Newton's Fractal (which Newton knew nothing about) 

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Who knew root-finding could be so complicated?
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Interactive for this video:

On fractal dimension:

Mathologer on the cubic formula:

Some articles on Newton's Fractal, and its cousins:

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

The insolubility of the quintic:

The math behind rasterizing fonts:

Viewer-made interactive:


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You can find code for specific videos and projects here:

Music by Vincent Rubinetti.

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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 UStotal, if you want to stay posted on new videos, subscribe: 3b1b.co/subscribe

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Published on


Oct 12, 2021




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Comments : 2 907   
Tommy G
Tommy G 23 hours ago
You find the Derivitave silly
Lv 45
Lv 45 Day ago
John 11:25-26 (KJV) Jesus said unto her, I am the resurrection, and the life: he that believeth in me, though he were dead, yet shall he live: And whosoever liveth and believeth in me shall never die. Believest thou this?
John 15:11-14 (KJV) These things have I spoken unto you, that my joy might remain in you, and that your joy might be full. This is my commandment, That ye love one another, as I have loved you. Greater love hath no man than this, that a man lay down his life for his friends. Ye are my friends, if ye do whatsoever I command you.
Rakkuunz 2 days ago
I don’t understand this but it’s somehow interesting
jacobepic 2 days ago
this is what acid looks like
potsdamcsc 2 days ago
Extremely cool
Eric F
Eric F 2 days ago
This is a mindblow fractal
GGiak 770
GGiak 770 4 days ago
Joseph Allan Ferriols
1:55 -tai
Zawiedek 5 days ago
2 philosophical takeaways from this video: a) Chaos takes place at the borders of more or less homogeneous areas; b) chaos is the outcome of extreme infinitesimal order ...
I like to thing I'm pretty smart but this shit is to goofy
LucidGamer137 6 days ago
We are standing on the shoulders of giants who were, often times, far more intelligent and wise than we are. Watching this video, I've found one of two things. Firstly, that I would have a brain aneurysm if my career involved performing mathematics much more complex than high school algebra. Secondly, I have found an inkling of understanding of the painfully unknowable vastness of eternity. It is at once the most inspirational yet intellectually deafening sort of thing to come across. Positively fascinating. The complexity of the reality we observe, how in one instance it appears limitless (like the borders of these complex fractals, webbing into an endless resolution of potential), and in the next very finite (the large portions where points are only drawn to a single origin). It speaks to me in a language I do not know, intellectually, but yet my very soul understands. I apologize for the strangeness of this comment, but I felt the need to express this to someone, somewhere.
Matthew Reid
Matthew Reid 6 days ago
Wtf is this
Potato Fury
Potato Fury 6 days ago
This is a amazing, now I’m happy I applied for that algorithm course
Slim Dim
Slim Dim 7 days ago
Get laid nerds.
Kutay 84
Kutay 84 7 days ago
5:27 How can I find this software and play and tweak with it? (Awesome short animation, 5 seconds to explain all French Mathematician Galois)
eWayxREAPzzz 7 days ago
I feel like I understand when you’re saying but the words don’t make sense.
Michael Sheely
Michael Sheely 8 days ago
Newton don't know nothing about this. You got nothing on this Newton!
:l 8 days ago
Is it just me? Or does anybody else just watch these videos, not knowing a single thing about what he's talking about?😅
I like to see if I can realize what's being said before the end
A A 8 days ago
I love this so much
Tiny Desk Engineer
So it's non-newtonian?
Reinaldo Favoreto
if you guess right the first guess, there will be no fractal formed. So fractal is the mind of the one that is infiitelly wrong all the time, no matter what guess one makes
Loris Z
Loris Z 9 days ago
Vortex 10 days ago
Can't quintic polynomials be solved using Horner's method? 🤔 Edit: Nevermind, this is too complicated for my brain.
gold nutter
gold nutter 10 days ago
omg and took him 3 seconds to get me to say poly know me all
gold nutter
gold nutter 10 days ago
geez took this wizard 0:24 to freeze my context pointer then at 1:00 yeah boring OH FFS !! YOU LOL made me press m 0 u see 2
gold nutter
gold nutter 10 days ago
TOP ology stop gap tech bottoms up lads.. xmas party gonna be w0 hex
gold nutter
gold nutter 10 days ago
Fibby nacci fibby fibby fibby you sing a song you get real high the end is nigh.. lolololn0
Captain Chaos
Captain Chaos 10 days ago
The "red, green and blue" paper actually being red, cyan and blue confused me for a bit. I wonder what was behind that. Is 3Blue1Brown colour blind perhaps?
Waffie 10 days ago
12:40 just broke my mind holy shit
Wutterly 10 days ago
do you guys wonder why some things exist? This is that thing
Random z Internetu
Random z Internetu 11 days ago
Now let's do Fibonacci tree
Cody Bertsch
Cody Bertsch 11 days ago
Isk why but the visuals of the graphs give me strong gravity field vibes
Popotato 11 days ago
Head hurts
Marcus WDF
Marcus WDF 12 days ago
I got hypnotised for a split second
Varunmuhil Viswanathan
At 6:12, I was literally screaming that I had learnt this in your ML videos.
Science AndHack
Science AndHack 12 days ago
Dear Grant, could you (or will) make a connection to the four-color set theorem? I thought about it about the middle of your video and was completely surprised at a suspected connection!
Xaphyr the First
Xaphyr the First 12 days ago
Do you have a clean picture of that 5-root polynomial near the end?? it looks sick!
afroninja234ya 12 days ago
Ms Hypocrisy
Ms Hypocrisy 12 days ago
This is not Newton's method. It it the Newton-Raphson method.
Christian Nersinger
Is an infinite grid, just an empty one, technically a fractal? genuine question
Noah Sabadish
Noah Sabadish 13 days ago
as someone with a very rudimentary understanding of fractals, i’m guessing yes
noam eyal
noam eyal 13 days ago
Grant you are the teacher of teachers :)
Daniel Pardo
Daniel Pardo 14 days ago
the fact that mandelbort set appeared at the end gave me some chills jajajaj
Saif islamic Teacher
‏طنز اور بحث سے رشتےکمزور ھو جاتے ھیں بس۔۔۔۔۔۔کبھی بھی اپنوں سے ایسی لڑائی مت لڑنا کہ۔۔۔۔۔۔۔ لڑائی تو جیت جاؤ لیکن اپنوں کو ہار جاؤ ....!!! 🌹🌸🌹
Vlad Koschenko
Vlad Koschenko 14 days ago
I had a task at work which essentially meant finding closest distance from a point to the Bezier curve. I carried out the math on paper and went "Aha, so I just have to find roots of this fifth-degree polynomial, so I'll code Newton's method, but now it's time for a break". So I went on UStotal, saw that you had a few new videos, clicked on the first one, and on 1:40 you literally started telling about the problem I was solving at the moment. What are the chances!? :D
L J Cohen
L J Cohen 14 days ago
Could you do a 3d tangent line for complex numbers?
ileiad 15 days ago
This is the weirdest political compass that I've ever seen
Flimsy Fox
Flimsy Fox 16 days ago
I now know why Voronoi noise looks the way it does.
Lamiah Hammam
Lamiah Hammam 18 days ago
22:04 - interesting part
Err0r 18 days ago
just a little thing that's been bothering me about fractals, there's other ways of looking at them. looking at the koch snowflake/curve, you could see the quarter thing, but when you see it to the limit, cutting it in half-s also viable, and honestly, idk how to get that down without showing you, but if you look at the snowflake, you see a hexagon, and that's weird, but if you take off a side of the hexagon, and flip it so that pointyt side up, you see the curve. cutting it back down, the curve can be divided into two, instead of just four.
bashful228 18 days ago
@13:40 dragging the root - amazing! how much computing when into that animation, real time or rendered @3Blue1Brown?
Fabio Vezzari
Fabio Vezzari 18 days ago
How did you compute the fractal? It would epic if I could compute fractals by feeding matlabs with complex polinomials and grids....I think I am going to do some reasearch on it
Ian B
Ian B 20 days ago
14:34 ah yes, the Orinoco Flow diagram. I know it well.
MW 20 days ago
What if we made a shade of colors based on the number of iterations that the point got closer to a root?
Benjamin Penazek
Benjamin Penazek 20 days ago
Wow! When you moved the roots around the complex plane and the limiting regions moved around accordingly--was that updating happening in real time? If so, how does one even begin to code that optimally? Or is it a prerender you coded up so the video would be smooth?
SeYTeR13 20 days ago
How do you begin to understand him?
violet_flower 21 day ago
Whoa, that was really cool to see voronoi diagrams pop out of fractal patterns
Anurag Kadam
Anurag Kadam 22 days ago
This video was crazy
Shahid Akhtar
Shahid Akhtar 23 days ago
You rekindled my love for maths.
Jan Hrdlička
Jan Hrdlička 23 days ago
When I first foud your videos about 3 years ago in a high school I had never ever thought about that my present teacher on linear algebra at university will use your videos as reccomended for understanding it. Also mind that I am from Czech republic
Harald Specht
Harald Specht 23 days ago
23:37 And you call them steamed hams despite the fact that they are obviously grilled
SpacemanCraig 23 days ago
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.
Anastasia Goold
Anastasia Goold 23 days ago
Ahh! It's the same kind as the magnetic pendulum fractal! But the potential field is different!! So cool!!!
Meher Baba is God in human form
alcyonecrucis 23 days ago
Did u plan this with that other guy?
Jaleel Bentley
Jaleel Bentley 25 days ago
All these squares make a circle.
Avanish Verma
Avanish Verma 25 days ago
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.
Ivan Morgun
Ivan Morgun 25 days ago
가위큐브_ 25 days ago
the eternal student
Hopefully, the possibility of hidden greatness in well-known formulas says something about the value of educating people with intellectual disabilities. It is discouraging to think that there are people who cannot contribute to their people's pursuit of knowledge.
Xenobeing 25 days ago
How about 3d ?
crimfan 26 days ago
Local convergence of root finding is weirder than I would have guessed! Plus points for the hardcore analog graphics.
Robson Reis
Robson Reis 26 days ago
Matilda the Hun
Matilda the Hun 27 days ago
me, chanting: let Grant curse!
Łukasz Orpik
Łukasz Orpik 27 days ago
You are simply amazing 💞
Le_ St0rm
Le_ St0rm 27 days ago
Outstanding work⭐
Kavishka Dilhara Wickramasinghe Abeywardhana
What if we take the value of a turning point as the initial value. The process terminates?
Zheqing Zhang
Zheqing Zhang 28 days ago
newton didn't have python or matlab. He of course didn't know this.
the eternal student
I do not appreciate the mixing of blasphemous or vulgar language with the moral purity of science, but so far I think you are gifted lecturer.
Oscar Keats
Oscar Keats 28 days ago
This bloke is a legend! Animations are OP
ganesh hegde
ganesh hegde 28 days ago
Michael Wicks
Michael Wicks 29 days ago
For some reason I'm drawing connections between this and organic chemistry where a genetic sequence is ultimately expressed unidimensionally (am I jumping too far ahead by already thinking of bordering points), yet it both expresses a representation of and remembers a series of interactions between individual molecules and an external mileu. However, those interactions have discrete (maybe unique is a better term) sets of causal factors that, on a population wide scale, determine the terms that are encoded into the genetic sequence. It's almost as if each iteration of the genetic sequence contains information that fails to memorise itself, is excessive material for the process of memorisation, or corresponds to interactions between the molecule and external mileu that contain information about the interaction. Could not a mathematical representation of this process be Newton's fractal (as genetic memorisation is iterative, and a process of "guessing") with each point harbouring information having to be in contact with n possible interactions between the organic chemical and its environment?
Steven Speck
Steven Speck 29 days ago
Ur the best
Jos Van den Eynde
Jos Van den Eynde 29 days ago
Great video. The application of fractal geometry discussed in the video was new to me. I've seen plenty of mandlebroth videos, but they all tend to discuss very similar topics. This was a new and insightful aspect of this field of mathematics. Fun to watch!
John 29 days ago
wouldnt this also mean that every given point would also always land on the same color? Say you had a guess that starts in a blue region, and after one iteration it landed in a red region. But then if we started from there, the dot would then by definition tend towards the red root. therefore, a dot in a given region will always land in the same color region after every iteration.
The Kwiatek
The Kwiatek Month ago
Is there a modified newton method that expoits that boundry proprty to find roots of huge polynomials? Eg. First find a boundry. 2nd seed a lot of starting points around that boundry with classic newton method hopeing to land close to all the roots.
ACoral Month ago
First step at 14:43 seems to be related to a problem of IRV voting system producing unexpected results.
Matt Buck
Matt Buck Month ago
18:20 - But WHY do they have that property! Let's find out!
Generous Month ago
I don't know why. But for some reason, complex math concepts intrigue me.
James Month ago
U’m what?
PM Month ago
I can't stop appreciating the amount of work put in these videos.
Noah Hradek
Noah Hradek Month ago
I was almost screaming out newton's formula by the middle of the video. Numerical analysis, good times.
Notmy F. Realname III
13:40 my computer just handed me a 2 weeks notice and a handwritten letter of appreciation.
willisverynice Month ago
I think the point is that we are all blobs on blobs on blobs
Jose V
Jose V Month ago
22:19 In the "uncountably infinite points inside some small disk" (area) thought experiment, I'm picturing the blob's area gets ripped apart into 5 areas that eventually settle on the 5 roots. But in the iterative process do those areas get ripped apart over and over again given they were originally on a the fractal boundary which has infinite detail.
Jose V
Jose V Month ago
14:07 it's interactive! so cool! 3b1b always has amazing visualizations.
Jake Levinson
Jake Levinson Month ago
I program this year‘s ago and I completely forgot about how cool it was
Alexander Bernauer
Keivan Monfared
Keivan Monfared Month ago
When I got the notification I was all: oh no, 3b1b is now following the same slippery slopes that vertasium etc have. But after watching the rest of UStotal and had nothing left to watch I came back and I was like: come on Ben you've got something for us, haven't you? And then he connects newton's method to bezier and voronoi and holomorphic dynamics etc. Well, just wanted to say, thanks Ben :)
GWS Month ago
*A wild Mandelbrot set has appeared* Those bugs are everywhere.