The Newton Fractal Explained! | Deep Dive Maths

A Newton fractal is obtained by iterating Newton's method to find the roots of a complex function. The iconic picture of this fractal is what I call The Newton Fractal, and is generated from the function f(z)=z^3-1, whose roots are the three cube roots of unity.

What is the history of this fractal? How does one create this fractal? And why does it look the way does?

00:10 Part 1: Introduction
05:00 Part 2: Newton's Method
09:40 Part 3: Computing the Newton Fractal
11:35 Part 4: The Julia Set
16:20 Part 5: The Fractal Structure
20:52 Part 6: Higher Roots of Unity

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Fibonacci Numbers and the Golden Ratio
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Thanks to Miriam Chasnov, Joshua Chasnov, Jonathan Wu, Dominic Pegler and Bee Dy for reviewing earlier versions of the video.

Here are some of the material I used in the video.

April 1896 - Empire Theater in Leicester Square, London
https://youtu.be/Zc4EX1pHcnA

1930 – London Street Scenes
https://youtu.be/DIv0a5xbHF8

mornings in paris
https://youtu.be/71NDjEnjdVA

The beauty and complexity of the Mandelbrot set
https://youtu.be/N3ah6iTupIg

Portrait of Newton at 46 (1689)
https://exhibitions.lib.cam.ac.uk/linesofthought/artifacts/newton-by-kneller/

Sir Isaac Newton (1702)
https://www.npg.org.uk/collections/search/portrait/mw04660/Sir-Isaac-Newton

Arthur Cayley FRS (before 1883)
https://en.wikipedia.org/wiki/Arthur_Cayley#/media/File:Arthur_Cayley.jpg

Mandelbrot zoom: seahorse tail
https://commons.wikimedia.org/wiki/File:Mandel_zoom_04_seehorse_tail.jpg

Mandelbrot sehttps://en.wikipedia.org/wiki/Mandelbrot_set#/media/File:Mandel_zoom_00_mandelbrot_set.jpg

HP65 calculator
https://www.hpmuseum.org/hp65.htm

Gaston Julia (woodblock print)
https://www.flickr.com/photos/mrvelocipede/2904088490

Gaston Julia (photograph)
https://fractalsaco.weebly.com/julia-set.html

Benoit Mandelbrot (photograph)
https://medium.com/swlh/the-wandering-scientist-turned-father-of-fractals-4dcdc867d4dd

Frost patterns on a glass
https://www.treehugger.com/amazing-fractals-found-in-nature-4868776

Lake Mead fractal
https://www.flickr.com/photos/buggs_moran/4516938146

Romanesco broccoli
https://en.wikipedia.org/wiki/Romanesco_broccoli#/media/File:Romanesco_broccoli_(Brassica_oleracea).jpg

Видео The Newton Fractal Explained! | Deep Dive Maths канала Jeffrey Chasnov