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 to these supporters: https://3b1b.co/lessons/holomorphic-dynamics#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:
Numberphile with Ben Sparks on the Mandelbrot set:
Excellent article on Acko.net, from the basics of building up complex numbers to Julia sets.
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:
I first saw Fatou’s theorem in this article:
Moduli spaces of Newton maps:
On Montel’s theorem:
On Newton’s Fractal:
These animations are largely made using a custom python library, manim. See the FAQ comments here:
You can find code for specific videos and projects here:
Music by Vincent Rubinetti.
Download the music on Bandcamp:
Stream the music on Spotify:
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: