Beautiful Fractal Pictures from Newton's Method

In the last section, we drew a map that colored each initial guess according to the root it eventually arrives at. For the real axis this map is relatively simple – but even still can have different regions mixed together. When we extend the “map of guesses” into the complex plane, something really surprising happens.

Let's examine the function

On the complex plane, this has four zeros, at each fourth root of 1: {1, -1, i, -i}.

Now let's take at each point in the complex plane, and color it by the root it eventually winds up at: purple for 1, red for i, green for -1, and blue for -i. The region covered by each color is the basin of attraction for the corresponding root. You might think that we'd get something like this, with simple basins of attraction separated by a crisp and well-defined boundary:

Not the map of guesses for z⁴-1=0

But you'd be wrong! In fact, it looks like this:

The Map of Newton's Method for z⁴-1=0
(Colored by eventual root, with brightness according to
how many iterations were needed to arrive at the root)

In fact the boundary between basins of attraction has a surprising and beautifully complex structure. This “hairyness” – the petals that bud off and split into smaller and smaller petals – occurs in a self-similar fashion at whatever large or small scale you choose. The boundary never becomes simple, no matter how close or far you look; and the structures you see at this scale are reproduced in a self-similar manner infinitely many times.

The boundary of all of the basins of attraction – the points that lie in between the red, blue, green and purple regions – form a type of fractal known as a Julia set fractal. An interesting fact about Julia Sets is that every point of the boundary touches all four basins of attraction. Notice that, in the wrong picture above, all four regions touch at the point x=0; at every other boundary point, only two regions touch. In the correct picture, every single point of the boundary is a star point like x=0: all four regions come together. This is a very bizarre and interesting property, and it leads directly to the fractal and self-similar structure of the picture.