An Introductory Study of Fractal Geometry
(excerpted from the printed manual)
Most people have probably seen the complex and often
beautiful images known as fractals. Their recent popularity
has made 'fractal' a buzzword in many circles, from
mathematicians and scientists to artists and computer
enthusiasts. Many books have been written on the subject,
but they were written primarily for high-level studies in
mathematics. This text and the companion computer programs
were prepared by a non-mathematician in an attempt to explain
and demonstrate, in simple terms, how some fractals are
generated. This is an informal introduction to fractal
geometry and is intended to provide a foundation for further
experimentation. We hope you will find it entertaining as
well as educational.
What is a fractal?
Many fractal images are very beautiful, almost magical,
and can stir the emotions. Because of their striking
appearance, we often think of them as works of art. We can
and often do appreciate them in this way, but they are much
more than art. They are not the result of 'electronic
Picassos' turned loose within a computer program. 'Fractal'
is not easily defined in layman's terms - the word 'fractal'
has its origin in the Latin word 'fractus', and loosely
translated means 'fragmented and irregular'. (see note) It is hoped
that the student will discover, through this text and the
exercises which accompany it, what fractals are.
The study of fractals is called fractal geometry.
Classical, or Euclidean geometry is familiar to most of us.
It deals with lines, polygons, circles, and other shapes and
objects. It has served us for centuries in the development
of science and technology, yet falls short of our needs when
we attempt to model many of the ordinary things around us.
Fractal geometry, on the other hand, is the 'geometry of
nature', and with it we can attempt to describe and mimic
nature in a way that was never before possible.
Fractal geometry was founded upon the work of many great
mathematicians of the last two centuries. Some of these early
works were considered radical, even dangerous, in their day.
The images they described were termed 'mathematical monsters'
because they did not fit into the classical tradition of
mathematics. A present-day mathematician, Benoit Mandelbrot,
has gathered these ideas together and developed a whole new
branch of mathematics - fractal geometry.
A Modern Tool
The
numerical methods used to produce fractals are not new, but
the ability to perform the computations and display the
results is made possible only by today's computers and
computer graphics. It might be reasonable to assume that the
mathematics involved in the study of fractals is abstract and
complicated, but you may be surprised at how incredibly
simple some of the methods can be. The difficulty in
producing fractal images lies not in the understanding of the
mathematics, but in the tedium of performing computations or
other operations thousands or millions of times. The
computer is well suited to this tedium.
A set
of computer programs is available that demonstrates the concepts that are
presented here. Most of the experiments involve use of these
programs.
Have fun!
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Copyright © 1995 Martin/EAL
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