Fractals are mathematical entities, but a lot more as well. The first examples of such objects were mathematical figures as the Cantor dust, the Koch curve (1904) and the Sierpinski triangle. Following those ones, which date from the late 19th and early 20th centuries, came the works of Gaston Julia and Pierre Fatou on Julia set fractals (1918-19), and some decades later the studies of Benoît Mandelbrot and other prolific scientists-matematicians on the Mandelbrot Set, strange attractors and bifurcations, among others. But fractals are everywhere. There are many "ordinary" objects that, because of their structure or behavior, are considered fractals in nature—although we don't usually recognize them as such. Clouds, mountains, coastlines, trees, and rivers are natural fractals. They differ from their mathematical counterparts for being finite entities instead of infinite ones. Other examples of fractals are the stock market and population growth.

Figure 16: Bifurcation.

Fractals have also crossed the border between science and art. Nowadays, many fractal artists produce gorgeous and skillfully worked-out representations of the mathematical counterparts. Fractal parameter (numerical) values can also be converted to sound notes in order to generate intriguing and fresh tunes. This has been termed as fractal music.

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fractal music sample.

More recently, experts have realized that fractals have been associated with art long before their mathematical evidence was established. For many centuries, man has used geometrical self-repeating patterns as decorative elements in vases, architecture, book illumination and many other representational arts that, in some ways, can be linked to fractal structures.

  
Figures 17, 18, 19
Hiberno-Saxon book illumination: detail of the Book of Kell (left);
Natural fractal structure: shell from a nautilus (center);
Gothic dome from the Ely Cathedral, UK (right).

Other studies have also demonstrated that many musical styles follow the ¹/_f  ratio associated with fractal frequencies in nature, as those found in noise interference and the flow of a river (Voss and Clark, 1975).

The Mandelbrot set was discovered by Benoît Mandelbrot, but named after him by Adrien Douady and J. Hubbard in 1982. The very peculiar figure has been reproduced in innumerable occasions since the first visual (computer) representation was rendered around 1980.

Figure 19: Colorful Mandelbrot Set.

The mathematical function that defines the Mandelbrot set can be expressed as the set of all c (c being a complex number) such that iterating z = z^2 + c (starting with z = 0) does not go to infinity. The equation in itself is very simple; the resulting graph, infinitely complex. A computer is the most practical tool we have to work with this fractal (as well as many others) due to its fast computational capabilities. If we were to try it by hand, we wouldn't be able to finish it in a lifetime.

Mandelbrot and Julia sets are closely related. The Mandelbrot set iterates z = z^2 + c with z starting at 0 and varying c. The Julia set, on the other hand, iterates the same function, but for fixed c and varying z values. Each point c in the Mandelbrot set specifies the geometric structure of the corresponding Julia set. If c is in the Mandelbrot set, then the Julia set will be connected. If not, the Julia set will be a collection of disconnected points plotted on a graph.

  
Figure 20: Mandelbrot-to-Julia transformation.