Fractals – What are they? And, what is fractal mathematics?

Fractals are shapes or behaviors that have similar properties at all levels of magnification or across all times. Most fractals are generated by a relatively simple equation where the results are fed back into an equation until it grows larger than some specified boundary. Graphs of fractal equations use complex numbers, in which the X-axis is the real number line and the Y-axis represents the imaginary number line.

Benoit Mandelbrot* (1924-) coined the term "fractal" in 1963 to describe a class of complex, self-similar objects that emerge out of simple, recursive rules. His early work in the 1950's and 1960's suggested that variations in stock market prices, the probabilities of words in English, and fluctuations in turbulent fluids might be modeled by exotic processes.

* Although Mandelbrot is generally considered the father of fractals, it was Gaston Julia (1893-1978) in 1918 who published the paper, Mémoire sur L'itération des Fonctions Rationnelles, dealing with the iteration of a rational function f, fⁿ(z), which stays bounded as n approaches infinity. Julia’s work was prominent in the 1920s, but was essentially forgotten until the 1970’s, when Mandelbrot was able to graphically depict Julia sets.

There are a variety of different kinds of fractals (e.g., divergent, convergent, iterated function system fractals, L-system, JuliaBrot and strange attractors). Julia sets and the Mandelbrot sets are examples of divergent fractals. You can see examples of various types of fractals at http://www.hiddendimension.com/Mathematics_Main.html .

A key feature of fractals is self-similarity, which Hall (1995) defines as: “The self-similar nature of a fractal implies that its irregularities are preserved at arbitrarily fine levels of detail. Indeed, this property is germane to the definition of a fractal, and a curve or surface does not possess fractal qualities if the property is violated” (p. 800).

Mandelbrot reminds us that this idea of "recursive self similarity" was originally developed by the philosopher Leibniz, and popularized by the writer Johnathan Swift in 1733 with the following verse:

So, Nat'ralists observe, a Flea
Hath smaller Fleas that on him prey,
And these have smaller fleas to bit 'em,
And so proceed ad infinitum.

Benoit B. Mandelbrot:
1) http://www.fractalwisdom.com/FractalWisdom/fractal.html
2) http://www.edge.org/3rd_culture/bios/mandelbrot.html
3) The Theory of Roughness: http://www.edge.org/3rd_culture/mandelbrot04/mandelbrot04_index.html
4) Web page at Yale: http://www.math.yale.edu/public_html/People/Mandelbrot.html
5) A charming conversation with Professor Mandelbrot: http://www.exploratorium.edu/origins/antarctica/ramfiles/live/antarctica-011214.ram

Julia, G. (1918). Mémoire sur l'itération des fonctions rationnelles, Journal de Math. Pure et Appl. (1918), 47-245.