Fract-ED from EALSoft

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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