The Mandelbrot Set

The Mandelbrot set is a type of infinitely complex mathematical object known as a fractal. No matter how much you zoom in, there is still more to see. Fractals are objects that display self-similarity at various scales. Magnifying a fractal reveals small-scale details similar to the large-scale characteristics. Although the Mandelbrot set is self-similar at magnified scales, the small scale details are not identical to the whole. In fact, the Mandelbrot set is infinitely complex. Yet the process of generating it is based on an extremely simple equation.




John Ewing has written:

Try these links:

Elementary introduction to the Mandelbrot Set.

Another introduction to the Mandelbrot Set.

A bit more advanced introduction.

Stunning images with large magnification.

An astonishing connection with pi!


Here's another fractal image. It summarizes the convergence of Newton's method (a technique one learns in elementary Calculus) applied to the equation z^3 + 1 = 0. The obvious root of this equation is -1, but there are two other roots in the set of complex numbers, which we will call w1 and w2. Points colored red converge to -1, those colored blue converge to w1, those colored green converge to w2. The shade of color shows the speed with which the point converges to the respective roots (lighter is faster). The infinite intracacy of the picture is an endless source of fascination.



This image was created with the computer algebra system Fermat written by Dr. Robert H. Lewis of Fordham.