All of the facts discussed in this paper may be discovered experimentally by students using a personal computer. The Mandelbrot Set Explorer is a tool designed to accomplish this. While proofs of many of these facts are well beyond the typical high school or undergraduate student, we have nevertheless found that these concepts serve to pique the students' interest in mathematics. Students first marvel that the counting or adding phenomenon is true. Soon they begin to wonder why it is true. It is our feeling that this is the beginning of a wonderful mathematical experience for the students, one that is very rarely enjoyed in the standard curriculum.