It is not often that an academic genius engages directly with the world beyond his ivory tower. Perhaps it is even more rare for a mathematician, whose knowledge is so arcane to the layperson, to do so. But until recently the world had such a man: Benoit Mandelbrot, a mathematician who engaged directly with the world.
And according to popular statistician Nassim Nicholas Taleb, Mandelbrot “had perhaps more cumulative influence than any other single scientist in history, with the only close second, Isaac Newton.”
Quite simply, Benoit Mandelbrot helped us to better understand the world through mathematics; and he helped non-mathematicians to understand mathematics as an art. By showing the geometry in the natural world, he showed geometry itself to be as beautiful as anything in nature.
“Why is geometry often described as cold and dry?” Mandelbrot asked in his book “The Fractal Geometry of Nature.” “One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline or a tree.” So he set about using geometry to describe such natural phenomena.
Mandelbrot found that many rough, jagged things like a coastline or a lightening bolt actually have patterns. By finding this, he founded what is known as fractal geometry, which is the first broad attempt to investigate quantitatively the ubiquitous notion of roughness.
Fractals are shapes that are irregular but repeat themselves at every scale: they contain themselves in themselves. As Mandelbrot explained at a talk before the Reality Club:
“The standard example is the cauliflower. One glance shows that it’s made of florets. A single floret, examined after you cut everything else, looks like a small cauliflower. If you strip that floret of everything except one ‘floret of a floret’ — very soon you must take out your magnifying glass — it’s again a cauliflower. A cauliflower shows how an object can be made of many parts, each of which is like a whole, but smaller. Many plants are like that.”
Fractals are a particular set of mathematical objects. They are infinitely repeating shapes that Mandelbrot found were not just pretty curiosities but actually provided the key to explaining rough objects and indeed complex data sets. These complex, apparently chaotic shapes are actually geometrically ordered, Mandelbrot showed, and so the Mandelbrot Set, a fractal design, was named for him.
So why is it important to know that a cauliflower is made up of lots of identical florets that are themselves the shape of a cauliflower? Well, Mandelbrot taught us that the world in which we live is not made up of regular shapes and smooth lines; it is made up of irregular mathematical shapes. It is made up of fractals.
“What is science?” Mandelbrot once asked. “We have all this mess around us. Things are totally incomprehensible. And then eventually we find simple laws, simple formulas. In a way, a very simple formula, Newton’s Law, which is just also a few symbols, can by hard work explain the motion of the planets around the sun and many, many other things to the 50th decimal. It’s marvelous: a very simple formula explains all these very complicated things.” This is perhaps the most important implication of Mandelbrot’s work: that very simple formulae could yield very complicated results and explain very irregular phenomena.
Benoit Mandelbrot was born to a Jewish family in Warsaw in 1924. His uncle Szolem Mandelbrot, a mathematician based in Paris, was a strong influence on him—”the love of his mind was mathematics,” Benoit later wrote—and Benoit soon displayed a passion for geometry. He also took an early interest in cartography, probably inspired by the many maps that his Lithuania-born father had around the house.
As Nazism threatened Poland, in 1936 Mandelbrot’s father decided to move the family to the ostensibly less at-risk France. But the Nazis soon came there, too. So when German troops invaded France in 1940, the Mandelbrots moved to the French “Free Zone.”
“Our constant fear was that a sufficiently determined foe might report us to an authority and we would be sent to our deaths,” Mandelbrot said. This was not unusual—as historian Bill Paxton first showed, the French were often very willing collaborators with Nazism, more than willing to betray the Jews of France. Nevertheless, Mandelbrot acknowledged that the family was safer in France than in Poland. “The fact that my parents, as economic and political refugees, joined Szolem in France saved our lives.”
The family survived and after the war, Mandelbrot earned a degree from Paris’ elite École Polytechnique, followed by a Master’s in aeronautics from Cal Tech and then a doctorate in mathematics back in Paris in 1952. He worked at the Institute for Advanced Study at Princeton and the Centre National de la Recherche Scientifique.
Mandelbrot was progressing well along an academic career track until he spent the summer of 1958 as an academic visitor at IBM—there he found the perfect environment in which to pursue his interest in computer-generated mathematics. ‘”For me the first step with any difficult mathematical problem was to program it, and see what it looked like,” he told a British documentary maker.
At IBM he began to think about irregular shapes like the squiggly coastline of Great Britain. He wondered how long the coastline was, and how one could go about finding out seeing as it was such a ragged, rough, squiggly thing. So in an essay titled “How long is the coast of Britain?” Mandelbrot asked himself that very question, and answered it.
Eight years later, in 1975, he invented the word fractal to describe his findings. From there he extended fractals into the plane of complex data. And he then used computers to plot fractals in a way that looked so good they became popular and made his name known beyond the world of mathematics. The posters, T-shirts, cards and so on that carried his fractals also may have been financially helpful.
Mandelbrot remained at IBM for 35 years. But when the company stopped funding his research division in 1987, he left for academia and joined the math department at Yale. There he became a tenured professor at the admirable age of 75 in 1999. He is the oldest professor to receive tenure in the history of Yale, where he remained until his retirement in 2005.
Today fractal geometry is used in many disparate fields, including work with marine organisms, vegetative ecosystems, earthquake data, the behavior of density-dependent populations, percolation and aggregation in oil research, and in the formation of lightning. As Mandelbrot’s TED talk bio explains, “Fractal math has brought new insight to the study of pretty much everything, from the behavior of stocks to the distribution of stars in the universe.”
Or as the former President of France Nicholas Sarkozy said, “His work, developed entirely outside mainstream research, led to modern information theory.”
In addition, Mandelbrot helped to show the beauty in nature. When he died in October 2010, at the age of 85, he left us a visual lexicon for our complex world, the geometry of fractals. As he once said, Benoit Mandelbrot showed that, “Bottomless wonders spring from simple rules, which are repeated without end.”