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:
If the entire Mandelbrot set were placed on an ordinary sheet of paper, the tiny sections of boundary we examine would not fill the width of a hydrogen atom. Physicists think about such tiny objects; only mathematicians have microscopes fine enough to actually observe them. -- "Can We See the Mandelbrot Set?", The College Mathematics Journal, v. 26, no. 2, March 1995.
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 intricacy of the picture is an endless source of fascination.