Mathematician Benoît B Mandelbrot died aged 85 on 14 October leaving an astonishing legacy. Described by The Economist as “the man who made maths pretty”, he showed how cauliflowers, coastlines and clouds could be described in mathematical terms.
Maths, prior to Mandelbrot’s time, had tended to concern itself with fairly artificial looking objects. It envisioned a world of smooth, protractor-drawn, regularly angled shapes: cones and cubes, spheres and squares, flat planes and perfect curves.
Mandelbrot’s genius was to impose some geometry on our real-life, rough-edged cosmos. As he wrote in his introduction to The Fractal Geometry of Nature (1982): “Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.”
Central to his enterprise were fractals. The most mesmeric of geometric patterns, these intricate forms do not become simpler as you zoom in on them, but rather repeat the same complexities on a smaller scale. Such patterning, thought Mandelbrot, was more than a mere curiosity. It was a fundamental natural structure. Coining the term in 1975, he spotted instances of fractals where nobody had previously thought to look.
When most people now think of fractals, they think of computer-generated images – a search for ‘self-hypnosis’ on YouTube will yield some fairly trippy specimens. One particularly psychedelic example is the Mandelbrot Set itself. Based on a simple algebraic equation, when the result is fed back into the equation repeatedly, and the results drawn by a computer, you get what looks like a fuzzy-edged blob. When you magnify this blob, however, whole galaxies of sub-blobs start appearing. You can magnify it an infinite number of times and you’ll still see the same formation.
Away from the computer, fractals can be found in places as quotidian as the veg aisle. If you take a cauliflower, and break off a chunk, you will observe how each of its sections resembles the cauliflower it came from. Look closer still, and you’ll see all the same loops and lumps and whorls and caulifloweresque convolutions – tinier and tinier cauliflowers blossoming before your eyes. This so-called ‘self-similarity’ is a feature of the most common class of fractals.
Mandelbrot became fascinated by the British coastline, whose squiggly shape, as seen on a map, turned out to be just as squiggly when you visited it. Nor were those squiggles straightened out when you peered in with a magnifying glass. It turns out that coastlines, like cauliflowers (like clouds, like ferns, like mountains…) follow fractal logic. This, said Mandelbrot, is a reflection of the processes by which they are formed.
Mandelbrot was a genuine polymath, who worked extensively in informatics and economics, and whose findings had a bearing on disciplines across the board. Chaos theory is based on fractals, as is current financial wisdom. In 2005, Mandelbrot published The Misbehavior of Markets, which showed how fractal geometry could be used to describe market workings. Presciently, he compared over-confident traders to “mariners who heed no weather warnings”, admonishing that classical models could not predict a storm.
Mandelbrot, who held dual French and American citizenship, spent most of his career working at IBM. With a spangle of awards and fellowships, and even an asteroid named in his honour, his immense contribution to the sciences – and arts too – did not go unheeded. When he died, he was described as “one of the most important figures of the last fifty years”, and an “icon who changed how we see the world”. Cosmologists, conceptual artists and greengrocers alike might agree.
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