Foreword by Lynn A. Steen
In 1980 Ronald Reagan was elected president of the
United States, IBM began an urgent program to develop
a personal computer for which Microsoft agreed to provide
the operating system, and Benoit Mandelbrot got
his first real look at the archetypal fractal, the eponymous
Mandelbrot set. One hundred years from now,
which of these events is more likely to be remembered
as having had the greatest influence on science and human
affairs?
At this moment, two decades later, personal computers
seem to be well in the lead. PCs are on every
desk, all now connected by the world-linking Internet.
The personal computer has truly transformed the way
the world works. No invention since the printing press
has created such a widespread impact and no human activity
has ever changed society so quickly.
However, when we examine Internet patterns in detail,
guess what we find just beneath the surface? The
footprints of fractals. These wondrous geometric objects,
discovered by Mandelbrot just over twenty years
ago, turn out to be the key to understanding the frenetic
behavior of signals linking the world's computers, as
well as the means to efficient compression that makes
it possible to transmit images over the Internet. Without
fractals, engineers would never have been able to make
the World Wide Web work as well as it does. And without
the Web, PCs would be just one more labor-saving
appliance.
Computers are important instrumentally; they provide
tools that enable us to work more efficiently, to
see patterns previously hidden, and to organize information
in new and revealing ways. In contrast, fractals
are important fundamentally; they provide elements of
a totally new geometry that offer a profoundly different
way to understand nature. In the long run, this new
understanding of nature will count for far more than momentary
advances in technology or politics.
The predominant Western view of the relation between
mathematics and nature is a legacy of Plato's distinction
between a world of ideals and a world of actualities.
Mathematics, in this view, belongs to the ethereal
world of ideals; nature, being earthly rather than
heavenly, belongs to a world of actualities that is both
imperfect and incomplete. Thus, reality is best understood
by approximation in terms of ideal mathematical
models.
Our most important inheritance from this tradition
is Euclidean geometry, the axiomatic study of lines, circles,
and triangles that form an ideal (and therefore
approximate) basis for understanding geography, mechanics,
astronomy, and everything real. From this perspective,
nature is like noisy mathematics--rough and
crumpled, slightly out of focus.
Fractals create an alternative to Euclidean geometry
whose elements are not lines and circles but iterations
and self-similarities, whose surfaces are not smooth but
jagged, whose features are not perfect but broken. Derived
from apparent pathologies that puzzled or affronted
traditional mathematicians, fractals reveal an entirely
new geometry that enables us to understand formerly inexplicable
real-world phenomena.
Fractals provide insight into the distribution of galaxies,
the spread of bacterial colonies, the grammar of
DNA, the shape of coastlines, changes in climate, development
of hurricanes, growth of crystals, percolation
of ground pollutants, turbulence of fluids, and the path
of lightning. They have been employed to create powerful
antennas, to develop fiber optics, to monitor financial
data, to compress images, and to produce artificial landscapes.
Their influence has been felt in art, architecture,
drama (Arcadia), film (Jurassic Park), music, and poetry.
Fractals have even penetrated the inner sanctum of
elite culture--New Yorker cartoons.
For many, this extraordinary utility would be a sufficient
warrant for fractals to be awarded a prominent
role in mathematics education. But reasons other than
utility can also be advanced, and it is these reasons,
not utility, that form the major thrust of this volume.
ix
M.L. Frame & B.B. Mandelbrot
Washington D.C. Mathematical Association of America, 2002
Fractals, Graphics, & Mathematics Education
x Foreword
Simply put, fractals enable everyone to enjoy mathematics.
Nothing else can make such a striking--and
important--claim.
Arguments about strategies for improving mathematics
education roil state and local school politics. Some
urge strict exam-enforced standards; others advocate
more inviting contexts for learning. Some promote traditional
curricula; others support enhanced or integrated
programs. Rarely do the protagonists in these "math
wars" stop to ask whether different mathematics might
yield increased learning.
But that is precisely the argument advanced by the
authors represented in this volume. They focus on teaching
fractals, not primarily because fractals are important
but because learning about fractals is, as one student put
it, "indescribably exciting . . . and uniquely intriguing."
It is easy to see why:
* The first steps are so much fun. Exploring fractals
creates unprecedented enthusiasm for discovery
learning among both teachers and students.
* Fractals are beautiful. Stunning visuals appeal to
the mind's eye and create contagious demand for
continued exploration.
* Anyone can play. Exploration of fractal geometry
appeals to students of every age, from primary
school through college and beyond.
* Fractals promote curiosity. Simple rules, easily
modified, create nearly uncontrollable temptations
to explore different options to see what surprising
patterns will emerge.
* Simple ideas lead to unexpected complexity. Fractals
are more life-like than objects studied in other
parts of mathematics; thus they appeal to many
students who find traditional mathematics cold
and austere.
* Many easy problems remain unsolved. Fractals
are rich in open conjectures that lead to deep mathematics.
Moreover, the distance from elementary
steps to unsolved problems is very short.
* Careful inspection yields immediate rewards. Insight
and conjectures arise readily when our welldeveloped
visual intuition is applied to fractal
images. In studying fractals, children can see and
conjecture as well as adults.
* Computers enhance learning. The visual impact
of computer graphics makes fractal images unforgettable,
while the unforgiving logical demands
of computer programs yield important lessons in
the value of rigorous thinking.
The history of mathematics education is long and
convoluted, reflecting both the changing nature of mathematics
and the evolving demands of society. Although
in the eighteenth and nineteenth centuries mathematics
was both experimental and theoretical, during much of
the twentieth century the theoretical aspect has dominated.
Much of mathematics education followed this
trend towards theory and abstraction, leading to alarming
reports of rising mathematical illiteracy not only in
the United States but in many other countries as well.
Fractals represent a rebirth of experimental
mathematics, enabled by computers and enhanced by
powerful evidence of utility. In the ebb and flow of
mathematical fashion, the struggle between theoretical
and experimental is once again more nearly in balance.
What remains is the challenge of restoring this balance
to mathematics education. It is to that important task
that this book is devoted.
Lynn Arthur Steen
St. Olaf College
Northfield, MN
April 13, 2000.