Chaos - James Gleick [44]
The paradoxical qualities of such constructions disturbed nineteenth-century mathematicians, but Mandelbrot saw the Cantor set as a model for the occurrence of errors in an electronic transmission line. Engineers saw periods of error-free transmission, mixed with periods when errors would come in bursts. Looked at more closely, the bursts, too, contained error-free periods within them. And so on—it was an example of fractal time. At every time scale, from hours to seconds, Mandelbrot discovered that the relationship of errors to clean transmission remained constant. Such dusts, he contended, are indispensable in modeling intermittency.
The Joseph Effect means persistence. There came seven years of great plenty throughout the land of Egypt. And there shall arise after them seven years of famine. If the Biblical legend meant to imply periodicity, it was oversimplified, of course. But floods and droughts do persist. Despite an underlying randomness, the longer a place has suffered drought, the likelier it is to suffer more. Furthermore, mathematical analysis of the Nile’s height showed that persistence applied over centuries as well as over decades. The Noah and Joseph Effects push in different directions, but they add up to this: trends in nature are real, but they can vanish as quickly as they come.
Discontinuity, bursts of noise, Cantor dusts—phenomena like these had no place in the geometries of the past two thousand years. The shapes of classical geometry are lines and planes, circles and spheres, triangles and cones. They represent a powerful abstraction of reality, and they inspired a powerful philosophy of Platonic harmony. Euclid made of them a geometry that lasted two millennia, the only geometry still that most people ever learn. Artists found an ideal beauty in them, Ptolemaic astronomers built a theory of the universe out of them. But for understanding complexity, they turn out to be the wrong kind of abstraction.
Clouds are not spheres, Mandelbrot is fond of saying. Mountains are not cones. Lightning does not travel in a straight line. The new geometry mirrors a universe that is rough, not rounded, scabrous, not smooth. It is a geometry of the pitted, pocked, and broken up, the twisted, tangled, and intertwined. The understanding of nature’s complexity awaited a suspicion that the complexity was not just random, not just accident. It required a faith that the interesting feature of a lightning bolt’s path, for example, was not its direction, but rather the distribution of zigs and zags. Mandelbrot’s work made a claim about the world, and the claim was that such odd shapes carry meaning. The pits and tangles are more than blemishes distorting the classic shapes of Euclidian geometry. They are often the keys to the essence of a thing.
What is the essence of a coastline, for example? Mandelbrot asked this question in a paper that became a turning point for his thinking: “How Long Is the Coast of Britain?”
Mandelbrot had come across the coastline question in an obscure posthumous article by an English scientist, Lewis F. Richardson, who groped with a surprising number of the issues that later became part of chaos. He wrote about numerical weather prediction in the 1920s, studied fluid turbulence by throwing a sack of white parsnips into the Cape Cod Canal, and asked in a 1926 paper, “Does the Wind Possess a Velocity?” (“The question, at first sight foolish, improves on acquaintance,” he wrote.) Wondering about coastlines and wiggly national borders, Richardson checked encyclopedias in Spain and Portugal, Belgium and the Netherlands and discovered discrepancies of twenty percent in the estimated lengths of their common frontiers.
Mandelbrot’s analysis of this question struck listeners as either painfully obvious or absurdly false. He found that most people answered the question in one of two ways: “I don’t know, it’s not my field,” or “I don’t