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. ------------------ 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 ... https://www.youtube.com/watch?v=LqbZpur38nw

The third and final part of the block collision sequence. Part 1: https://youtu.be/HEfHFsfGXjs Part 2: https://youtu.be/jsYwFizhncE Home page: https://www.3blue1brown.com Brought to you by you: http://3b1b.co/clacks-thanks Error correction: I wrote the answer as floor(pi/theta), when really it should be ceiling(pi/theta) - 1 t account for values of theta perfectly dividing pi. For example, the case of equal masses gives an angle of pi/4, and 3 total clacks. This beautiful result, and the solution shown here, are due to Gregory Galperin: https://www.maths.tcd.ie/~lebed/Galperin.%20Playing%20pool%20with%20pi.pdf ... https://www.youtube.com/watch?v=brU5yLm9DZM