Supplemental Information

The word 'fractal' was coined by Benoit B. Mandelbrot. It was his desire to give new names to the new concepts found in fractal geometry, rather than reuse old terms and create confusion. It is also he who has declared fractal geometry the 'geometry of nature'.


          Mandelbrot gives us his definition for a fractal:
          a set for which the Hausdorff Besicovitch dimension
          strictly exceeds the topological dimension. 1

     It can further be stated that "every set with a
          noninteger [dimension] is a fractal". 1

In Euclidean geometry we think of dimensions having integral values. Points, lines, polygons, and polyhedra have topological dimensions of 0, 1, 2, and 3, respectively. Fractals, however, generally have non- integer dimensions, even though they may be entirely represented on a two-dimensional medium. 'Fractal dimension' will be mentioned in a later study, but a thorough discussion of it is beyond the scope of this text.

Another visual phenomenon . . .

Fractal designs are appearing in commercial products. Notebook covers, pencils, T-shirts and other products can be found bearing a variety of colorful fractal patterns. There is another visual phenomenon which is even more popular than fractals. This form of art has been given many names, but is most commonly called random dot stereography.

Random dot stereograms (RDSs) are those curious pictures of seemingly chaotic dots or patterns containing a three-dimensional image which can only be seen when the image is stared at in cross-eyed fashion. Some people may confuse the RDS with a fractal; however, it is not. An RDS is produced by an entirely different and unrelated process.

Briefly, an RDS can be produced by first generating a vertical stripe consisting of a random array of dots or other patterns. This stripe is repeated horizontally a number of times. A three-dimensional image is then superimposed on this by shifting certain dots within the outline of the image. Because of the initial randomness, the superimposed image is discernible only through conscious effort. With practice, one can easily focus on the image, however, it is believed that a small percentage of the population is unable to 'see' an RDS image.

The stereo effect is similar to that produced by stereo glasses, in which two images differing only by a slight change in viewing angle are viewed through a pair of special glasses resembling binoculars.

Thomas Malthus and the Equation

Some have expressed confusion as to why Malthus is even mentioned in this text. He knew nothing of fractals. He did not express his ideas as an equation - that was done by others much later. I only wished to point out that fractals can be derived from something once believed to be far removed from mathematics - nature.

Thomas Malthus was not a mathematician, but an economist and preacher, among other things. He was prominent in his day, and his famous "Essay on Population" was controversial and hotly debated. His writings on the subject were first published in 1798 and influenced England's social laws throughout the 19th century.

The "Malthusian Theory" encompasses many ideas on the subject of population growth as it relates to economics, social reforms, and the poor. Malthus proposed that population is dependent on many social factors - the economic outlook, food supply, disease, etc, and that under natural conditions the population would stabilize. He believed that legislative tampering, such as welfare programs for the poor, family planning, disease control, etc, would upset the natural checks on population, allowing the population to increase beyond the capacity of the planet to sustain it. This idea is expressed in the population equation,

                y = x + k * x * (1 - x)

where x is the present generation, y is the future generation, and k is the combined effect of all the factors which influence growth. There should be (and is) a value for k in which the population reaches a point of equilibrium, that is, where no growth or decline occurs. Note that the equation above is not THE population equation, but is an example of one. Many similar equations are used in population studies; this happens to be one chosen for this text.

By the early 20th century, it was generally believed that Malthus was wrong. He did not foresee the increases in productivity caused by the Industrial Revolution, which enabled population growth and economic expansion that would not otherwise have been possible. Today, however, the Malthusian Theory is being reconsidered by some. Population has become a major global issue, as evidenced by the international population conference of September 1994.

Suggestion: It may be helpful to work through the solution to this equation and plot a number of points on a graph. Also, show that the equation can be written in another perhaps more familiar form, as in

                 y = x + kx - kx2

It is fitting that a population equation be used to study iteration, since population growth is itself an iterative process - each generation is derived from the preceding one. Malthus could not have conceived that a product of his theory would one day be looked upon in such an entirely different context. The strange behavior of the equations which produce the bifurcation diagram was not discovered until 1971 by Robert May. 3 It has since become a popular model for studies of chaos and fractals.

The teacher and students are encouraged to perform experiment 2.1. Using a calculator may at first seem easy, but it doesn't take long to realize what a mess it can become. This experiment was designed for two reasons: to develop a better understanding of iteration by mechanically working through the process, and to gain an appreciation for the power of the computer.

Suggestion: Compare the results from different students or groups who used a different number of significant figures in their calculations. Point out the difference in error (if any) that can accrue due to this difference of precision.

One main reason why fractals weren't "discovered" before computers is because of the tediousness of using mechanical methods. Few would bother to do all that work, especially when there was no reason to believe it would lead anywhere. The computer enables one to experiment with little effort.
When the students attempt to identify the attractors using IT, challenge them to find values for k which have no attractor. Assume that an attractor would be evident after at least 1000 iterations.

Additional exercise: when a K-value with no attractor is identified, plot the y-values on a line. Note: for some values there may be definite ranges in which all values fall, though no two values will be the same. For the time being, just have the students take note of the fact - it will be become more significant later. (see Strange Attractor, later in this text)

For maximum benefit, the students should be encouraged to follow the steps of experiment 2.3 exactly as written. Jumping ahead into more complex bifurcation displays may spoil some of the fun. The intent of step 3 is to see if one can predict the future behavior of the function based on what was seen in the first five iterations. In the author's opinion, the behavior becomes less and less predictable as more iterations are performed.
Experiment 2.4, like the one before it, was intended to help correlate the numerical data from IT with the graphic images of BIF. BIF would simply produce pretty pictures, not necessarily with understanding, without this correlation. It may be necessary to spend extra time to relate these two representations of the same data.
The real fun with BIF comes with zooming. The concept should be easily understood. It can be compared to a camera's zoom lens or a magnifying glass. Most students are probably already familiar with the term.


          Bifurcation: a fork, or split, as in a branch

Bifurcation is a fairly common mechanism. Going back to our earlier discussion about population, bifurcation is exactly what happens when we examine a person's genealogy - the family tree splits with each generation of ancestry. It is also conceivable that in generations past branches of the family tree had been crossed, producing a chaotic intermixing of ancestry much like the chaos region of the bifurcation diagrams.

Chaos itself comprises an entire field of study, and is applied to the analysis of many phenomena, such as fluid flow, aerodynamics, and weather. Anyone who has seen the popular movie "Jurassic Park" has heard of chaos and the famous "butterfly effect", whereby the flapping of a butterfly's wings could ultimately cause a thunderstorm on another continent.

Self-similarity, the "order within chaos", is one of the most striking characteristics of the bifurcation diagrams and many other fractals. It is a characteristic we will see again and again in further fractal studies.

Additional exercise: have the students 'explore' and identify self-similar regions not mentioned elsewhere and report their zoom coordinates.

Atoms resemble solar systems? I'll probably get pounced on by a physicist for that statement! I know the atomic model has changed since I was a kid, but this traditional view of the atom is still held by many, and it is the first picture that comes to my mind when I think of an atom. My only argument is - nobody really knows what an atom looks like.