Where Newton meets Mandelbrot (Holomorphic dynamics)
How the right question about Newton's method results in a Mandelbrot set.
Video on Newton's fractal: https://youtu.be/T_S2j5GaLRQ
Special thanks: https://3b1b.co/lessons/newtons-fractal#thanks
Extra special thanks to Sergey Shemyakov, of Aix-Marseille University, for helpful conversations and for introducing me to this phenomenon.
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Introduction to Fatou sets and Julia sets, including a discussion of Montel's theorem and its consequences:
http://www.math.stonybrook.edu/~scott/Papers/India/Fatou-Julia.pdf
Numberphile with Ben Sparks on the Mandelbrot set:
https://youtu.be/FFftmWSzgmk
Excellent article on Acko.net, from the basics of building up complex numbers to Julia sets.
https://acko.net/blog/how-to-fold-a-julia-fractal/
Bit of a side note, but if you want an exceedingly beautiful rendering of the quaternion-version of Julia fractals, take a look at this Inigo Quilez video:
https://www.youtube.com/watch?v=rQ2bnU4dkso
I first saw Fatou's theorem in this article:
https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-91/issue-2/On-the-iteration-of-a-rational-function--computer-experiments/cmp/1103940533.pdf
Moduli spaces of Newton maps:
https://arxiv.org/pdf/1512.05098.pdf
On Montel's theorem:
https://people.ucsc.edu/~fmonard/Sp17_Math207/lecture11.pdf
On Newton's Fractal:
https://core.ac.uk/download/pdf/1633889.pdf
------------------
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
3:02 - Rational functions
4:15 - The Mandelbrot set
8:12 - Fixed points and stability
12:51 - Cycles
16:25 - Hidden Mandelbrot
21:17 - Fatou sets and Julia sets
26:24 - Final thoughts
------------------
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
Video on Newton's fractal: https://youtu.be/T_S2j5GaLRQ
Special thanks: https://3b1b.co/lessons/newtons-fractal#thanks
Extra special thanks to Sergey Shemyakov, of Aix-Marseille University, for helpful conversations and for introducing me to this phenomenon.
------------------
Introduction to Fatou sets and Julia sets, including a discussion of Montel's theorem and its consequences:
http://www.math.stonybrook.edu/~scott/Papers/India/Fatou-Julia.pdf
Numberphile with Ben Sparks on the Mandelbrot set:
https://youtu.be/FFftmWSzgmk
Excellent article on Acko.net, from the basics of building up complex numbers to Julia sets.
https://acko.net/blog/how-to-fold-a-julia-fractal/
Bit of a side note, but if you want an exceedingly beautiful rendering of the quaternion-version of Julia fractals, take a look at this Inigo Quilez video:
https://www.youtube.com/watch?v=rQ2bnU4dkso
I first saw Fatou's theorem in this article:
https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-91/issue-2/On-the-iteration-of-a-rational-function--computer-experiments/cmp/1103940533.pdf
Moduli spaces of Newton maps:
https://arxiv.org/pdf/1512.05098.pdf
On Montel's theorem:
https://people.ucsc.edu/~fmonard/Sp17_Math207/lecture11.pdf
On Newton's Fractal:
https://core.ac.uk/download/pdf/1633889.pdf
------------------
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
3:02 - Rational functions
4:15 - The Mandelbrot set
8:12 - Fixed points and stability
12:51 - Cycles
16:25 - Hidden Mandelbrot
21:17 - Fatou sets and Julia sets
26:24 - Final thoughts
------------------
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
3Blue1Brown
3blue1brown, by Grant Sanderson, is some combination of math and entertainment, depending on your disposition. The goal is for explanations to be driven by animations and for difficult problems to be made simple with changes in perspective.
Contact and F...