Fractal Theory: Introduction

What is the Mandelbrot set? A mathematician might say it was the locus of points, C, for which the series Zn+1 = Zn * Zn + C, Z0 = (0,0) is bounded by a circle of radius two, centered on the origin. For most of us who aren't mathematicians, the Mandelbrot set is:

One of the fascinating things about the Mandelbrot set is the seeming contradiction in it. It is said to be the most complex object in mathematics, perhaps the most complex object ever seen. But at the same time, it is generated by an almost absurdly simple formula. Multiply Z by itself. Add C. The answer is the new value for Z. Repeat until the absolute value of Z is greater than two, or until our counter expires. If Z exceeds two, then the point is not in the Mandelbrot set. If it doesn't exceed two after a large number of iterations, then it is assumed to be in the Mandelbrot set.