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