They Erased the Most Interesting Part of the Mandelbrot Set

In 1980 a computer at IBM printed a strange black shape surrounded by specks, and the specks were cleaned off as printer dirt. They were the most interesting part of the picture. This video builds the Mandelbrot set from one instruction: square a number. First on the number line, then on the plane, then with one extra step, plus c. Along the way: Julia sets and the two French mathematicians who described them without ever seeing one, why a computer can decide "forever" with a circle of radius two, why the colours in every picture are counts, chaos at the edge, a zoom of ten trillion times, and why the shape is a map of every Julia set at once. Every fractal in this video is computed from the formula. The c values on screen are the c values being drawn. Pick a number of your own, anywhere on the plane. Does it stay, or does it leave? Write your number and your guess in the comments. Subtitles in English and Russian are built into the video. 0:00 Square it 0:31 The specks 1:00 The number line 2:16 The plane 4:00 Plus c 5:03 Gaston Julia and Pierre Fatou 6:02 Mandelbrot's question 6:26 Spirals, stars and scribbles 7:41 He was not the first 8:07 The point of no return 9:06 Carving the shape 9:24 A walk along the edge 9:52 Chaos at the edge 10:38 Ten trillion times closer 11:07 A map of Julia sets 11:57 The dust 12:57 Pick a number Adapted from two Numberphile videos, which are well worth watching: Ben Sparks, What's so special about the Mandelbrot Set?    • What's so special about the Mandelbrot Set...   Holly Krieger, The Mandelbrot Set:    • The Mandelbrot Set - Numberphile   Sources Gaston Julia: MacTutor History of Mathematics Brooks and Matelski, The dynamics of 2-generator subgroups of PSL(2,C), 1978 (published 1981), Annals of Mathematics Studies 97 Mandelbrot, Fractal aspects of the iteration of z → λz(1 − z) for complex λ and z, 1980, Annals of the New York Academy of Sciences 357 Douady and Hubbard, Comptes Rendus, 1982 Dudko, survey for ICM 2026, arXiv:2512.24171 Photos Gaston Julia (with Gustav Herglotz): Konrad Jacobs, Oberwolfach Photo Collection, CC BY-SA 2.0 de, cropped and converted to grayscale Pierre Fatou: public domain Benoit Mandelbrot at TED 2010: Steve Jurvetson, CC BY 2.0, cropped and converted to grayscale IBM Thomas J. Watson Research Center: Simon Greig, CC BY 2.0, cropped and converted to grayscale French soldiers returning from the trenches, 1915: Agence Rol, BnF Gallica, public domain Julia's 1918 memoir, cover sheet: Numdam and BnF Gallica