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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 where discontinuous, with no tangent at a gap or an isolated point. Some 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 modelling nature.
Beginning in the early 1870s, though, a 50 - year crises transformed mathematical thinking. Weierstrass described a function that was continuous but nondifferentiable (no tangent could be described at any point). Cantor showed how simple, repeated procedure could turn a line into a dust of scattered points, Peano generated a convoluted curve that eventually touches every point on a place. 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 analyse 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 where 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 the Browian motion in fluids) found that traditional models could not 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 tantalising 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.
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