Fractals, Chaos and the Mandelbrot Set Workshop - Pt. 3

This short explanation of fractals comes from the excellent book:
Fractal Programming in Turbo Pascal by R. T. Stevens
It is copyrighted by Henry Holt & Co., 115 West 18th St., N.Y., N.Y. 10011 and is used here with permission.

What Are Fractals?

When I tell people that I have been writing a book on fractals, they usually respond with two questions. The first is "What are fractals?" and the second is "What are fractals good for?" If I am feeling in an ornery mode, I respond to the first question with Mandelbrot's classical definition: "A fractal is a curve whose Hausdorff-Besicovitch dimension is larger than its Euclidian dimension." But more than this is really required in explaining fractals, so let's start at the beginning.

The Beginning of Fractal Curves

Draw a line on a sheet of paper. Euclidean geometry tells us that this is a figure of one dimension, namely length. Now extend the line. Make it wind around and around, back and forth, without crossing itself, until it fills the entire sheet of paper. Euclidean geometry says that this is still a line, a figure of one dimension. But our intuition strongly tells us that if the line completely fills the entire plane, it must be two-dimensional.

Such thinking started a revolution in mathematics about a hundred years ago. Mathematicians such as Cantor, von Koch, Peano, Hausdorff, and Besicovitch drew curves that were called "monsters," "psychotic," and "pathological" by traditional mathematicians. A new type of dimensioning was proposed, in which a curve could have a fractional dimension, not just an integer one. Recursive techniques and iterated expressions were found that could describe curves that have fractional dimensions. But without high speed digital computers, the actual drawing of such curves was a long and tedious process. So, little progress was made in this unusual field for nearly a hundred years.

The advent of digital computers made the investigation of such curves a fruitful field. From the early investigations, we could understand what we were trying to do. We wanted to draw curves that appeared to have more complex dimensional characteristics than were explained by traditional geometry. Computers were turned loose on very simple mathematical iterated expressions, in which the next state of a parameter depends solely on a simple relationship to the current state of the parameter. The iteration was performed many times and the resulting location of the parameter at each state was plotted. The resulting plots turned out to have many interesting characteristics. For one thing, They never repeated themselves. Furthermore, they tended to have the characteristic of self-similarity. In other words, if a small portion of the plot was enlarged, its shape was very much like a large portion of the original plot. Finally, the plots turned out to have shapes of great interest and extreme beauty.

The curves still didn't make much sense in terms of traditional mathematics, and consequently remained an anathema to traditional mathematicians. Dr. Benoit Mandelbrot was the first person to make use of a digital computer to investigate fractals in depth, and his results were not welcomed warmly by traditional mathematicians.

How Are Fractals Used?

Explaining the use of fractals is a little more difficult. Mandelbrot contends that just as the shapes of traditional geometry are the natural way of representing man-made objects (squares, circles, triangles, etc.), fractal curves are the natural way of representing objects and as a means of representing natural scenes. Moreover, fractals occur naturally in the expressions for mathematical phenomena as varied as the prediction of weather systems, the describing of turbulent flow of liquid, and the growth and decline of populations. Finally, fractals are useful in dimensional transformations that can be used for expressing and compressing graphical data. Ignoring the artistic value, the best answer to the question, "What are fractals good for?" is the reply "Fractals appear to provide solutions to many previously unanswered questions at the frontiers of the physical sciences." Consequently, to work at the frontiers of physical science, one needs to understand what fractals are and how to work with them.

Basic Considerations

Let's establish some points of orientation that will be useful in practical investigations of the chaotic field of fractals. First, intuition leads us to believe that fractal curves should have a dimensions greater than their traditional geometric dimension. Second, there is now sound mathematical grounding for accepting this premise. Third, fractal curves are associated with many physical and natural phenomena. Fourth, fractals often possess a rare and unusual beauty. No doubt, this is partly true because fractals correspond to the way in which nature produces those shapes that we are most familiar with and that basically define our ideas of "the beautiful." Finally, fractals have the unusual characteristic that they can be defined totally by relatively simple mathematical equations, yet they are not periodic. Thus the progression of the fractal curve differs widely if we start at just slightly different points in space, so unless we can measure where we are with absolute precision, we cannot be sure just what the progression of the curve will look like. This is in spite of the fact that the curve is defined through all of its wanderings by very simple iterated expressions. Most fractals are self-similar, so that the shape that we identify in the plot of a fractal curve repeats itself on a smaller and smaller scale as we enlarge the image further and further.