Speaking loosely without technical terms such as the Hausdorff-Besicovitch dimension and topological dimension of a set of points, a fractal, coined by Benoit Mandelbrot, is a geometric shape that is self-similar, i.e., a large part of it contains a great many smaller parts that resemble the large part in some way; see Figure 1 below.
Nature is filled with fractals as seen in such objects as mountains, shorelines, trees (branches, barks and roots), ferns, fluid flow patterns, cloud formations (with or without lightning), blood vessels and mycelium strands. The idea of fractal was already conceived by some mathematicians like Cantor in the 19th century, and about a hundred years ago, a group of mathematicians represented by Fatou and Julia studied certain fractals generated by the so-called “dynamical systems.” So, the fractals are an old subject which did not particularly excite mathematicians for about fifty years until Mandelbrot showed the Mandelbrot set plotted by a computer using some of the theories developed earlier by Fatou and Julia. Stimulated by the novelty, beauty and complexity of the Mandelbrot set, some of the mathematicians reinvigorated the subject and developed deeper and more extensive theories that contributed to fractal geometry.
On the other hand, “chaos,” often associated with fractals, was basically born as a brand new subject in 1974 from biologist Robert May’s computer simulations of population dynamics through the dynamical system called “logistic equation,” although some chaos-related phenomena such as the sensitivity to a small change in the initial situation that might cause a hugely unexpected outcome or the “butterfly effect” had been observed earlier by some mathematicians and physicists. “Chaos” was then welcomed with great sensations after American Mathematical Monthly published “Period Three Implies Chaos” by T.Y. Li and James Yorke in 1975. Younger mathematicians were especially excited to see the fact that there appeared very little difference between chaotic and random outcomes even though the former resulted from the deterministic process through the logistic equation.
Now, fractals provide not only mathematical insights for natural objects and phenomena that are beyond the reach of Euclidean geometry but also numerous applications in surprisingly many areas in sciences, mathematics and art especially when it is tied with chaos.
Yahoo!, we can find thousands of Internet fractal galleries, many of which display stunningly beautiful computer-generated fractal art images. This indicates that a very large population not only appreciates the digital art form but also participates in the eye-opening creative activity. Written below is a guide on how to program a computer and plot popular types of fractals that are closely tied to chaos. Because of the close association to chaos, a slight change in input values (intentional or accidental) may greatly affect the pattern of the output fractal and this unpredictability provides an added excitement while running the computer program. The subject is kept less formal and mathematical with light topics added here and there in an attempt to avoid it to become cut-and-dried. The audience, who may be merely intrigued by the general idea behind fractal plotting, is encouraged to try it. Although the basic process is not based on higher mathematics, it certain takes intelligence and patience to be successful in the outcomes. Therefore, the experience may at some point start to stir your deep imaginations in the part of mathematics that is quite profound and involved. Best of all, though, it is plain fun and worth the effort.