Mandelbrot's Complex Visions: A New Way of Looking at Mathematics


Benoit Mandelbrot, whose unorthodox visions have helped reshape modern science, sat in an office at Johns Hopkins University this week and poked his finger at a picture of what looked like a cave of stubby multicolored stalagmites and stalactites linked by gossamer filaments.

The 67-year-old Yale professor is currently studying a branch of methematics dealing with "diffusion limited aggregations," or DLAs, which can be used to describe, among many other things, the paths that water takes when it is pumped into an oil well to force more oil out.

"It is one of nature's favorite forms for reasons that are not completely understood, which is quite exciting," said the scholar, who came to Hopkins to deliver a lunchtime lecture.

The illustration he was pointing to was a picture of a DLA equation Dr. Mandelbrot drew using fractal geometry, a branch of mathematics he developed and named in the mid-1970s.

Fractals grew out of the mathematician's conviction that there was a order lurking behind the seeming randomness and chaos of many irregular structures found in nature -- ranging from the outline of mountains and clouds to the branching of broccoli spears and air sacks in the lung to the rise and fall of the Nile over two thousand years.

Fractals have been phenomenally successful. They are now routinely used in research in a staggering variety of scientific disciplines: physics, physiology, probability, metallurgy, materials science, seismology, chemistry and economics.

Dr. Mandelbrot's basic insight was that many seemingly random parts of nature share a common organizing principle: self-similarity at increasingly smaller scales. That means, for example, that a mile section of the coast of Britain will demonstrate a similar overall pattern to a 100-mile stretch. A twig resembles a branch, which resembles the entire tree.

As he peered through wire-frame glasses at the illustration, Dr. Mandelbrot's expression shimmered between that of a cherub in a Renaissance painting and that of shrewd shopkeeper sizing up a shipment of goods.

"These lines are extremely significant," he said, pointing to a detail in the DLA illustration. "My colleagues, my friends were writing learned articles about the shape of these lines, but they'd never seen them. And I was always shocked. How did they know? Now we've drawn these lines. And they turned out to be very different from what people expected."

Dr. Mandelbrot seems to delight in using his curious way of looking at things to point out what most of the rest of us have missed.

"I have helped the eye return into science," he says.

The mathematician was born in Warsaw in 1924 into a Lithuanian Jewish family -- his father was a clothing wholesaler, his mother a dentist. When he was 12, the family moved to Paris. Several years later, with war looming, they moved again to south central France.

As a student in Paris and later at the California Institute of Technology, he said, he had trouble with algebra but loved the curves and shapes of geometry. Even today, he said, he cannot remember a telephone number after hearing it. He must write it down "and then I remember forever."

But pre-computer mathematics, he said, was an austere and abstract discipline with little room for Dr. Mandelbrot's fascination with the shapes of nature. By his own description, he became the intellectual equivalent of a homeless person, rummaging around in trash cans for neglected or forgotten mathematical theories.

His search frequently led him to stray outside his discipline. Besides teaching math, Dr. Mandelbrot has taught economics at Harvard, engineering at Yale and physiology at the Einstein College of Medicine.

When he was at IBM's Thomas J. Watson Research Center in Yorktown Heights, N.Y. in the early 1970s, Dr. Mandelbrot cooked up a formula for the shape of mountain peaks.

"Twenty years ago the idea of a formula for mountains was an absurd idea, a poetic idea," he said with a smile. The only way to devise the formula, he said, was with computer graphics -- which basically had not yet been invented. Fortunately, he and his colleagues had the latest IBM equipment at hand.

"It was very klunky, very unreliable, very awkward to use, but marvelous," he said. "Each time we did a picture and walked down the hall, each time people would stop us and [say] 'My goodness; it's beautiful! What is it?' So we would say, 'Well, guess.' And they guessed all kinds of crazy things which we didn't intend."

Mostly, he said, they guessed that the shape had something to do with their particular field of research -- even though the Watson center sheltered scholars from many disciplines.

"All kinds of new research projects started because people stopped us in the hall and thought that this reminded him of his own work," he said. "We went to see him or her and we saw that, indeed, they were dealing with fractals also. That's why the idea of fractals has penetrated so many domains. Because people recognize their own pictures, it resonates."

VTC Dr. Mandelbrot does not pretend that fractals can be found everywhere in everything: there is probably a deep disorder in nature, he agreed, that is fundamentally patternless.

"When I was a young person my teachers told me mathematics described almost everything in nature," he said. "I never believed them, because I looked at mountains and clouds and it didn't describe them. But that's a matter of time. In due time I would describe mountains and clouds [mathematically]."

Later, he added: "I never claimed that this was a kind of all-purpose thing. In fact, I make the comparison often that this is like a clearing in the jungle. It is a clearing in the jungle, but the jungle is very, very thick."

Dr. Mandelbrot's best-known achievement is probably the Mandelbrot set, which some call the most complex object in mathematics. It is a relatively simple formula that, when plugged into a computer and graphed on a complex plane (in which one dimension is described by real numbers and the other by imaginary numbers), creates a stunningly beautiful, fantastically complicated structure of filigree and swirls. The figure is also infinitely deep -- it can be magnified indefinitely and still display intricate patterns.

As might be expected from something created by Dr. Mandelbrot, the structure exhibits striking self-similarity at every scale -- including the recurrence of a pattern that resembles the Michelin-tire man. Studying pictures of the Mandelbrot set, you start looking for the figure the way you look for Alfred Hitchcock in one of the director's movies.

Dr. Mandelbrot's ideas, while not widely known among laymen, have made him something of a minor celebrity among many students, scientists and people with an amateur interest in science.

The only explanation for this, he says, is that his work "has an element of what I call white magic. There's nothing magical in the sense of being unexplained. It's repeatable. Anybody can learn the tricks. But something which has for humans something that is an attraction in every culture I've visited."

"It tells us something about humans," he said. "What it tells us I don't know exactly. But I can sort of speculate. The world around us -- mountains, clouds the rivers and so on -- is full of fractals." And people respond to those patterns in a way they don't respond, say, to the austere straight lines and flat planes of classical geometry, or of modern architecture and art.

"Humans have to be in harmony, if you wish, with these shapes because you see them always," he said.

But he shrugs and laughs. This is one subject that his vision cannot penetrate. "I am no more knowledgable about that than anybody else," he said.

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