Newton's Method Fractals

Newton's method is a way to solve equations, invented by the famous Isaac Newton. When it works, it works quickly - and it usually works. It works equally well for equations in complex numbers. However it doesn't always work.

To make the movies on this page I applied Newton's method to the equation (z2+1)(z-a) for different complex numbers a. The colour depends on which of the three complex solution the method converged to, and how fast. As well as movies, you can also get posters of these fractal images.

I find it amazing that within these fractals you can find the familiar Mandelbrot set in all its glorious complexity!


Here's a poster showing what you see as you slowly zoom in on the Newton's method fractal.

If you prefer, here's the same pictures, with a different colour scheme.


Here's a few animations I've uploaded to Youtube. The first one shows what you see when you zoom in... the starting point here is chosen to be a/3.

In the next few, a changes with time, and the starting point depends on the position in the picture.