Beyond the Mandelbrot set, an intro to holomorphic dynamics

From 3Blue1Brown.

An intro to holomorphic dynamics, the study of iterated complex functions.
Video on Newton’s fractal:
Special thanks to these supporters:

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:

Ben explains how he made the Geogebra files on his channel here: (part 1) (part 2)

Excellent article on, 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:–computer-experiments/cmp/1103940533.pdf

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:

Various social media stuffs: