Beyond the Mandelbrot set, an intro to holomorphic dynamics

An intro to holomorphic dynamics, the study of iterated complex functions.

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:

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:

Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld

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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:

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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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Various social media stuffs:

[DrTrefor]

I enjoy how on twitter you asked recently whether we preferred two 17 minute videos or one 34 minute video. Instead you seem to have given two ~30 min videos:D Best of both worlds:D

[tgs]

As a kid of the 80s who iterated Mandelbrot sets on an i386 and would wait patiently for hours to see patterns emerge, I have to draw attention to the computational miracle you’re looking at... Julia sets being near instantly populated with the waive of a mouse!

[BlockOfRed]

A second part! It was announced and actually came!

Edit: Thank you for all these wholesome videos. Waiting for them is always worth it, no matter what the topic is!

[DOSWR]

I recently got my Ph. D. in holomorphic dynamics. We often refer to the "stuff goes everywhere principle" as the "explosion property" of Julia sets. In fact, for the higher dimensional generalization of holomorphic dynamics (known as quasiregular dynamics), this explosion property is used as the definition of a Julia set.

[Nirmanyu]

Yeah, I used to think it was just recreational... then I started doin' it during the week... you know, simple stuff: differentiation, kinematics. Then I got into integration by parts... I started doin' it every night: path integrals, holomorphic functions. Now I'm on diophantine equations and sinking deeper into transfinite analysis. Don't let them tell you it's just recreational.

Fortunately, I can quit any time I want.

[NovaWarrior]

I personally will always be easy on 3blue1brown about his deadlines.

That's because..these videos are hard to make, and I mean at every single step.

It's hard to write a nonfiction narrative that's correct, then harder to write a narrative people can learn from and harder still to write a weaved story where listeners can come away feeling like they've seen something beautiful, which is of course what we want to communicate as artists: to convey our personal sense of beauty to someone we don't know.

Right now, I'm making an "explainer" with Manim because it looks incredible when it's done. But rendering and working and trying to make Manim work for me has been both fun and developmental because it's a test in both your fundamental programming, and your ability to articulate your math knowledge to a rigid computer. It's not harder than anything I've ever done. But it takes time, especially when you're caught up with other facets of life. Take it easy on yourself Grant!

[thom]

@3:42 - "I think this distinguishes Julia as one of the greatest mathematicians of all time who had no nose."
Newton: Thank you for adding that critical qualifier at the end of your statement.

[rv]

Fractals such as the Mandelbrot and Julia sets are one of the things that, when I was in high school, convinced me that I would've done math at university. The others were chaos theory, non Euclidean geometries, and Simon Singh's book on Fermat's Last Theorem. Crucially, none of the books that got me interested in mathematics in high school were school books.

[BobWidlefish]

I quite like this description:
“The Mandelbrot Set is a geography of iterative stability.”

[DeclanMBrennan]

When I was taught Newton Raphson many year ago, I was told: "Make sure your initial guess for the root is good, otherwise it doesn't always work." *Who knew "doesn't always work" was code for all this incredible beauty?* Thank you very much for being our guide.

[thisisnotmyrealname]

I almost started writing my homework essay, thanks for showing me cool fractals instead

[jriceblue]

It's obvious that a LOT of work went into this video, and I want to appreciate that. Thanks.

[adriancarpio]

This can't be happening! Two great uploads in the span of less than a week!

[mikoajgutowski]

Method for solving Exercise 3 at 19:10:

Let r, t, y denote roots of polynomial. Then we can write it as (x-r)(x-t)(x-y). Then multiply it all out and take second derivative.

[aminoacids]

I'm just loving what's happening nowadays. For your new initiative on submitting quality math videos, my YouTube recommendation is blessed!

[AmoghA]

*"Mathematics is like a very good detective novel. At first everything is shrouded in mystery and nothing is clear. But as you dive deeper to understand more, the plot gets crystal clear."*

Mathematics is honestly, truly amd genuinely very beautiful and I've fallen in love with this channel.

[SumanthVepa]

This an outstanding and brilliant exposition. I am absolutely gobsmacked at the painstaking effort and talent it must have taken to produce a video of this quality on this subject.

[crisisv]

@3:44 "I think this distinguishes Julia as easily being one of the greatest mathematicians of all time who had no nose." Actually, his nose was just in a different plane, so he was perfectly capable of detecting complex smells.

[JBOboe]

"Gaston Julia is one of the greatest mathematicians of all time who had no nose"
*Sad Tycho Brahe noises*

[santiagoerroalvarez]

I'm at a loss for words. What a fantastically coherent, clear, beautiful and exciting video. And, by the way, I really loved the exercises on this one. Never thought I would get to understand why the Mandelbrot set, of all things, has the shape it does. I felt like I was starting to _actually_ understand the topic, so don't ever feel like you're assigning "too many" exercises, please.

Thank you for your work, Grant.