Where Newton meets Mandelbrot (or more fancifully, “holomorphic dynamics”)

Where Newton meets Mandelbrot (or more fancifully, “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 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.

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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:

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:

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

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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:

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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

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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

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