Speaking loosely without technical terms such as the Hausdorff-Besicovitch dimension, a fractal 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, blood vessels and mycelium strands. Benoit Mandelbrot coined the term "fractal" in 1975 and published a book entitled "The Fractal Geometry of Nature" in 1982.
The idea of fractal was not particularly new for the computer age as some mathematicians like Cantor already conceived it in the 19th century. About a hundred years ago, a group of mathematicians represented by Pierre Fatou and Gaston Julia studied certain fractals generated by so-called "dynamical systems," and about fifty years later Mandelbrot plotted the "Mandelbrot set" using a computer and a simple dynamical system. The novelty, beauty and complexity of the computer-generated fractal reinvigorated and stimulated mathematicians to develop further theories in 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 condition that might cause a hugely unexpected outcome or the "butterfly effect" had been observed earlier by some mathematicians and physicists. "Chaos" received great sensations when 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. It is now known that fractals together with chaos provide not only mathematical insights into natural objects and phenomena but also surprisingly many applications in sciences, mathematics and art.
Google we can find lots of Internet fractal galleries, many of which display stunningly beautiful computer-generated fractal art images. This indicates that a 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 tied to chaos. Because of chaos involved, a slight change in input values (intentional or accidental) may greatly affect the pattern of the fractal output 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 so as not to be 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 certainly takes intelligence and patience to get successful outcomes. The experience may at some point start to stir the participants' deep imaginations in the part of mathematics that is in fact quite profound. Best of all, though, it is plain fun.