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** The Fractal Geometry of the Mandelbrot Set**

** I. The Periods of the Bulbs**

Robert L. Devaney

Department of Mathematics

Boston University

Boston, MA 02215 USA

One of the most intricate and beautiful images in all of mathematics is the Mandelbrot set, discovered by Benoit Mandelbrot in 1980. Most people within the mathematics community, and many people outside of the discipline, have seen this image and have marveled at its geometric intricacy. Unfortunately, only a few of these people are acquainted with the equally beautiful mathematics that lurks behind this image.

In this document we present a few of these ideas in an elementary setting. All of these ideas were presented to high school students who participated in a ``Chaos Club'' organized by Jonathan Choate, Mary Corkery, Beverly Mawn, and the author at Boston Technical High School during the 1991-93 academic years. The goal of the club was to introduce inner city high school students to some of the beauty and excitement of contemporary mathematics. This paper describes some of the computer experiments that were performed by the participating students. In a later paper [7] we describe some of the elementary geometry and number theory that students may discover by viewing this set.

It is a pleasure to thank Alex Kasman for his assistance in getting this paper up on the Web and for numerous insightful comments which greatly improved the presentation.

- 1 Iteration
- 2 The Mandelbrot Set
- 3 Periods of the Bulbs
- 4 Julia Sets
- 5 The fundamental dichotomy
- 6 Back to M
- 7 Classroom Activities
- 8 Summary
- References

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Prof. Robert L. Devaney (Boston University)