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** 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

** F**_{c}(x) = x^{2} + c

from the
geometry of the Mandelbrot set. In this paper we extend these ideas to
show further relations between the dynamics of **x**^{2} + 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].

**1 Basic Definitions**
(Next Section)