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What Is a Fractal?

And who is this guy Mandelbrot?

Images and text by Alan Beck

The word "fractal" was coined less than twenty years ago by one of history's most creative mathematicians, Benoit Mandelbrot, whose seminal work, The Fractal Geometry of Nature, first introduced and explained concepts underlying this new vision. Although prior mathematical thinkers like Cantor, Hausdorff, Julia, Koch, Peano, Poincare, Richardson, Sierpinski, Weierstrass and others had attained isolated insights of fractal understanding, such ideas were largely ignored until Mandelbrot's genius forged them at a single blow into a gorgeously coherent and fruitful discipline.

Lamp (63 k / jpg)

Mandelbrot derived the term "fractal" from the Latin verb frangere, meaning to break or fragment. Basically, a fractal is any pattern that reveals greater complexity as it is enlarged. Thus, fractals graphically portray the notion of "worlds within worlds" which has obsessed Western culture from its tenth-century beginnings.

Traditional Euclidean patterns appear simpler as they are magnified; as you home in on one area, the shape looks more and more like a straight line. In the language of calculus such curves are differentiable. The trajectory of an artillery shell is a classic example. But fractals, like dendritic branches of lightning or bumps of broccoli, are not differentiable: the closer you come, the more detail you see. Infinity is implicit and invisible in the computations of calculus but explicit and graphically manifest in fractals.

Serpentes (97 k / jpg)

Whether generated by computers or natural processes, all fractals are spun from what scientists call a "positive feedback loop." Something--data or matter--goes in one "end," undergoes a given, often very slight, modification and comes out the other. Fractals are produced when the output is fed back into the system as input again and again.

LarvasLeaf (105 k/ jpg)

Fractals show us that the simplest engines of change often produce exquisitely elaborate patterns. Such systems are at work all around us, from the stock market to the stars. And to the fractal artist, Mandelbrot's insights echo Kandinsky's assertion that "the process of creation is the same in art and nature."