The Fractal Microscope

 

A Distributed Computing Approach to Mathematics in Education

The Fractal Microscope is an interactive tool designed by the Education Group at the National Center for Supercomputing Applications (NCSA) for exploring the Mandelbrot set and other fractal patterns. By combining supercomputing and networks with the simple interface of a Macintosh or X-Windows workstation, students and teachers from all grade levels can engage in discovery-based exploration. The program is designed to run in conjunction with NCSA imaging tools such as DataScope and Collage. With this program students can enjoy the art of mathematics as they master the science of mathematics. This focus can help one address a wide variety of topics in the K-12 curriculum including scientific notation, coordinate systems and graphing, number systems, convergence, divergence, and self-similarity.

 

Why Fractals?

Many people are immediately drawn to the bizarrely beautiful images known as fractals. Extending beyond the typical perception of mathematics as a body of sterile formulas, fractal geometry mixes art with mathematics to demonstrate that equations are more than just a collection of numbers. With fractal geometry we can visually model much of what we witness in nature, the most recognized being coastlines and mountains. Fractals are used to model soil erosion and to analyze seismic patterns as well. But beyond potential applications for describing complex natural patterns, with their visual beauty fractals can help alter students' beliefs that mathematics is dry and inaccessible and may help to motivate mathematical discovery in the classroom.

A popular representation of fractal geometry lies within the Mandelbrot set, named after its creator Benoit B. Mandelbrot who coined the name "fractal" in 1975 from the Latin fractus or "to break" (Jürgens et al., 1990). The Mandelbrot set (figure 1) is the set of all points that remain bounded for every iteration of z = z*z + c on the complex plane, where the initial value of z is 0 and c is a constant (Jürgens et al., 1992).

Figure 1. The Mandelbrot set visualized and shaded in blue.

But we can appreciate the beauty of the fractals encompassed in the Mandelbrot set without the specific mathematics behind it. With the help of an NCSA supercomputer and two programs written by Michael South and Dr. Robert M. Panoff working with the Education Group at NCSA, it is possible to explore many common elementary mathematical principles while examining the Mandelbrot set. In fact, some students from Wiley Elementary School in Urbana, Illinois have done just that. One program, the Fractal Microscope, allows anyone to zoom in and out of the Mandelbrot set quickly (in a few seconds, as opposed to a few hours with most home computers) and easily by simply pointing and clicking within the Macintosh environment. The other program, Starstruck, visualizes the path produced through the Mandlebrot set by each iteration.

As an independent project, Rennes University in France has set up a gallery of fractal images.

 

Fractals in the Classroom

With this "fractal microscope" students can go anywhere in the set they wish while sitting in the classroom. The natural beauty of the fractal gives students incentive to explore coordinate systems, counting schemes, pattern development, integer arithmetic, the concept of infinity, and other topics in the mathematics and science curriculum.

 

Why Supercomputers?

There are definitely uses for fractals within the classroom, such as introducing similarity (although the Mandelbrot set is only quasi self-similar), density, infinity, vector addition, division and reduction of fractions, scale and magnification, and pattern discovery. The obstacle for most teachers is time. Programs for fractal generation that run on personal computers take as long as a class period (or even overnight!) to generate pieces of the Mandelbrot set when the zoom factor exceeds a thousand or so. Many of the exciting aspects of the structure of fractals only appear at much greater magnifications. By accessing supercomputing resources over the Internet, speed increases of 500-1000 times have been realized.

 

Bibliography

Consult the bibliography for references and more information.

 


Copyright 1993, University of Illinois Board of Trustees
National Center for Supercomputing Applications, Education Group
rpanoff@ncsa.uiuc.edu and jgasaway@ncsa.uiuc.edu