8.1.1 Before Mandelbrot
Like new forms of life, new branches of mathematics and science don't
appear from nowhere. The ideas of fractal geometry can be traced to the
late nineteenth century, when mathematicians created shapes -- sets of
points -- that seemed to have no counterpart in nature. By a wonderful
irony, the "abstract" mathematics descended from that work has now
turned out to be MORE appropriate than any other for describing many
natural shapes and processes.
Perhaps we shouldn't be surprised. The Greek geometers worked out the
mathematics of the conic sections for its formal beauty; it was two
thousand years before Copernicus and Brahe, Kepler and Newton overcame
the preconception that all heavenly motions must be circular, and found
the ellipse, parabola, and hyperbola in the paths of planets, comets,
and projectiles.
In the 17th century Newton and Leibniz created calculus, with its
techniques for "differentiating" or finding the derivative of functions
-- in geometric terms, finding the tangent of a curve at any given
point. True, some functions were discontinuous, with no tangent at a
gap or an isolated point. Some had singularities: abrupt changes in
direction at which the idea of a tangent becomes meaningless. But these
were seen as exceptional, and attention was focused on the "well-
behaved" functions that worked well in modeling nature.
Beginning in the early 1870s, though, a 50-year crisis transformed
mathematical thinking. Weierstrass described a function that was
continuous but nondifferentiable -- no tangent could be described at any
point. Cantor showed how a simple, repeated procedure could turn a line
into a dust of scattered points, and Peano generated a convoluted curve
that eventually touches every point on a plane. These shapes seemed to
fall "between" the usual categories of one-dimensional lines, two-
dimensional planes and three-dimensional volumes. Most still saw them as
"pathological" cases, but here and there they began to find
applications.
In other areas of mathematics, too, strange shapes began to crop up.
Poincare attempted to analyze the stability of the solar system in the
1880s and found that the many-body dynamical problem resisted
traditional methods. Instead, he developed a qualitative approach, a
"state space" in which each point represented a different planetary
orbit, and studied what we would now call the topology -- the
"connectedness" -- of whole families of orbits. This approach revealed
that while many initial motions quickly settled into the familiar
curves, there were also strange, "chaotic" orbits that never became
periodic and predictable.
Other investigators trying to understand fluctuating, "noisy" phenomena
-- the flooding of the Nile, price series in economics, the jiggling of
molecules in Brownian motion in fluids -- found that traditional models
could not match the data. They had to introduce apparently arbitrary
scaling features, with spikes in the data becoming rarer as they grew
larger, but never disappearing entirely.
For many years these developments seemed unrelated, but there were
tantalizing hints of a common thread. Like the pure mathematicians'
curves and the chaotic orbital motions, the graphs of irregular time
series often had the property of self-similarity: a magnified small
section looked very similar to a large one over a wide range of scales.