How can starting with a guess and using Isaac Newton's iterative solving method to find the roots of polynomial equations produce fractals?
3Blue1Brown's Grant Sanderson explains in the following 26 minute video. That may sound like a lot, but it is one of the more visually captivating maths explainers we've come across in quite a while, which really takes off at about the halfway point when chaotic boundaries emerge.
Newton's method is also known as the Newton-Raphson method, which is one of the tools that make it possible to come up with approximate numerical solutions to equations where direct, exact solution aren't an option. Here's a nice written introduction to how the method works, where step one involves making your best guess at the answer!
Update 16 October 2021: The sequel, Where Newton meets Mandelbrot (or more fancifully, “holomorphic dynamics”), is out!