
Fractal GeometryFractal geometry is the geometry of chaos theory in the sense that fractal geometry may be used to visually illustrate the behavior of chaotic dynamical systems. But fractal geometry is much more than this. It was developed quite apart from chaos theory by a different cast of characters. Its "father" is indisputably Benoit Mandelbrot who coined the word "fractal" in 1975, the same year in which James Yorke gave chaos theory its name. Mandelbrot got part of his inspiration from the brilliant, but obscure, work of the French mathematician Gaston Julia who, about sixty years earlier, had developed a theory about iterating functions of complex variables. Julia could only begin to draw by hand some of the incredible images that would light up computer screens around the world in the 1980s and 90s, but Mandelbrot would ultimately bring the expertise of some of IBM's most brilliant computer programmers to bear on Julia's mathematics. Mandelbrot had a second source of inspiration from the nineteenth
century. Certain mischievous mathematicians of that era had grown weary
of the neverending touting of the calculus of Newton and Leibniz as the
beall and endall of mathematics. Do not misunderstand. It is impossible
to overstate the power that the differential and integral calculus placed
in the hands of 18th and 19th century physicists  it was the most revolutionary
event in the history of mathematics and science. To this day, the calculus
ranks with humankind's greatest intellectual achievements. But the view
that every problem in the real world could be solved by the methods of
calculus was wrong. A parade of 19th century mathematicians  Weierstrass,
Hilbert, Cantor, von Koch, Sierpinski, and others  produced fantastic
curves that were continuous everywhere, meaning no "gaps" or "breaks",
but differentiable nowhere, meaning essentially that these curves were
all "corners"  they were not "smooth" anywhere. (For a more detailed
account of two such curves, click here.) The standard curves of Euclidean geometry  lines,
parabolas, sine waves, and the like  are all both differentiable (smooth)
and continuous (unbroken), at least almost everywhere. Such curves hold
no secrets that cannot be divined by the power of the calculus. On the
other hand, curves which are everywhere continuous and nowhere differentiable
rise up in defiance of any attempt to be analyzed by Newton's handy set
of instructions. In the 19th century these curves were called "pathological"
and "monstrous" and mainstream mathematicians had better things to do
than to waste time dealing with bizarre constructions that had no relevance
to the real world anyway. In the 1970s, Mandelbrot had a different point
of view. He saw in these rough, jagged shapes the potential for
modeling natural objects and phenomena more accurately than Euclidean
geometry ever could. "Mountains are not cones, clouds are not spheres,
bark is not smooth, nor does lightning travel in a straight line," said
Mandelbrot. Far from being pathological, these were the shapes of everday
experience. So Mandelbrot named them "fractals" and wrote the seminal
work on fractal geometry called The
Fractal Geometry of Nature in which he introduced the concept
of fractal dimension, a way of characterizing the roughness of fractal
shapes. 
