Mandelbrot fractals

A loud coloring of points outside the Mandelbrot set.

The Mandelbrot set is a set of points that fail to escape under an iterated point process. Readers may want to review the complex arithmetic rules in the articles on Newton's method fractal. To decide if a complex point z=x+yi is in the Mandelbrot set, generate the following sequence of complex points. The first is z itself. The next point is the square of the current point plus z. The first part of this sequence is thus:

The Mandelbrot sequence for a point z.

If a point, under the operation of this iterated squaring, gets more than a distance of 2 from 0+0i then it is not in the Mandelbrot set. Click on the example above and you will see that this set of non-escaping points has a very complex shape. Later we will show that there is added complexity visible as we zoom in at each level of the set. New features appear forever.

The set above is colored using a rainbow palette and choosing the color by the number of iterations required for the moving point to escape. This is a fairly standard algorithm for coloring the points not in the Mandelbrot set. The flat, cosine, and clarity methods from the Newton's method article can also be used.