The Father of Fractals

The pioneering mathematician Benoit Mandelbrot spotted a common thread in complex shapes such as clouds, coastlines and Romanesco broccoli.

By STEPHEN WOLFRAM

In nature,technology and art the most common form of regularity is repetition: a single element repeated many times, as on a tile floor. But another form is possible, in which smaller and smaller copies of a pattern are successively nested inside each other, so that the same intricate shapes appear no matter how much you "zoom in" to the whole. Fern leaves and Romanesco broccoli are two examples from nature.
One might have thought that such a simple and fundamental form of regularity would have been studied for hundreds, if not thousands, of years. But it was not. In fact, it rose to prominence only over the past 30 or so years—almost entirely through the efforts of one man, the mathematician Benoit Mandelbrot, who died in 2010 just before completing this autobiography.

The Fractalist

By Benoit Mandelbrot
Pantheon, 321 pages, $30

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Romanesco broccoli has a fractal structure: Each cone-like head consists of smaller, similarly shaped cones, which in turn consist of even smaller cones. In a graph of a fractal equation, this 'nesting' process would continue infinitely.

Born to a Jewish family in Poland in 1924, he was the son of a dentist mother and a businessman father. One of his uncles, Szolem Mandelbrojt, was a fairly successful pure mathematician in France, and it was there that the Mandelbrots fled in 1936 as the situation for Jews in Poland worsened.
After hiding out during the war in the countryside, Mandelbrot wound up as an outstanding math student at a top technical college in Paris. At the time, French mathematics was dominated by the highly abstract Bourbaki movement, which was named for the collective pseudonym with which its members signed their work. When Mandelbrot graduated in 1947, instead of joining them, he decided to study aeronautical engineering at Caltech. He was determined, he writes, to stay away from established mathematics so that he could "feel the excitement of being the first to find a degree of order in some real, concrete, and complex area where everyone else saw a lawless mess."
Returning to France two years later, he got involved with information theory and, from that, statistical physics and the structure of language. He ended up writing his Ph.D. on "Games of Communication," notably investigating how the frequencies with which different words were used in a text followed a distribution known as a "power law." He did military service as a kind of science research scout, spent time at a Philips color television lab, and then began life as a "wandering scientist," partly supported by the French government.
His early destinations included Princeton (where he worked with the game-theory pioneer John von Neumann) and Geneva (with the psychologist Jean Piaget). And then, in 1958, he visited IBM, ostensibly to work on a new automated-translation project—and ended up staying for 35 years.
IBM saw its research division, in part, as a way to spread its reputation, and it allowed Mandelbrot to continue wandering and spend time as a visitor at universities like Harvard and Yale (where he eventually spent the last years of his career). His work moved from the statistics of languages to the statistics of economic systems and, by the mid-1960s, to hydrodynamics and hydrology. His typical mode of operation was to apply fairly straightforward mathematics (typically from the theory of random processes) to areas that had barely seen the light of serious mathematics before. He calls himself a "would-be Kepler of complexity," evoking (as he does continually) Johannes Kepler, the 17th-century scientist who determined the laws that describe the movement of the planets.
But in the early 1970s a mathematician friend (Mark Kac) made a crucial suggestion: Stop writing lots of papers about strangely diverse topics and instead tie them together in a book. The unifying theme could have been a technical one—in which case few people today would have probably ever heard of Benoit Mandelbrot.
But instead, perhaps just through the very act of exposition (his autobiography does not make it clear), Mandelbrot ended up doing a great piece of science and identifying a much stronger and more fundamental idea—put simply, that there are some geometric shapes, which he called "fractals," that are equally "rough" at all scales. No matter how close you look, they never get simpler, much as the section of a rocky coastline you can see at your feet looks just as jagged as the stretch you can see from space. This insight formed the core of his breakout 1975 book, "Fractals."
Before this book, Mandelbrot's work had been decidedly numerical, with simple graphs being about as visual as most of his papers got. But between his access to computer graphics at IBM and publishers with distinctly visual orientations, his book ended up filled with striking illustrations, with his theme presented in a highly geometric way.
There was no book like it. It was a new paradigm, both in presentation and in the informal style of explanation it employed. Papers slowly started appearing (quite often with Mandelbrot as co-author) that connected it to different fields, such as biology and social science. Results were mixed and often controversial. Did the paths traced by animals or graphs of stock prices really have nested structures or follow exact power laws? Even Mandelbrot himself began to water down his message, introducing concepts like "multifractals."
But independent of science, fractals began to thrive in computer graphics, particularly in making what Mandelbrot called "forgeries" of natural phenomena—surprisingly lifelike images of trees or mountains. Some mathematicians began to investigate fractals in abstract terms, connecting them to a branch of mathematics concerned with so-called iterated maps. These had been studied in the early 1900s—notably by French mathematicians whom Mandelbrot had known as a student. But after a few results, their investigation had largely run out of steam.
Armed with computer graphics, however, Mandelbrot was able to move forward, discovering in 1979 the intricate shape known as the Mandelbrot set. The set, with its distinctive bulb-like lobes, lent itself to colorful renderings that helped fractals take hold in both the popular and scientific mind. And although I consider the Mandelbrot set in some ways a rather arbitrary mathematical object, it has been a fertile source of pure mathematical questions—as well as a striking example of how visual complexity can arise from simple rules.
In many ways, Mandelbrot's life is a heroic story of discovery. He was a great scientist, whose virtues I am keen to extol. For all his achievements, however, he could be a difficult man, as I discovered in interactions over the course of nearly 30 years. He was constantly seeking validation and constantly fighting to get his due, a theme that is already clear on the first page of his autobiography: "Let me introduce myself. A scientific warrior of sorts, and an old man now, I have written a great deal but never acquired a predictable audience."
He campaigned for the Nobel Prize in physics; later it was economics. I used to ask him why he cared so much. I pointed out that really great science—like fractals—tends to be too original for there to be prizes defined for it. But he would slough off my comments and recite some other evidence for the greatness of his achievements.
In his way, Mandelbrot paid me some great compliments. When I was in my 20s, and he in his 60s, he would ask about my scientific work: "How can so many people take someone so young so seriously?" In 2002, my book "A New Kind of Science"—in which I argued that many phenomena across science are the complex results of relatively simple, program-like rules—appeared. Mandelbrot seemed to see it as a direct threat, once declaring that "Wolfram's 'science' is not new except when it is clearly wrong; it deserves to be completely disregarded." In private, though, several mutual friends told me, he fretted that in the long view of history it would overwhelm his work.
Every time I saw him, I would explain why I thought he was wrong to worry and why, even though in the "computational universe" that my book described fractals appear in just a small corner, they will nevertheless always be fundamentally important—not least as perhaps the unique intermediate step between simple repetitive regularity and the apparent randomness of more complex computational processes.
I saw Mandelbrot quite frequently in the years before his death, and many questions he answered with the phrase, "Read my autobiography." And indeed it does answer some questions—especially about his early life—though it leaves quite a few unanswered, such as how exactly his breakthrough 1975 book came together. But knowing more about Benoit Mandelbrot as a person helps to illuminate his work and to illustrate what it takes for great new science to be created. "The Fractalist" is a well-written tale of a scientific life, complete with first-person accounts of a surprising range of scientific greats.
—Mr. Wolfram is the founder of
Wolfram Research and the author of "A New Kind of Science."