THE FRACTAL GEOMETRY OF THE MANDELBROT SET 2: HOW TO COUNT AND HOW TO ADD

Abstract

In a previous paper in this series,1 we showed how one may read off dynamical information about orbits of 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 show that Mandelbrot set may be used as a vehicle to show students how to “count” and how to “add” geometrically. 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.