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He had realized that, given an analytic problem, he could almost always think of it in terms of some shape in his mind.
Perhaps nowhere but in France, with its love of authoritarian academies and received rules for learning, could Bourbaki have arisen. It began as a club, founded in the unsettled wake of World War I by Szolem Mandelbrot and a handful of other insouciant young mathematicians looking for a way to rebuild French mathematics. The vicious demographics of war had left an age gap between university professors and students, disrupting the tradition of academic continuity, and these brilliant young men set out to establish new foundations for the practice of mathematics.
In part, Bourbaki began in reaction to Poincaré, the great man of the late nineteenth century, a phenomenally prolific thinker and writer who cared less than some for rigor. Poincaré would say, I know it must be right, so why should I prove it? Bourbaki believed that Poincaré had left a shaky basis for mathematics, and the group began to write an enormous treatise, more and more fanatical in style, meant to set the discipline straight.
In the United States, too, mathematicians were pulling away from the demands of the physical sciences as firmly as artists and writers were pulling away from the demands of popular taste.
Every serious mathematician understands that rigor is the defining strength of the discipline, the steel skeleton without which all would collapse. Rigor is what allows mathematicians to pick up a line of thought that extends over centuries and continue it, with a firm guarantee.
For a mathematician, the choice was clear: he would abandon any obvious connection with nature for a while. Eventually his students would face a similar choice and make a similar decision. Nowhere were these values as severely codified as in France, and there Bourbaki succeeded as its founders could not have imagined. Its precepts, style, and notation became mandatory.
This nomad-by–choice, who also called himself a pioneer-by–necessity, withdrew from academe when he withdrew from France, accepting the shelter of IBM’s Thomas J. Watson Research Center. In a thirty-year journey from obscurity to eminence, he never saw his work embraced by the many disciplines toward which he directed it.
Engineers were perplexed by the problem of noise in telephone lines used to transmit information from computer to computer. Electric current carries the information in discrete packets, and engineers knew that the stronger they made the current the better it would be at drowning out noise. But they found that some spontaneous noise could never be eliminated. Once in a while it would wipe out a piece of signal, creating an error.
By talking to the engineers, Mandelbrot soon learned that there was a piece of folklore about the errors that had never been written down, because it matched none of the standard ways of thinking: the more closely they looked at the clusters, the more complicated the patterns of errors seemed.
His description worked by making deeper and deeper separations between periods of clean transmission and periods of errors. Suppose you divided a day into hours. An hour might pass with no errors at all. Then an hour might contain errors. Then an hour might pass with no errors.
But suppose you then divided the hour with errors into smaller periods of twenty minutes. You would find that here, too, some periods would be completely clean, while some would contain a burst of errors. In fact, Mandelbrot argued—contrary to intuition—that you could never find a time during which errors were scattered continuously.
This highly abstract description had practical weight for scientists trying to decide between different strategies of controlling error. In particular, it meant that, instead of trying to increase signal strength to drown out more and more noise, engineers should settle for a modest signal, accept the inevitability of errors and use a strategy of redundancy to catch and correct them.
The Noah Effect means discontinuity: when a quantity changes, it can change almost arbitrarily fast.
Prices can change in instantaneous jumps, as swiftly as a piece of news can flash across a teletype wire and a thousand brokers can change their minds. A stock market strategy was doomed to fail, Mandelbrot argued, if it assumed that a stock would have to sell for $50 at some point on its way down from $60 to $10.
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 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.
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.
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.
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?”
In fact, he argued, any coastline is—in a sense—infinitely long. In another sense, the answer depends on the length of your ruler. Consider one plausible method of measuring. A surveyor takes a set of dividers, opens them to a length of one yard, and walks them along the coastline. The resulting number of yards is just an approximation of the true length, because the dividers skip over twists and turns smaller than one yard, but the surveyor writes the number down anyway.
An observer trying to estimate the length of England’s coastline from a satellite will make a smaller guess than an observer trying to walk its coves and beaches, who will make a smaller guess in turn than a snail negotiating every pebble.
And in fact, if a coastline were some Euclidean shape, such as a circle, this method of summing finer and finer straight-line distances would indeed converge. But Mandelbrot found that as the scale of measurement becomes smaller, the measured length of a coastline rises without limit, bays and peninsulas revealing ever-smaller subbays and subpeninsulas—at least down to atomic scales, where the process does finally come to an end. Perhaps.
SINCE EUCLIDEAN MEASUREMENTS—length, depth, thickness—failed to capture the essence of irregular shapes, Mandelbrot turned to a different idea, the idea of dimension. Dimension is a quality with a much richer life for scientists than for non-scientists.
The three dimensions are imagined as directions at right angles to one another. This is still the legacy of Euclidean geometry, where space has three dimensions, a plane has two, a line has one, and a point has zero.
In reality, of course, road maps are as three-dimensional as everything else, but their thickness is so slight (and so irrelevant to their purpose) that it can be forgotten. Effectively, a road map remains two-dimensional, even when it is folded up. In the same way, a thread is effectively one-dimensional and a particle has effectively no dimension at all. Then what is the dimension of a ball of twine?
From a great distance, the ball is no more than a point, with zero dimensions. From closer, the ball is seen to fill spherical space, taking up three dimensions. From closer still, the twine comes into view, and the object becomes effectively one-dimensional, though the one dimension is certainly tangled up around itself in a way that makes use of three-dimensional space.
From closer still, one is enough—any given position along the length of twine is unique, whether the twine is stretched out or tangled up in a ball. And on toward microscopic perspectives: twine turns to three-dimensional columns, the columns resolve themselves into one-dimensional fibers, the solid material dissolves into zero-dimensional points. Mandelbrot appealed, unmathematically, to relativity: “The notion that a numerical result should depend on the relation of object to observer is in the spirit of physics in this century and is even an exemplary illustration of it.”
Mandelbrot moved beyond dimensions 0,1,2,3…to a seeming impossibility: fractional dimensions.
Fractional dimension becomes a way of measuring qualities that otherwise have no clear definition: the degree of roughness or brokenness or irregularity in an object.
Mandelbrot specified ways of calculating the fractional dimension of real objects, given some technique of constructing a shape or given some data, and he allowed his geometry to make a claim about the irregular patterns he had studied in nature.
coastline,” in Mandelbrot’s words. To construct a Koch curve, begin with a triangle with sides of length 1. At the middle of each side, add a new triangle one-third the size; and so on. The length of the boundary is 3 × 4/3 × 4/3 × 4/3…—infinity. Yet the area remains less than the area of a circle drawn around the original triangle. Thus an infinitely long line surrounds a finite area.
On reflection, it becomes apparent that the Koch curve has some interesting features. For one thing, it is a continuous loop, never intersecting itself, because the new triangles on each side are always small enough to avoid bumping into each other. Each transformation adds a little area to the inside of the curve, but the total area remains finite, not much bigger than the original triangle, in fact.
The gasket is the same but with equilateral triangles instead of squares; it has the hard-to–imagine property that any arbitrary point is a branching point, a fork in the structure. Hard to imagine, that is, until you have thought about the Eiffel Tower, a good three-dimensional approximation, its beams and trusses and girders branching into a lattice of ever-thinner members, a shimmering network of fine detail.
Fractional dimension proved to be precisely the right yardstick. In a sense, the degree of irregularity corresponded to the efficiency of the object in taking up space.
The three-dimensional analogue is the Menger sponge, a solid-looking lattice that has an infinite surface area, yet zero volume.
In pursuing this path, Mandelbrot had two great advantages over the few other mathematicians who had thought about such shapes. One was his access to the computing resources that go with the name of IBM. Here was another task ideally suited to the computer’s particular form of high-speed idiocy.
His studies of irregular patterns in natural processes and his exploration of infinitely complex shapes had an intellectual intersection: a quality of self-similarity. Above all, fractal meant self-similar.
Self-similarity is symmetry across scale. It implies recursion, pattern inside of pattern. Mandelbrot’s price charts and river charts displayed self-similarity, because not only did they produce detail at finer and finer scales, they also produced detail with certain constant measurements.
Self-similarity is an easily recognizable quality. Its images are everywhere in the culture: in the infinitely deep reflection of a person standing between two mirrors, or in the cartoon notion of a fish eating a smaller fish eating a smaller fish eating a smaller fish.
While mathematicians and theoretical physicists disregarded Mandelbrot’s work, Scholz was precisely the kind of pragmatic, working scientist most ready to pick up the tools of fractal geometry. He had stumbled across Benoit Mandelbrot’s name in the 1960s, when Mandelbrot was working in economics and Scholz was an M.I.T. graduate student spending a great deal of time on a stubborn question about earthquakes.
Like a few counterparts in a handful of other fields, particularly scientists who worked on the material parts of nature, Scholz spent several years trying to figure out what to do with this book. It was far from obvious. Fractals was, as Scholz put it, “not a how-to book but a gee-whiz book.” Scholz, however, happened to care deeply about surfaces, and surfaces were everywhere in this book.
The unifying ideas of fractal geometry brought together scientists who thought their own observations were idiosyncratic and who had no systematic way of understanding them. The insights of fractal geometry helped scientists who study the way things meld together, the way they branch apart, or the way they shatter.
Within the top of the solid earth are surfaces of another kind, surfaces of cracks. Faults and fractures so dominate the structure of the earth’s surface that they become the key to any good description, more important on balance than the material they run through.
Geophysicists looked at surfaces the way anyone would, as shapes. A surface might be flat. Or it might have a particular shape. You could look at the outline of a Volkswagen Beetle, for example, and draw that surface as a curve.
Scholz found that fractal geometry provided a powerful way of describing the particular bumpiness of the earth’s surface, and metallurgists found the same for the surfaces of different kinds of steel. The fractal dimension of a metal’s surface, for example, often provides information that corresponds to the metal’s strength.
Fractal descriptions found immediate application in a series of problems connected to the properties of surfaces in contact with one another.
Contacts between surfaces have properties quite independent of the materials involved. They are properties that turn out to depend on the fractal quality of the bumps upon bumps upon bumps. One simple but powerful consequence of the fractal geometry of surfaces is that surfaces in contact do not touch everywhere. The bumpiness at all scales prevents that. Even in rock under enormous pressure, at some sufficiently small scale it becomes clear that gaps remain, allowing fluid to flow.
It is why two pieces of a broken teacup can never be rejoined, even though they appear to fit together at some gross scale. At a smaller scale, irregular bumps are failing to coincide.
HOW BIG IS IT? How long does it last? These are the most basic questions a scientist can ask about a thing. They are so basic to the way people conceptualize the world that it is not easy to see that they imply a certain bias. They suggest that size and duration, qualities that depend on scale, are qualities with meaning, qualities that can help describe an object or classify it.
Imagine a human being scaled up to twice its size, keeping all proportions the same, and you imagine a structure whose bones will collapse under its weight. Scale is important.
