Professor Benoit Mandelbrot: Mathematician whose development of fractal geometry increased our understanding of nature's complexity
Thursday 21 October 2010
The mathematician Benoît Mandelbrot is best known for his work in the field of fractal geometry. He told me that when he was seeking a word to describe his geometry, thumbing through his son's Latin dictionary he came across the word fractus, which translated as "broken, fractional, irregular". The most famous fractal is the Mandelbrot set, which generates a complex structure from simple rules.
The world we live in is not naturally smooth-edged and regularly shaped like the familiar cones, circles, spheres and straight lines of Euclid's geometry. The real world is rough-edged, wrinkled, crinkled and irregularly shaped. Until recently we didn't have a geometry to describe the natural world, but fractal geometry, which gave us the tools, is now used in the study of marine organisms, vegetative ecosystems, earthquake data, the behaviour of density-dependent populations, percolation and aggregation in oil research, in medical imaging and in the formation of lightning.
Fractals, defined by Mandelbrot as "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole" – the notion of self-similarity – are also aesthetically pleasing, frequently revealing stunning beauty in the most subtle ways. Although Mandelbrot was not the first mathematician to study fractals, the computer power available to him as a researcher at IBM enabled him to become a powerful advocate for this new field, especially in two influential books, Fractals: Form, Chance and Dimension and The Fractal Geometry of Nature.
Benoît Mandelbrot was born in Warsaw in November 1924 to Jewish Lithuanian parents. He showed an early love for geometry and excelled at chess. He recalled that he did not think the game through logically, but geometrically. In 1936 his family fled Poland to escape the Nazi persecutions, moving to southern France. Mandelbrot became an apprentice toolmaker, – he did well because he had firm hand and a good feeling for the shape of things.
At the end of the Second World War Mandelbrot returned to Paris for college entrance examinations, which he passed with distinction, winning a place at the Ecole Normale then moving to the Ecole Polytechnique. From there he went on to Caltech in the US to study turbulence. He gained a masters degree in aeronautics and, after returning to Paris, a doctorate in mathematics in 1952.
He then spent a year at the Institute for Advanced Study at Princeton, in 1953-54, and returned to France, where he became a professor at the University of Lille. During the 1960s Mandelbrot studied galaxy clusters, applying his ideas on scaling to the structure of the universe. Searching through forgotten and obscure journals he found a half-forgotten paper by the mathematician and meteorologist Lewis Fry Richardson. Richardson loved asking questions others considered worthless, and one of his papers, "Does the wind possess a velocity?" anticipated work by Edward Lorenz and other founders of Chaos Theory.
One of Richardson's great insights was a model of turbulence as a collection of ever-smaller eddies. Mandelbrot was struck, too, by Richardson's 1961 observations on the lengths of coastlines, and published a paper called "How long is the coast of Britain?" This apparently simple question of geography reveals, on close inspection, some of the essential features of fractal geometry.
In 1975 Mandelbrot published the first edition of The Fractal Geomtery of Nature, which made many waves in the world of mathematics. In my film about Mandelbrot, Clouds Are Not Spheres, Professor Ivar Giaever, the Nobel Prize-winning physicist, observed, "I like the way he started his book: 'Why is geometry often described as cold and dry? One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline or a tree'. And so what Mandelbrot has done is to develop a new geometry, which allows us to describe nature as we actually see it. He has made people think in a different way."
On 1 March 1980 at IBM's Thomas J Watson Research Center Mandelbrot discovered what came to be called the Mandelbrot set, described by Sir Arthur C. Clarke in my film The Colours of Infnity as "one of the seven wonders of the world ... What is so remarkable, in fact astounding, about the Mandelbrot set is that, although it is infinitely complex, it's based on incredibly simple principles – unlike almost everything in modern mathematics. Anybody who can add and multiply can understand the principles on which it's based. In fact it could have been discovered any time in human history. But the problem was this, although it's only based on adding and multiplying, you have to carry out those operations millions, billions of times, to create a complete set."
The seeds of this discovery had been sown decades previously. In Paris in 1917, two mathematicians, Gaston Julia, a student of Henri Poincaré, and Pierre Fatou, published papers connected with complex numbers. The results of their endeavours eventually became known as Julia sets, though it wouldn't be until the advent of modern computers that Julia sets could be seen for the first time.
It was Mandelbrot's mathematician uncle, Szolem Mandelbrot, who initially directed him to the work of Julia and Fatou on self-similarity and iterated functions. In Clouds are not Spheres he told me, "For me the first step, as with any difficult mathematical problem, was to programme it, and see what it looked like. We started programming Julia sets of all kinds. It was extraordinarily great fun! And in particular, at one point, we became interested in the Julia set of the simplest possible transformation. Z goes to Z squared plus C. So Z times Z plus C (z² + c). I made many pictures of it. The first ones were very rough. But the very rough pictures were not the answer. Each rough picture asked a question. So I made another picture, another picture. And after a few weeks we had this very strong, overwhelming impression that this was a kind of big bear we have encountered!"
He continued, "I think the most important implication is that, from very simple formulas you can get very complicated results. It's fundamental from the viewpoint of the very basis of science. Because, what is science? 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 marvellous: a very simple formula explains all these very complicated things.'
Apart from being a visionary and a maverick, Mandelbrot was also a kind and generous man, always patient and at pains to make his ideas clear. He received 15 honorary doctorates and among many other awards he won the Wolf Prize for Physics in 1993, the Honda Prize in 1994, the Médaille de Vermeil in 1996, the John Scott Award in 1999, the Lewis Fry Richardson Medal in 1999 , and the William Procter Prize for Scientific Achievement in 2002. The citation for the Wolf Prize thanks Mandelbrot "for changing the way we look at the world through the concept of Fractal Geometry".
Benoît B Mandelbrot, mathematician: born Warsaw 20 November 1924; married Aliette Kagan (two sons); died Cambridge, Massachusetts 14 October 2010.
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