Tuesday, October 19, 2010

Revealing the universe's playful irregularity

There are many internally consistent mathematical systems. Few, however bear a real resemblance to the structure of the natural world. Bucky Fuller and Benoît Mandelbrot were in pursuit of that system of systems. Both may have found it. (GW)

Benoît B Mandelbrot: the man who made geometry an art

By Jonathan Jones
The Guardian
October 19, 2010

The sphere of maths has borne few as provocative as the man whose 'fractals' demonstrated the universe's playful irregularity

Few recent thinkers have woven such a beautiful braid of art and science as Benoît B Mandelbrot, who has died aged 85 in Cambridge, Massachusetts. (The B apparently doesn't stand for anything. He just felt like adding it.) Mandelbrot was a provocative mathematician, a subversive geometer. He left a beautiful legacy in visual art, for Mandelbrot was the man who named and explained fractals – those complex, apparently chaotic yet geometrically ordered shapes that delight the eye and fascinate the mind. They are icons of modern understanding of the universe's complexity.

The Mandelbrot set, one of the most famous fractal designs, is named after him. With its fizzing fringe of crystal-like microforms blossoming out of a conjunction of black circles, this fractal pattern looks crazy but is the outcome of geometrical calculations.

Geometry, said Mandelbrot, is seen as "dry" because it can only explain regular shapes like the square, the cylinder and the cone. Such shapes have been analysed mathematically since the time of the ancient Greeks, which is why traditional geometry is known as Euclidean geometry. But in the 19th and 20th centuries, physicists and mathematicians started to think beyond Euclid and his regular universe. Mandelbrot was not the first, but with his startling fractals concept he created a visual manifesto for a non-Euclidean universe.

Fractals – and I'd be delighted if mathematicians can give a better explanation below– are shapes that are irregular but repeat themselves at every scale: they contain themselves in themselves. Mandelbrot used the example of a cauliflower which, like a fern, is a fractal found in nature; if you look at the smallest sections of these vegetable forms, you see them mirroring the whole.

Mandelbrot, who worked at IBM before becoming a professor at Yale, started thinking about irregular shapes by looking at maps of Great Britain. The squiggly shape of the UK mainland fascinated him and he wondered whether it was possible to make a mathematical model of its perimeter. Can you measure the British coastline? He discovered that you can at a distance, but that then the closer you look, the more you find. In a sense, the British coastline is "infinite".


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