In this lecture we describe via a combination of computer experiments, animations, and mathematical techniques the beautiful mathematics that lies behind the Mandelbrot set. While many people have seen the intricate and beautiful patterns that form the boundary of this set, few realize that each of these patterns has its own specific mathematical meaning and that mathematicians can understand this complexity in full detail. The only prerequisites for this talk is a working knowledge of complex numbers. Toward the end of the talk, things will get pretty "advanced" as we will show you the "real" way to count and to add.