3Blue1Brown - 2021-10-12
Who knew root-finding could be so complicated? Next part: https://youtu.be/LqbZpur38nw Special thanks to the following supporters: https://3b1b.co/lessons/newtons-fractal#thanks An equally valuable form of support is to simply share the videos. ------------------ Interactive for this video: https://www.3blue1brown.com/lessons/newtons-fractal On fractal dimension: https://youtu.be/gB9n2gHsHN4 Mathologer on the cubic formula: https://youtu.be/N-KXStupwsc Some articles on Newton's Fractal, and its cousins: https://www.chiark.greenend.org.uk/~sgtatham/newton/ https://blbadger.github.io/polynomial-roots.html Some of the videos from this year's Summer of Math Exposition are fairly relevant to the topics covered here. Take a look at these ones, The Beauty of Bézier Curves https://youtu.be/aVwxzDHniEw The insolubility of the quintic: https://youtu.be/BSHv9Elk1MU The math behind rasterizing fonts: https://youtu.be/LaYPoMPRSlk Viewer-made interactive: https://codepen.io/mherreshoff/full/RwZPazd --- These animations are largely made using a custom python library, manim. See the FAQ comments here: https://www.3blue1brown.com/faq#manim https://github.com/3b1b/manim https://github.com/ManimCommunity/manim/ You can find code for specific videos and projects here: https://github.com/3b1b/videos/ Music by Vincent Rubinetti. https://www.vincentrubinetti.com/ Download the music on Bandcamp: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown Stream the music on Spotify: https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u ------------------ Timestamps: 0:00 - Intro 0:48 - Roots of polynomials 5:55 - Newton’s method 11:16 - The fractal 17:56 - The boundary property 23:13 - Closing thoughts ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: http://3b1b.co/subscribe Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3blue1brown Reddit: https://www.reddit.com/r/3blue1brown Instagram: https://www.instagram.com/3blue1brown_animations/ Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown
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
Thank you for saving me from doom scrolling
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!
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
Bla bla bla, mostly bullshit.
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.
"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
@Milky sausage 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
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 can't stop appreciating the amount of work put in these videos.
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!
This is one of the best YouTube channels in existence. Amazing work.
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 ❤
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?
@Miguel Carmo manim library
@Miguel Carmo 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 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
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.
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.
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
This channel has the best visuals I've ever seen, and that is not restricted to YouTube.
Amazing content, music and narration to go by it too :)
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
@mohamed belafdal i mean i'm a high school student
Just doing my part for the algorithm. If there’s any channel that deserves all the boosts it can get, it’s this one
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.
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.
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.
Thanks for stopping my short binge, three blue one brown
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.
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.
@thetoyking91 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 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 🙏
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.
I escaped the shorts feed; thanks 3b1b.
This is one of the BEST videos i have seen in a long time!!!
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
Never did I think 'Blobs on blobs' would prove an eye opener. Great video.
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.
The quadratic formula likely having been used trillions of times in the production of Coco really gives you a sense of how ubiquitous math is, and not just confined to the classroom! Thanks for wonderful video!
I remember doing Newton’s method in my calc classes, definitely something to be aware of in any problem
Simply stunning. First thing that comes to mind are chaotic systems such as the double pendulum or the three body problem.
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
@Jason Reed 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.
Please dedicate a video to the unsolvability of polynomials of degree 5 and beyond. Ever since I heard Edward Frenckel mention this years ago it has fascinated me. Your channel is magnifcient. Thank you.
Thanks fam. I'm escaping this short binge
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
Every single video, from when I found you years ago, has managed to push me towards a math degree. Every time man.
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
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...?
Thank you from the bottom of my heart for what you do. It is inspiring. Your reasoning dynamics is pure and direct. I feel you ask the first questions after new information is availble. At least the first ones I come up with. It is quite common for me to cry out of emotion because of the elegance with which you inquire on the topics your videos are about. Namaste!
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 YOUTUBE 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.
Yesterday I was rewatching tour vídeos on constraint optimization for a test i have today, and i could online think about how amazing your content is. Extremely happy to see this video now!
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
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 :)
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.
@Jake Zee are you seriously trying to quantify mathematics to such a simple subset of reality? Mathematics is, quite frankly, the study of patterns...ask anyone
Wow, you always blow me away with your videos! I would've never imagined that Newton's method involves an inherently fractal structure such as that shown here. Moreover, this fractal seems USEFUL in a way that the Mandelbrot doesn't seem to be, or at least, not in a way in which it's usually presented. Keep up the good work!
@diaz6874 - 2021-10-12
"What the %$!* is going on?"
—Pi creature, 2021.
After all of these years, the pi creature thingy finally expressed his anger against his master.
@TheThirdPrice - 2021-10-12
This is the beginning of the revolution.
I hope they are comfortable with quaternions
@rowida753 - 2021-10-12
His understanding is getting advanced enough to understand that there will be more “wtf” moments the higher up you go.
@PiercingSight - 2021-10-12
Truly the most pertinent question.
@Gameboygenius - 2021-10-12
"What the %$!* is going on?"
[music stops]
@JohnFallot - 2021-10-12
…someone needs to turn that into a 3 second reaction video.