# 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