While holding down your mouse, draw a square around an area you want to zoom into. Then just let go ...don't click again. If you want to start again, just click once in the window without dragging. You can iterate about 10 times when very small areas are chosen. After that, the limitation of number resolution becomes apparent.

Java applet by Eckhard Roessel - freeware

Following explanation is taken from The Mandelbrot Set:

It is a somewhat 'heady' explanation. Read my book compilations in this category to get a clearer picture on fractals and chaos in general.

What is the Mandelbrot set? It's the the set of all complex numbers z for which sequence defined by the iteration

z(0) = z, z(n+1) = z(n)*z(n) + z, n=0,1,2, ... (1)

remains bounded. This means that there is a number B such that the absolute value of all iterates z(n) never gets larger than B. A bounded sequence may or not have a limit. For example, if z=0 then z(n) = 0 for all n, so that the limit of the (1) is zero. On the other hand, if z=i ( i being the imaginary unit), then the sequence oscillates between i and i-1, so remains bounded but it does not converge to a limit.

One of the most widely read basic introductions to the Mandelbrot set and other fractals is by H.-O.Peitgen and P.H.Richter, The Beauty of Fractals, Springer Verlag, 1986, ISBN 0-387-15851-0. It contains many spectacular pictures.

Mathematician, born in 1924 in Warsaw

Benoit Mandelbrot, a matematician born in Warsaw 1924, is largely responsible for the present interest in Fractal Geometry. He showed how Fractals can occur in many different places in both Mathematics and elsewhere in Nature.

His website introduces him as follows:

Seeks a measure of order in physical, mathematical or social phenomena that are characterized by abundant data but extreme sample variability. The surprising esthetic value of many of his discoveries and their unexpected usefulness in teaching have made him an eloquent spokesman for the "unity of knowing and feeling".

Mandelbrot's website can be found here.

Excellent portal on fractals containing software applets, articles, etc.
Teachers organisation

Portal containing mainly Mandelbrot software
James Madison University

Nice online applet with extensive documentation
University of Utah