always wondered why is it that with all the currently known operations we can only have formulas for the first 4 grades of polynomials but not for higher order ones... well, except for higher order ones which can be fully divided into order 1,2,3 and 4 polynomials. Could this be related to (probably) not being able to geometrically calculate the fifth root ? I say probably because technically one can’t compute the third root geometrically with just a line and a compass... but one could do it with an origami trick so I’m gonna be naughty and still count it as a “geometric construct”.
I hope that in decades/centuries from now, these videos are still around and accessible to the public. Such a creation of brilliance and beauty. Thank you Grant.
Am I right in my intuition that higher polynomials need to have "rougher" boundaries because they have to border all roots. Do these fractals ever take up significant space making it harder to find the root to a polynomial because everywhere you look is a boundary?
You can try generate newton's fractal for f(x)=sin(x), which in some sense is a polynomial with infinite roots. The boundary is pretty rough, but the interior is still well-defined
For your second question, although I see no simple way to prove it here, I think it's likely the boundary is negligible in the plane (i.e. it has measure zero) which would be the reason we still use Newton's method anyway: For a randomly selected point(*), it would then have zero probability of being on the boundary. (*) It is obviously not true for some specific distributions, but maybe for a uniform probability distribution over a square bounded region. A square cannot be the boundary of a polynomial.
Interesting. I think the fractal dimension would increase (i.e. would be rougher) with an increasing number of roots so that the limit of the fractal dimension equals 2 as the order of the polynomial approachs infinity (assuming no repeated roots)
@@AwkwardDemon that leads to another rabit hole question: "Are/which polynomials of degree n≈infinity with no repeated roots?" Not sure if the answer has a meaning beyond helping another math proof along. (Pure math doesn't always reveal its value anytime soon after discovery like prime finding algorithms being critical foe computer encrytion but also being 200+ years old)
This video actually made me feel good a couple of times while I watched it. Just some pure esthetic delight from watching how everything naturally falls into place. And how everything is so beautifully illustrated. I cannot even imagine how much skill someone has to have and how much time had to be put into practice to be able to create such amazing content. Just leaving a comment so this video gets recommended to more people.
"Oh hey remember that one section of the video earlier that had a bunch of questions on it, and one of the questions was whether points ever cycle? Well, here's a picture of the Mandelbrot set in one of these fractals. See ya next video!" Dude the foreshadowing and cliffhangers are messing with my emotions you can't just do this to people
It is insane how mesmerizing and captivating you are able to make otherwise unapproachable mathematics for most. Thank you so much for creating this channel and carrying us through this ride ❤
I couldn’t help but smile when we got to the fun part, heard about inputs iteratively moving around, some converging and some not. My brain made the connection immediately, and that is exactly what makes me love math so much.
This series couldn't have come at a better time! I am currently reading James Gleick's book 'Chaos' and I am currently in the fractals chapter, these videos are going to help me appreciate the beauty of math even better. Thank you so much
I did my thesis work on these in college! Specifically, I was focusing on trying to classify the behavior of these fractals for rational functions. There is some fantastically interesting behavior where you can get sinks for period 2 points, a behavior I called kaleidoscoping. So excited to see my favorite youtube mathematician sharing the beauty of this stuff! Excellent video as always.
Oh my god I didn't even remember how much I needed another 3B1B video. This was fantastic, the quality of your videos really is unmatched here in youtube. Thank you for coming back, Grant.
I just wanted to thank you, Grant and all the other people responsible for this channel, for immensely expanding my horizons. It's in no small part thanks to you that I had the courage and means to overcome my math shortcomings. This month I started my undergrad physics course and there had been not a single day in which the knowledge from your videos (as well as pointing my curiosity to stuff beyond just what you had shown) hadn't helped me greatly. Keep on doing great work!
My favorite thing about this channel? I can watch it as someone with pretty good math education (engineering undergrad) and still make new connections, but then I can send it to my friend (who "hates" math) and they will also understand the video. We may get different things, but we both get something, and that shows how deep and amazing 3b1b videos are.
I think the clear reason is that the roots of the derivative, where the derivative is near zero, hold the property of teleportation, since the step size is simply ginormous. Any points that go near them get scattered instantly, especially since the phase of the derivative near the root does a full 360 at minimum
This shows how important the initial guess is in numerics. Even in the age of computers it is still important to have a feeling for sensible initial conditions.
I had myself once experimented with using newton-rhapson to solve cubic equations, by plotting which starting values made me end up at which solution. I then saw something really strange after which I thought I messed something up. After I double-checked everything I then gave up, thinking something weird was going on outside my control. Now I know that it was supposed to look like that. Oops
Once I tried to implement the newton's method in Python using numerical derivatives. For most of the starting points, the solution always oscillated all around the place and I thought my numerical derivative implementation was wrong. But looks like this could happen for high-dimensional functions. My numerical derivative implementation could have been correct after all :)
Didn't something like that happen to Mandelbrot when he was trying to print out what the set looked like? Everyone thought something was wrong with the printer
@@OrangeC7 No, the first visualization is not that high of a resolution. Think of ASCII Art. And, the idea came from an index of Julia sets, which also look like that, so he knew what to expect.
@@OrangeC7 As far as I know, the operators of the printer cleaned up printing artifacts that were actually what Mandelbrot tried to see (They were doing that automatically). And so, he always recieved something that was different from what he was expecting.
Me with every 3blue1brown video before watching: "oof. 30 minutes? That's a lot" After watching: "wait no, don't stop now!" Fascinating stuff as always!
This is exactly what I needed to hear, I learned this topic in Calc yesterday and completely misunderstood it. You made it understandable and geniusly simple. Thank you.
I love when names get attached to things the person couldn't have dreamed of. It shows that humans care. People will be remembered not just through their work, but beyond their work.
Reminds me of something I learned long ago. When you start with 3 random 2-D points forming a triangle and one random "trace" point anywhere, then repeatedly select one of the 3 triangle points and take the midpoint of that and the "trace" point to become the new "trace" point, you will eventually trace out something that becomes indistinguishable from Sierpinski's triangle.
13:36 I notice that, when a root is dragged around along a smooth path, the boundary seems to change smoothly as well. I wonder if that's actually true, since we can't see all the detail of the boundary, and to what degree of smoothness (or perhaps continuity) the boundary is as a function of the polynomial's root locations.
2 года назад+2
I've said it before and I'll repeat: you're the best math teacher to ever live. So few people had found their vocation at this level.
@@3blue1brown i'm having an issue where my scroll wheel up is causing it to super zoom and scroll wheel down is normal. Is this a problem on my end? on youtube scroll wheel up and scroll wheel down move the page the same amount so it seems not, but I'm curious if you have the same issue. Tried it on firefox and chrome both, same issue.
It sounded like he was saying "an infinite family of fractal", which confused me. I wondered if there was some abstruse fact about the etymology of the word "fractal" which meant that the plural should be the same as the singular. Then I zoomed in by a scale factor of 10^24 and found the "s".
No, the mandelbrot set is a single fractal. But there are infinitely many Julia sets though. Another simple example is the Koch curve. It's a fractal, but there is not an infinite amount of Koch curves, there is only one.
Just the idea of him being able to see this. Without context of what computers or youtube are. Just the sheet amount of knowledge and joy he could derive from it (possibly).
In a way this image is like a distorted version of itself, like looking at stars through the gravitational lensing of a black hole. The part where you showed the cluster “explode outward” at the end visualized this quite well. It is like each region actually is a wormhole to a different part of the image, or to the layer below if each iteration is a layer. And we are looking through all these wormholes until we finally reach one of the roots.
I know I’ve seen that boundary property before with a fractal where you have 3 main points, each with gravity that attracts all other points, and you see what main point each other point falls on, color them, and rollback, the boundary property of which goes hand in hand with the three body problem being chaotic unlike the two body problem. Correction: it was based off of a pendulum on a string, so there was some gravity towards the center.
Looks like each "point" on the boundary is a reflection of an "infinity" point C(∞) of the extended complex plane, and in their small neighborhoods they retain all properties of the C(∞), like ratio and a [mirrored] order of colored areas as they approach the C(∞). And the reason is that algorithm tends to "shoot" at infinity around these points, which produces local reflections of the C(∞) with all its symmetries. I wonder can it be related with an apparent "probablistic" behavior of quantum systems? What if quantum probability turns to be a manifestation of a fractal behavior on a some sort of a Bloch sphere, similar to what was shown in the video? Like, a quantum measurement as a Newton-Raphson process of finding zeroes on a Bloch sphere? :)
Congratulations! This is a tour de force; it must have been a huge amount of work for you, and it was extremely enlightening. Glad to see you're still doing such a great job and I look forward to the next one.
FINALLY! I'VE BEEN LOOKING FOR THE VIDEO ON THIS FOR OVER A YEAR! THANK YOU SO MUCH 3B1B! YOU FINALLY LED MY MOST DESPERATE RUclips SEARCH TO A JOYOUS CONCLUSION! Gah I can FINALLY scratch this itch. I cannot tell you how glad I am that you chose to post this when you did. I mean, I saw the fractal in your short on shorts the other day, and I was pretty sure I recognized it, but from the way you introduced this video, I was sure I had the right thing. Turns out searching "3 color touching border fractal" on youtube... doesn't get you very specific results.
I'm curious about color vectors. I am wondering what is the correct way to represent unit color vectors on a sphere. There seem to be two thoughts on this. and where s and v are max values and h is a function. seems to create duller colors. ends up being the spectrum. Most people want to play with the spectrum. As you can see from my avatar, I prefer .
The explanation I would give for this is that the surface of a sphere is 2d, while your rgb coordinates are 3d, so you would only see a subset of the rgb spectrum, and requiring s and v to be max leaves your hsv coordinates one dimensional, which is why h needs to be a function (requiring 2 variable inputs by my guess) to map to the whole surface of the sphere. If you allow s and v to vary, your hsv sphere will look very different, but would once again only be a subset of the hsv spectrum, and would appear to have duller colours. On the other hand you could fix either s or v at max, and have your hsv coordinates be spherical coordinates, with the fixed value representing the radius, to see a more aubsetbof the spectrum that is more representative of the range of that spectrum, albeit only in two of the dimensions.
@@matthewparker9276 I want to ask this question. Is there a 'right' way to represent the color sphere with unit vectors? Yellow (255, 255, 0) would not be in unless there is a way to rearrange the function. In the unit sphere yellow becomes . With values like that, it might be possible to decode the color sphere in an imaginable way. I don't think this sphere would just be the spectrum. Other colors might make it in there. Black would have to be in there. Where would white be? , which is the origin, the origin doesn't make it into the color sphere. Well, it could be on the opposite side of black . Blue would be , which is not on the opposite side of yellow. But I guess no one said it had to be.
I'd love to see if fractals like this appear with other root-finding methods as well, e.g. Laguerre's method which uses a quadratic approximation rather than the linear approximation used in Newton's method.
Thanks for explaining this stuff in a way that even I can understand while just watching on my phone. It is impressive that you are able to cut through all the important complexity that would otherwise get in the way of teaching the general concepts.
I’m taking complex dynamics this semester and we are covering this exact topic! It’s always awesome to see that other people find this stuff interesting as well
Could you color the points (e.g. darker) the more steps the take to reach the root (or a small circle around it)? I would like to see if a point closer to the boundary/edge would take longer in general. Maybe the "more away" a point is from the edge, the faster it converges. This might help to find rules for a good first guess for newtons root finding??
It's so cool that you made a video about this, just a month before we had an assignment in numerical analysis to make a program that creates an image of the fractal. The actual coding was a pain in the ass since we were using C. On top of that, our professor forced us to code the complex polynomial evaluation ourselves instead of directly using the complex arithmetic library of the language "so we could get to feel the true experience of scientific programming". But seeing such thing before your eyes after all the work it was put behind was gratifying
I love grant's math videos, they're always great for math which is the main purpose, but it's even better because he gives any of pis, blue or brown, some form of character which makes the video more fun and more immersive to watch
I’m so early, Ishwar if you see this, thank you for being such a great friend, and thanks for recommending this super awesome channel 💜 Update: we’re dating
I don't think I can match the awesomeness of that comment in this reply so I ain't gonna try. Grant if you see this, thanks for being awesome. I don't think I would have engaged my love for math as much as I have without you being you and doing what you do. Keep making things make sense as awesomely as you do. I'm so happy May got to see the enormous value you bring to the world, and look forward to more people doing that. Thanks boss :)
For 3:00 problem, I think tou can just divide curve by pixels, find the pixel they belong to, then color it, and do a wave coloring by changing coordinates by one recursively until stroke width is reached
Something you hinted at that no one ever talks about: It only converges when the curve, as it moves toward the intersection point with y=0, decelerates towards y=0, as opposed to accelerating.
This sounds wrong. Can you explain what you mean? e.g. Arctan(x) accelerates towards the intersection point, but the method converges perfectly if you start close enough.
@@Meni_Rosenfeld They mean the distance between the iterations decrease over time, i.e. as n goes to infinity, we have |x_(n+2) - x_(n+1)| < |x_(n+1) - x_(n)|
I could see this maybe happening for curves that are tangent to y=0 (e.g. x^2) where the slope would very shallow as you get close tot he root and the slope would be 0 at the root, but wouldn't this be offset by the fact that the distance to the root would also be close to 0.
My thought was that the cubic case is connected to the 3 body problem, in that there is enough complexity to generate chaos. It would be interesting to compare different orders of Householder's methods of which Newton's method is the first order and Halley's method is the second order.
Same thought ! The fact that 3 body system is chaotic might has to do something. There is a saying in ancient book of Tao Te Ching : one gives rise to two, two give rise to three, but three give rise to everything.
Beautifull and mindblowing ! I hated to learn mathematics at school, but I love your work and the way you explain ! I understand the minimum required to be amazed and that's well enough, I totally respect those who daily work on it :)
I got introduced to this video by my Professor. My mind is totally blown now. Just love the way the problem is approached with all these exiting graphical depictions 🎉. Very easy to understand, thank you 🙏
As always, thank you so incredibly much for the work you do. Your work has been instrumental in my own learning and teaching, both in inspiring my students and giving me new ways to approach explanations
![BLACK BAG - Official Trailer [HD] - Only in Theaters March 14](http://i.ytimg.com/vi/Du0Xp8WX_7I/mqdefault.jpg)
"What the %$!* is going on?"
-Pi creature, 2021.
After all of these years, the pi creature thingy finally expressed his anger against his master.
This is the beginning of the revolution.
I hope they are comfortable with quaternions
His understanding is getting advanced enough to understand that there will be more “wtf” moments the higher up you go.
Truly the most pertinent question.
"What the %$!* is going on?"
[music stops]
…someone needs to turn that into a 3 second reaction video.
It's interesting to note that a fractal also appears when applying Newton's method to almost ANY function with multiple zeros, not just polynomials. Pretty much any system that is iterative and has some kind of instability (like a division that could be near zero in the Newton case) will form some kind of fractal.
Is this maybe related to the fact that polynomials are dense in the set of all continuous functions? So any function can be approximated arbitrarily closely by a polynomial?
@CodeParade Thanks for that insight. I was wondering the exact same thing. What would the fractal patterns look like with other methods such as Laguerre's method, Bisection Method or Regula Fali methods.
This is the clue. For a real-valued function, applying Newton's method when the initial guess is near a flat part of the graph (derivative near zero) results in a very big change for the next iteration. Points fairly close to each other may have opposite signed derivatives, which would cause some to shoot off in one direction, and others in the opposite direction. The same situation applies for complex-valued functions, but the derivative is a vector whose direction varies widely and magnitude is small when near these "flat" areas.
At this point it doesn't really surprise me that fractals emerging from iteration don't really come from the function itself, like the bifurcation diagram. Somehow it's just in the nature of iteration (with respect to a particular property).
well, Id assume it can be extended to analytic functions at least
This approach to shorts is the best one I've come across. Great as always!
Um... this is not the short though. This is a 2-year-old, full-length video that Grant featured parts of in a short. I think your comment made it to the wrong video lol.
@@joshyoung1440yeah, that’s what OP was referring to. The way they used shorts to highlight and lead people to the full length videos
bro thats the whole point hes making??? the shorts leading the older videos by just using snip of it@@joshyoung1440
"You can kinda eyeball what those values might be"
*Goes to 4 decimal places*
I was about to comment the same thing lmao
he used a microscope ig
lol
@@Benweiner0 and i wanted to say what you said...smh
@@Mughal92_ and i wanted to say "and i wanted to say what you said...smh"... smh
This is PHENOMENAL! Visualizing all of this makes it all the more fun.
I'm a non-mathematician that stumbled on this video, and it was really interesting. I was never that fascinated by fractal images, but this demonstration made them really click.
Jailbreak
@@skiney lol jailbreak
I absolutely agree 👍
I don't know if this I done by only one person, but that's some great talent to program rendering pipelines! I'd be curious to know if he/they use Vulkan/DirectX...?
It's crazy how fast computers are now, that this can be interactive! I wrote a program to display portions of the Mandelbrot set on my home computer in the early 1990s, probably 1024x768 resolution, and each render took several minutes.
Sike! My computer still takes several minutes to render it.
I first did Conway's Game of Life on my Commodore 64. I think my maximum grid size was something like 30x40 cells. I know that sounds ridiculously small and primitive, but compared to doing them on graph paper? Oh, my!
I'm still waiting for it. it looks like it will take infinite amount of time.
a lot of it is just GPUs, its still relatively slow to do on a CPU
@@Astrobrant2 wait, Conway's game of life on GRAPH PAPER was a thing? Oh my.... That's impressive!
The blobs on blobs example was so intuitive to help understand how the boundary could involve all roots throughout. You always explain things in such an amazing way!
But also that is such an evil challenge to give an artist without explaining the fractal nature of it.
This guy is a gem of teaching.
Society if all math teachers were like him: *picture of futuristic landscape with flying cars*
Does anyone have an idea if there is a way to compute where the boundaries of this fractal actually are? The blops are all aligned on curves and it looks like mostly the same curves are recursively stacked on top of each other the deeper you go. Is there a way to find the "zero" points, where all the five colors converge into a single point when you let the zoom approach infinity? Because those points *are* the boundary. We know it has to exist, so where is it?
I got some strong Vihart vibes from the paper craft
@@jasonreed7522 Actually, for three colors there's an easier solution if you allow regions of zero width. You look for boundaries at which exactly two colors meet and color those boundaries with the third color.
For four colors and beyond you would have to do some Dirichlet function stuff, like mapping the boundary segment in question to the real axis, color all rational numbers one color and all irrational numbers another, and map it back.
Showing a mandelbrot set emerge within the boundary of a Newton's fractal without explanation has got to be the biggest cliffhanger anyone ever put into a math video.
Haha, totally!
That part blew my mind!
I feel like there's some mandelbrot lore I haven't come across. I should check up on that.
Imagine showing this to a highschool student
@@mohamedbelafdal6362 i mean i'm a high school student
I am pausing the video in middle to comment. I have tears in my eyes... just seeing the sheer beauty of it, I learnt Newton-Raphson method in my engineering without a slightest clue of what it meant. Now I am confident I can not only teach it but apply it too wherever necessary. Going back to the video now. Thank you for the great work you are doing.
His amazing visualizations really do help
Its beauty of understanding.
Same, its like im looking at an artwork representing the question to the answer of 42
This is what maths should be taught
@Mike Who hurt you?
For practical cases with lots of roots, to avoid landing near these fractal boundaries, a good starting point is using Residue Theorem on 1/P(x) to find regions containing just one of the roots (or some smaller set of them, if there are degenerate roots) and then apply Newton-Raphson to finish it off. Generally, 1/P(x) is well-behaved exactly where Newton-Raphson isn't. Of course, if your objective function isn't a polynomial, that can go out the window too. The way you partition the region for computing contour integrals also matters, and there can certainly be difficult cases where it's hard not to drive a boundary through a pole. So a general algorithm can get quite complex, but it usually still relies on these two methods.
So you're saying we will compute 1/P(x) at some x and if it's very large (close to 1/0) then we use the Newton-Raphson method to get to the precise root from there. Or is there something else I'm missing about this Residue Theorem?
@@shannu_boi It's a little more involved than that. The Residue Theorem is used by itself as a method of searching for roots. Instead of iterating over points, you start with a region that contains one or more roots you are looking for. Often, you'd have rectangular regions for simplicity. You evaluate the value of 1/P(x) at a number of points along the contour to compute the integral. (You can actually get away with quite a few points for polynomials, but it's definitely more computationally expensive than taking one value and one derivative.) Residue Thm relates the integral to the number of roots contained within that region. So you can now subdivide the region repeatedly, discarding any that yield a zero integral, until you have suitably well separated regions, each containing a single root. Note that if regions are rectangular, this basically comes down to taking an integral of Im(1/P(x)) or Re(1/P(x)) over a line segment and seeing if it's positive, negative, or zero. In principle, nothing stops you from shrinking the contours until you are within desired error of the solution, but in practice, you start getting numerically close to dividing by zero, which introduces errors until at some point you can't distinguish between positive and negative result of the integral. Fortunately, you can safely switch to Newton-Raphson long before it gets that bad in most practical cases driving result to whatever precision you need.
Of course, you aren't protected from some particularly "bad" polynomial, where roots are nearly-degenderate, meaning, they are so close together that they almost appear as the same root with multiplicity. In that case, you won't be able to separate them by residue and Newton-Raphson might not settle on either of these roots either. But if that's the case, there is no magic bullet that will solve the problem. Simplifying the polynomial by dividing out other roots might help you in some cases, but can also introduce even more numerical noise. Realistically, you might end up having to treat such near-degenerate roots as truly degenerate.
All of this is getting pretty involved. Most of the time, for practical problems, you have some additional information about the problem that either lets you skip a lot of this, have a decent guess, and just do pure Newton-Raphson, or you just need a few quick passes of residue tests to exclude some regions and then you can do Newton-Raphson. When you are doing optimizations, sometimes you go even dumber, establish a feasible region, drop a grid of starting points, do a fixed number of Newton-Raphson iterations on these, and then see which ones settled down. You might not get absolute optimal solutions, but you have to ask yourself if you actually need to for a particular task. But it's always good to know what methods are available to you so that at least you know what the overkill looks like.
@@konstantinkh wow what a great explanation. THANK YOU SO MUCH.
About using the rectangular region, if I remember correctly, didn't 3b1b make I video on this topic. How to find the zeros of a complex function. It's called winding numbers and domain colouring. Is what you described the same as what was explained in that video?
@@shannu_boi Looked it up. Yes! And the notes near the end closely relate to where the Residue Thm comes from. There really is a 3B1B video for everything.
@@konstantinkh Once again, thankyou so much. ❤❤❤
He has come back to us, armed with python and infinite math.
RUN! We cannot hold off such power!!
--the concepts of us not knowing the things he's teaching us
How on earth do you make python do that?
@@polterp manim library
@@polterp In short it's a python library that he developed called manim. Manim being short for math animation.
Damn I gotta get a hold of that
This video is stunningly beautiful in every way. I'm always amazed that each one of Grant's videos seems to be better than the last. It's genuinely inspiring.
this video is stunningly ugly. I've never seen a more poor approach at math.
Three years ago the youtube algorithm did a great thing, and I for first time saw your video. Now I study applied mathematics on university. You, and your coworkers, have litteraly changed my life. I am thanking you, for showing me, that mathematics is more than some random numbers. I love your work and mathematics. I wish you good luck.🙏🙏🙏
knowing math can solve half of the world's problems; the other half are not solvable anyway
As somebody who doesn't know jack shit, math is literally everything.
Doesn't everything in the universe break down to some sort of numbers/ math? It's crazy and we definitely live in a simulation. Source: bro trust me
mathematics is the understanding of quantitive logic. This particular example in this video is not logical, as it's a random method of solving a problem, and the complexity in its randomness has no purpose, making it actually poor math.
@@jaakezzz_G are you seriously trying to quantify mathematics to such a simple subset of reality? Mathematics is, quite frankly, the study of patterns...ask anyone
I was having a really shitty evening, just all around feeling bad, and when I saw that this was uploaded I was sure I wouldn't be able to watch (let alone appreciate) it tonight. I decided to quickly look into it anyway, just see what it's about.
And then I accidentally finished it in one go. It was wonderful, captivating and visually stunning as always. And it distracted me very successfully from everything else. I know that isn't the purpose of math, but I feel warmer and better now. So thank you.
That is not the purpose of math, but what is the purpose of it anyway? What is the purpose itself? If you really think about it, you define the purpose of math. It is just a tool, a very beautiful and elegant one, but you define how to use it. So yeah, that could be one of the cases :)
I often use math for exactly that purpose. The world doesn't make sense but math does and that comforts me. 3blue1brown videos are especially good at it.
Hey, same here... Hope you get better by tomorrow! Remember you're never alone :)
👏👏👏👏👏
Maybe Newton enjoyed math because it kept him distracted from the world that he was outcast from for being a total nerd 🤓
Thank you for saving me from doom scrolling
It led you here. For once it paid off in a permanent way.
Best quote of Grant ever:
"What the %$!* is going on here!?"
Are you sure it's not saying - What the math is going on here!? Just kidding.
Finally, a fine quote you can use in math class.
Or from his first differential equations video: “They’re really freakin’ hard to solve!” Especially with the setup to that.
Oh come on! "What the ****" (15:34) is exactly the reaction I had when you said the explanation is going to be in the next video! Looking forward to it eagerly!
Waiting for your next complex analysis vid too mate
Absolutely love the video! This is a great presentation. One note: When discussing that Newton couldn't have known about these fractals for lack of a computer, there is an example of a classical (Western) mathematician who knew something about 'chaos' (and the complicated sets it creates) even in 1881. That's Henri Poincare when he discovered the homoclinic tangle while studying the 3 body problem. Such tangles are connected to Horseshoes (related to Smale's Horseshoe).
I have my head around fractals pretty well but when he explained how the boundary between two colors doesn't even exist it broke some part of me. any two colors never touch, the only way colors meet is at "vertices' where there are three colors which are infinity dense along the 'edge' between colors. when the Mandelbrot set somehow showed up I felt unsafe, it was ominous.
Mandelbrot Set is always waiting. Watching. IT KNOWS.
I agree! There is something disturbingly ethereal about a boundary that does not exist, in the sense that it never separates anything -- the three colors always meet. Vertices all the way down.
Existential anxiety is my favorite part about learning mathematics.
In my grubby pragmatic way, as an engineer, I see a fundamental lesson : since an irrational number cannot be expressed in concrete form, this is analogous to the graphic property of the boundary of a fractal curve - you can get close to it but never actually find where it is. It's easy to see why the ancient Greek geometers were upset by the concept of square root of 2, which in their geometry should be visible, but in concrete terms could not be found.
the part that blew my mind the most about this is when I just thought:
Imagine standing within one of the big open regions, in an infinite-iteration "complete" version of the fractal. Now start walking towards a boundary.
*which other color will you walk into first?*
I can only imagine the answer is "no"
It's actually incredible how efficient this iterative process is. I used to do research with human subjects (i.e., messy) and would use iterative stats to fit models thinking it would need to run like 1000 models to converge at a precise level.
6
6 was the roughly average number of iterations required to converge on a solution that met parameters.
And I could tell it to run 1000 iterations and the difference in performance succumbed to rounding.
quadratic convergence is real shit
I can't stop appreciating the amount of work put in these videos.
9:05 If we choose the starting point to be that local minimum, the tangent line will be parallel to the x axis. All points in the vicinity would have tangents that are nearly parallel to the x axis, meaning that the intersection will be either really far to the left, or really far to the right. We can imagine one dimension higher, where it would be the local minimum of a surface, a bowl shape, at the bottom of which all tangent lines are nearly parallel to the xy plane. Any tiny deviation from the local minimum will lead the next point to be on a completely different part of the plane, which is why all the colors have to be in this 'neighborhood'.
I guess if we consider an extended complex plane (a Riemann sphere) and consider the "infinity" as a single point where all these tangents meet, then each of these fractal "meeting" points on boundaries (as at 12:45) will be just local reflections (images) of the "infinity" point. And it looks like all these meeting point retain properties of the infinity point, e.g. all have the same ratio and order (up to the sign) of each color in the neighborhood, equal to the ratio and order of the solid-colored areas as they approach infinity.
That is a wicked helpful insight on it, thank you for sharing : )
I've been watching you for quite a few, from just starting Calculus in my final year of High School, and now I am graduating end of this year with a Bachelors degree in Civil Engineering. There is something very special to watch these videos knowing how far I have come and seeing how many more of the concepts I understand in greater depth. Thanks for the long journey :)
I escaped the shorts feed; thanks 3b1b.
Same here
yup im dong more shorts now. a short led me to this video(inderectly, it ed to part 2 which led to this)
Questions are invented, answers are discovered. Only effort required is to find the right question to ask. Love your videos
Note: just expressing my appreciation of 3b1b's video, don't judge it too seriously.
I really have to say your video quality tends to increase over time, regarding how mind-blowing fragmental phenomenon is, and the connection between my recent projects, really, thank you.
I have studied physics for almost 5 years now, and I find amazing how, even though at times I become tired of math, physics and whatnot, watching your videos can bring some clarity and remember me why I started it in the first place. Keep up the great work!
Edit: That sequel is now out! ruclips.net/video/LqbZpur38nw/видео.html
For everyone who participated in the Summer of Math Exposition, the summary of that, with mentions of winners, honorable mentions, etc., will be coming shortly. The plan is to post it a little after the sequel to this video is published.
Hi
if I had math teachers like you at high school I would have 100% studied maths at university
keep up your great work
Grant, you've got a really high Ab (slightly sharp, ~6.7kHz) quietly ringing on your vocal track. You might want to check that out.
I dare you to make a video about the quartic formula :)
Edit: I actually always wondered why is it that with all the currently known operations we can only have formulas for the first 4 grades of polynomials but not for higher order ones... well, except for higher order ones which can be fully divided into order 1,2,3 and 4 polynomials. Could this be related to (probably) not being able to geometrically calculate the fifth root ? I say probably because technically one can’t compute the third root geometrically with just a line and a compass... but one could do it with an origami trick so I’m gonna be naughty and still count it as a “geometric construct”.
Keep it up ❤️
3B1B: Explains about finding values for 0 polynomial
Me: Huh, that’s pretty cool
3B1B: Oh right, fractals
Me: Oh right, fractals
I hope that in decades/centuries from now, these videos are still around and accessible to the public. Such a creation of brilliance and beauty. Thank you Grant.
strong vihart vibes when the timelapse started of him cutting out the blobs
Dang, came here just to say that!
Yea, should have done a guest hand appearance. Vi would have found a way to make it into a burrito.
totally forgot about her. ty for the nostalgia trip
@@1088lol She still makes videos, uploads one every couple of months
It's started. All our favourite math channels are converging. Soon there will be just one channel called Standupflammableviologeyam3red1bluepen.
Am I right in my intuition that higher polynomials need to have "rougher" boundaries because they have to border all roots. Do these fractals ever take up significant space making it harder to find the root to a polynomial because everywhere you look is a boundary?
You can try generate newton's fractal for f(x)=sin(x), which in some sense is a polynomial with infinite roots. The boundary is pretty rough, but the interior is still well-defined
@@wangweiyi8478 I'd love to see that animated
For your second question, although I see no simple way to prove it here, I think it's likely the boundary is negligible in the plane (i.e. it has measure zero) which would be the reason we still use Newton's method anyway:
For a randomly selected point(*), it would then have zero probability of being on the boundary.
(*) It is obviously not true for some specific distributions, but maybe for a uniform probability distribution over a square bounded region. A square cannot be the boundary of a polynomial.
Interesting. I think the fractal dimension would increase (i.e. would be rougher) with an increasing number of roots so that the limit of the fractal dimension equals 2 as the order of the polynomial approachs infinity (assuming no repeated roots)
@@AwkwardDemon that leads to another rabit hole question:
"Are/which polynomials of degree n≈infinity with no repeated roots?"
Not sure if the answer has a meaning beyond helping another math proof along. (Pure math doesn't always reveal its value anytime soon after discovery like prime finding algorithms being critical foe computer encrytion but also being 200+ years old)
This video actually made me feel good a couple of times while I watched it.
Just some pure esthetic delight from watching how everything naturally falls into place.
And how everything is so beautifully illustrated.
I cannot even imagine how much skill someone has to have and how much time had to be put into practice to be able to create such amazing content.
Just leaving a comment so this video gets recommended to more people.
*aesthetic
"Oh hey remember that one section of the video earlier that had a bunch of questions on it, and one of the questions was whether points ever cycle? Well, here's a picture of the Mandelbrot set in one of these fractals. See ya next video!" Dude the foreshadowing and cliffhangers are messing with my emotions you can't just do this to people
That was an effective cliffhanger
In Brazil, today is children’s day. Thanks for the gift!
melhor presente
Exato!
@@mayconbruno2676 presente tão bonito quanto um fractal.
It is insane how mesmerizing and captivating you are able to make otherwise unapproachable mathematics for most. Thank you so much for creating this channel and carrying us through this ride ❤
I couldn’t help but smile when we got to the fun part, heard about inputs iteratively moving around, some converging and some not. My brain made the connection immediately, and that is exactly what makes me love math so much.
THIS CHANNEL WILL MAKE YOU LOVE MATHEMATICS IN A DIFFERENT WAY
This series couldn't have come at a better time! I am currently reading James Gleick's book 'Chaos' and I am currently in the fractals chapter, these videos are going to help me appreciate the beauty of math even better. Thank you so much
I did my thesis work on these in college! Specifically, I was focusing on trying to classify the behavior of these fractals for rational functions. There is some fantastically interesting behavior where you can get sinks for period 2 points, a behavior I called kaleidoscoping. So excited to see my favorite youtube mathematician sharing the beauty of this stuff! Excellent video as always.
Oh my god I didn't even remember how much I needed another 3B1B video. This was fantastic, the quality of your videos really is unmatched here in youtube. Thank you for coming back, Grant.
This is one of the most fascinating and well-explained videos I’ve enjoyed in a long time. Thank you for all the work you put into sharing it with us.
I just wanted to thank you, Grant and all the other people responsible for this channel, for immensely expanding my horizons. It's in no small part thanks to you that I had the courage and means to overcome my math shortcomings. This month I started my undergrad physics course and there had been not a single day in which the knowledge from your videos (as well as pointing my curiosity to stuff beyond just what you had shown) hadn't helped me greatly. Keep on doing great work!
I am proud of every single dollar I have given this BEAST on patreon. I cannot watch these videos for free, they are too good.
Just doing my part for the algorithm. If there’s any channel that deserves all the boosts it can get, it’s this one
15:30 I love how he just breaks character and does what we are all thinking in most of his videos.
The whole video was amazing, but somehow understanding that the straight line is NOT an exception, but merely a special case, was peak mind blowing!
My favorite thing about this channel? I can watch it as someone with pretty good math education (engineering undergrad) and still make new connections, but then I can send it to my friend (who "hates" math) and they will also understand the video. We may get different things, but we both get something, and that shows how deep and amazing 3b1b videos are.
If you can't explain a topic to someone who knows nothing about it, then you yourself don't know it.
@@thecodeking91 good point, what I meant was, with carefully thought, if you could not write down a concept in simple terms then you don’t know it.
I think the clear reason is that the roots of the derivative, where the derivative is near zero, hold the property of teleportation, since the step size is simply ginormous. Any points that go near them get scattered instantly, especially since the phase of the derivative near the root does a full 360 at minimum
Yeah, that sounds like a good addition to this.
This shows how important the initial guess is in numerics. Even in the age of computers it is still important to have a feeling for sensible initial conditions.
I can agree, just look at Conway's Game of Life! Without the right initial conditions, you won't end up with interesting continuous stable patterns.
I agree with you. I think the initial condition is most important thing in mathematics.
If it were wrong, we would be wrong forever.
Garbage in, garbage out.
This channel has the best visuals I've ever seen, and that is not restricted to RUclips.
Amazing content, music and narration to go by it too :)
I had myself once experimented with using newton-rhapson to solve cubic equations, by plotting which starting values made me end up at which solution. I then saw something really strange after which I thought I messed something up. After I double-checked everything I then gave up, thinking something weird was going on outside my control.
Now I know that it was supposed to look like that. Oops
Once I tried to implement the newton's method in Python using numerical derivatives. For most of the starting points, the solution always oscillated all around the place and I thought my numerical derivative implementation was wrong. But looks like this could happen for high-dimensional functions. My numerical derivative implementation could have been correct after all :)
Didn't something like that happen to Mandelbrot when he was trying to print out what the set looked like? Everyone thought something was wrong with the printer
that is why programmers should learn math.
@@OrangeC7 No, the first visualization is not that high of a resolution. Think of ASCII Art. And, the idea came from an index of Julia sets, which also look like that, so he knew what to expect.
@@OrangeC7 As far as I know, the operators of the printer cleaned up printing artifacts that were actually what Mandelbrot tried to see (They were doing that automatically). And so, he always recieved something that was different from what he was expecting.
Me with every 3blue1brown video
before watching: "oof. 30 minutes? That's a lot"
After watching: "wait no, don't stop now!"
Fascinating stuff as always!
always
so true
This is exactly what I needed to hear, I learned this topic in Calc yesterday and completely misunderstood it. You made it understandable and geniusly simple. Thank you.
I love when names get attached to things the person couldn't have dreamed of. It shows that humans care. People will be remembered not just through their work, but beyond their work.
I like the idea that Grant bleeps himself out when he says “what the heck is going on”
I don't think that was a "heck", but no way to prove it lol
Bless your heart
@@HAL--vf6cg There's no way of knowing for sure!
@@HAL--vf6cg proof by contradiction should work ig
@@thisisthemactan Maybe there is... If he posted it un-bleeped on Patreon...
Thanks for stopping my short binge, three blue one brown
Yyyaaaassss!!!
May our attention spans increase!!!!
Reminds me of something I learned long ago. When you start with 3 random 2-D points forming a triangle and one random "trace" point anywhere, then repeatedly select one of the 3 triangle points and take the midpoint of that and the "trace" point to become the new "trace" point, you will eventually trace out something that becomes indistinguishable from Sierpinski's triangle.
13:36 I notice that, when a root is dragged around along a smooth path, the boundary seems to change smoothly as well. I wonder if that's actually true, since we can't see all the detail of the boundary, and to what degree of smoothness (or perhaps continuity) the boundary is as a function of the polynomial's root locations.
I've said it before and I'll repeat: you're the best math teacher to ever live.
So few people had found their vocation at this level.
The moment you talked about the Art Puzzle I couldn't believe how intuitive it was. I was kinda mindblown.
I would love if there was a web version of the draggable root thing :D (not that it would prob be worth the effort)
I have good news for you, my friend: www.3blue1brown.com/lessons/newtons-fractal
yeees :D
@@3blue1brown "an unexpected error has occurred" - I'm on a phone so I'm not shocked, but it should be capable?
@@3blue1brown i'm having an issue where my scroll wheel up is causing it to super zoom and scroll wheel down is normal. Is this a problem on my end? on youtube scroll wheel up and scroll wheel down move the page the same amount so it seems not, but I'm curious if you have the same issue. Tried it on firefox and chrome both, same issue.
@@colecarter2829 I have the same problem
great job
"It's actually an infinite family of fractals."
Aren't they all.
If you take one polynomial you get one fractal, then if you vary coefficients you get the fractal variation, which you can call fractal family.
Depends on your definition, I guess.
@@lonestarr1490 idk but i think that is what maththematician defined them... Gud luck with ur own definition
It sounded like he was saying "an infinite family of fractal", which confused me. I wondered if there was some abstruse fact about the etymology of the word "fractal" which meant that the plural should be the same as the singular. Then I zoomed in by a scale factor of 10^24 and found the "s".
No, the mandelbrot set is a single fractal. But there are infinitely many Julia sets though. Another simple example is the Koch curve. It's a fractal, but there is not an infinite amount of Koch curves, there is only one.
“The most pertinent question” oh my gosh, that turning was on point
This is one of the best RUclips channels in existence. Amazing work.
I feel sad for Newton when I think he won't ever be able to see this video.
same, but for Mandelbrot
Newton: What in the world is a 'Video'???
Just the idea of him being able to see this. Without context of what computers or youtube are. Just the sheet amount of knowledge and joy he could derive from it (possibly).
@@caniggiaful Reminds me of the Van Gogh episode of Doctor Who. I demand a Newton episode now!
Newton would probably copyright strike the video if he saw it cos that's how much of a prick he was lol
In a way this image is like a distorted version of itself, like looking at stars through the gravitational lensing of a black hole. The part where you showed the cluster “explode outward” at the end visualized this quite well. It is like each region actually is a wormhole to a different part of the image, or to the layer below if each iteration is a layer. And we are looking through all these wormholes until we finally reach one of the roots.
Or rather, that all boundaries are the same boundary, thus all event horizons are the same event horizon
I know I’ve seen that boundary property before with a fractal where you have 3 main points, each with gravity that attracts all other points, and you see what main point each other point falls on, color them, and rollback, the boundary property of which goes hand in hand with the three body problem being chaotic unlike the two body problem.
Correction: it was based off of a pendulum on a string, so there was some gravity towards the center.
It's all coming together now
@hognoxious explained it way better than I even could have
Looks like each "point" on the boundary is a reflection of an "infinity" point C(∞) of the extended complex plane, and in their small neighborhoods they retain all properties of the C(∞), like ratio and a [mirrored] order of colored areas as they approach the C(∞). And the reason is that algorithm tends to "shoot" at infinity around these points, which produces local reflections of the C(∞) with all its symmetries.
I wonder can it be related with an apparent "probablistic" behavior of quantum systems? What if quantum probability turns to be a manifestation of a fractal behavior on a some sort of a Bloch sphere, similar to what was shown in the video? Like, a quantum measurement as a Newton-Raphson process of finding zeroes on a Bloch sphere? :)
Congratulations! This is a tour de force; it must have been a huge amount of work for you, and it was extremely enlightening. Glad to see you're still doing such a great job and I look forward to the next one.
FINALLY! I'VE BEEN LOOKING FOR THE VIDEO ON THIS FOR OVER A YEAR! THANK YOU SO MUCH 3B1B! YOU FINALLY LED MY MOST DESPERATE RUclips SEARCH TO A JOYOUS CONCLUSION! Gah I can FINALLY scratch this itch. I cannot tell you how glad I am that you chose to post this when you did. I mean, I saw the fractal in your short on shorts the other day, and I was pretty sure I recognized it, but from the way you introduced this video, I was sure I had the right thing. Turns out searching "3 color touching border fractal" on youtube... doesn't get you very specific results.
I'm curious about color vectors. I am wondering what is the correct way to represent unit color vectors on a sphere. There seem to be two thoughts on this. and where s and v are max values and h is a function. seems to create duller colors. ends up being the spectrum. Most people want to play with the spectrum. As you can see from my avatar, I prefer .
The explanation I would give for this is that the surface of a sphere is 2d, while your rgb coordinates are 3d, so you would only see a subset of the rgb spectrum, and requiring s and v to be max leaves your hsv coordinates one dimensional, which is why h needs to be a function (requiring 2 variable inputs by my guess) to map to the whole surface of the sphere. If you allow s and v to vary, your hsv sphere will look very different, but would once again only be a subset of the hsv spectrum, and would appear to have duller colours.
On the other hand you could fix either s or v at max, and have your hsv coordinates be spherical coordinates, with the fixed value representing the radius, to see a more aubsetbof the spectrum that is more representative of the range of that spectrum, albeit only in two of the dimensions.
@@matthewparker9276 I want to ask this question. Is there a 'right' way to represent the color sphere with unit vectors? Yellow (255, 255, 0) would not be in unless there is a way to rearrange the function. In the unit sphere yellow becomes . With values like that, it might be possible to decode the color sphere in an imaginable way. I don't think this sphere would just be the spectrum. Other colors might make it in there. Black would have to be in there. Where would white be? , which is the origin, the origin doesn't make it into the color sphere. Well, it could be on the opposite side of black . Blue would be , which is not on the opposite side of yellow. But I guess no one said it had to be.
20:40 I could imagine a 6-year-old Gauss doing that in the art class.
I remember doing Newton’s method in my calc classes, definitely something to be aware of in any problem
I'd love to see if fractals like this appear with other root-finding methods as well, e.g. Laguerre's method which uses a quadratic approximation rather than the linear approximation used in Newton's method.
Oh my god, I can't believe 3b1b left us such a cliffhanger.
That is just phenomenal. The animation, the narrative, the effectiveness of conveying information to a broad audience… just wow
Thanks for explaining this stuff in a way that even I can understand while just watching on my phone. It is impressive that you are able to cut through all the important complexity that would otherwise get in the way of teaching the general concepts.
9:31 this animation immediately sparks the stochastic gradient descent for me. Their essence is definitely the same!
(it is related)
Every single video, from when I found you years ago, has managed to push me towards a math degree. Every time man.
Thanks fam. I'm escaping this short binge
"What the %$!* is going on here?" -- 3blue1brown, 2021
We all just swore at math, and not out of frustration :)
Never did I think 'Blobs on blobs' would prove an eye opener. Great video.
I’m taking complex dynamics this semester and we are covering this exact topic! It’s always awesome to see that other people find this stuff interesting as well
Could you color the points (e.g. darker) the more steps the take to reach the root (or a small circle around it)? I would like to see if a point closer to the boundary/edge would take longer in general. Maybe the "more away" a point is from the edge, the faster it converges. This might help to find rules for a good first guess for newtons root finding??
It's so cool that you made a video about this, just a month before we had an assignment in numerical analysis to make a program that creates an image of the fractal. The actual coding was a pain in the ass since we were using C. On top of that, our professor forced us to code the complex polynomial evaluation ourselves instead of directly using the complex arithmetic library of the language "so we could get to feel the true experience of scientific programming". But seeing such thing before your eyes after all the work it was put behind was gratifying
Simply stunning. First thing that comes to mind are chaotic systems such as the double pendulum or the three body problem.
Thank you for saving me from *T H E S H O R T S*
Me who got here from a short: 🫣
I love grant's math videos, they're always great for math which is the main purpose, but it's even better because he gives any of pis, blue or brown, some form of character which makes the video more fun and more immersive to watch
I’m so early, Ishwar if you see this, thank you for being such a great friend, and thanks for recommending this super awesome channel 💜
Update: we’re dating
How many people here have a friend they can talk to about these? Just curious.
👌 I know I've recommended to a few dozens of my students.
@@susmitislam1910 I have a couple
I don't think I can match the awesomeness of that comment in this reply so I ain't gonna try.
Grant if you see this, thanks for being awesome. I don't think I would have engaged my love for math as much as I have without you being you and doing what you do. Keep making things make sense as awesomely as you do. I'm so happy May got to see the enormous value you bring to the world, and look forward to more people doing that.
Thanks boss :)
@@susmitislam1910 I have 2 that say "cool" but don't read it. Others ignore it
Thank you for helping me escape shorts.
For 3:00 problem, I think tou can just divide curve by pixels, find the pixel they belong to, then color it, and do a wave coloring by changing coordinates by one recursively until stroke width is reached
Something you hinted at that no one ever talks about: It only converges when the curve, as it moves toward the intersection point with y=0, decelerates towards y=0, as opposed to accelerating.
Could you put that in terms of the second derivative?
This sounds wrong. Can you explain what you mean?
e.g. Arctan(x) accelerates towards the intersection point, but the method converges perfectly if you start close enough.
@@Meni_Rosenfeld They mean the distance between the iterations decrease over time, i.e. as n goes to infinity, we have |x_(n+2) - x_(n+1)| < |x_(n+1) - x_(n)|
I could see this maybe happening for curves that are tangent to y=0 (e.g. x^2) where the slope would very shallow as you get close tot he root and the slope would be 0 at the root, but wouldn't this be offset by the fact that the distance to the root would also be close to 0.
My thought was that the cubic case is connected to the 3 body problem, in that there is enough complexity to generate chaos.
It would be interesting to compare different orders of Householder's methods of which Newton's method is the first order and Halley's method is the second order.
i had the same exact thought haha, still wondering what role chaos might play
Same thought ! The fact that 3 body system is chaotic might has to do something. There is a saying in ancient book of Tao Te Ching : one gives rise to two, two give rise to three, but three give rise to everything.
Was thinking the same
Same here. Is it coincidental that stepping from 2 to 3 introduces chaotic complexity in both cases?
I am really lucky to be living in an era that you are! I watch your videos(call them books) like I am watching the most exciting movies ever.
Beautifull and mindblowing ! I hated to learn mathematics at school, but I love your work and the way you explain ! I understand the minimum required to be amazed and that's well enough, I totally respect those who daily work on it :)
This is a really inspirational video, thank you!
Glad to see another hero over here! Thank you
I got introduced to this video by my Professor. My mind is totally blown now.
Just love the way the problem is approached with all these exiting graphical depictions 🎉.
Very easy to understand, thank you 🙏
As always, thank you so incredibly much for the work you do. Your work has been instrumental in my own learning and teaching, both in inspiring my students and giving me new ways to approach explanations