The Fractal Geometry of the Mandelbrot Set

II. How to Count and How to Add

Robert L. Devaney
Department of Mathematics
Boston University
Boston, MA 02215
e-mail: bob@math.bu.edu

In a previous paper in this series [7], we showed how one may read off dynamical information about orbits of the complex function

Fc(x) = x2 + c

from the geometry of the Mandelbrot set. In this paper we extend these ideas to show further relations between the dynamics of x2 + c and the Mandelbrot set. In particular, we use the Mandelbrot set as a vehicle to teach students how to "count" and how to "add."

The ideas in this paper arose from a series of experiments conducted by high school students in "chaos clubs" organized in the Boston public schools by Jonathan Choate, Mary Corkery, Beverly Mawn, and the author. The goal of these clubs was to expose young students to contemporary ideas in mathematics. Students discovered various facts about the Mandelbrot set using a combination of computer experiments and group projects. This paper presents a summary of some of the students' findings.

Since this paper was written, we have developed an interactive website called the Mandelbrot Set Explorer that allows anyone to perform the experiments described above. We heartily recommend using this site before settling down to reading this paper. Also, much of the material in this paper can be found in expanded fashion in the book [3], which has also appeared well after this site was developed.

It is a pleasure to thank Alex Kasman for his help in getting this and the other papers in this series up on the Web. A more detailed version of this paper may be found in [8]. For mathematical details, see [9].


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