Mathematics
A cloud Atlas
Memoirs of a European mathematician
The Economist, Oct 27th 2012 | from the print edition
Swirling sets
The Fractalist: Memoir of a Scientific Maverick. By Benoit Mandelbrot. Pantheon Books; 324 pages; $30. Buy from Amazon.com
HOW long is Britain’s coastline? Ask the same question of a piece of string; it depends on the string. But for an island, the tape measure matters too. As a cartographer zooms in, new inlets and promontories appear. Time and money, if not waves and tides, will eventually call a halt to such diligence. But it is a fact that the shorter the ruler, the longer the coastline.
This is the result of “self-similarity”: small stretches of coastline look much like bigger ones. Without buildings or trees to fix the scale, a photograph of a one-kilometre bay could easily be confused with that of a 100km one. Such “self-similarity” is everywhere once you start to look. Focus in on a cloud and wisps resolve themselves into new, smaller wisps that look strangely familiar. Blood vessels branch, and their branches branch in turn.
Mathematicians have been constructing abstract shapes with this property since the 19th century. Giuseppe Peano built an infinitely wiggly curve by adding detours—and then detours on those detours, and so on for ever. Georg Cantor cut chunks out of a line, and chunks out of what remained, to make a self-similar “dust”. Waclaw Sierpinski did the same to two- or three-dimensional shapes, leaving a lace or foam. Such constructs seemed unclassifiable, somewhere between solid and gossamer—and impervious to the traditional tools of geometers.
This book is the autobiography of Benoit Mandelbrot, the mathematician who finally recognised the utility and ubiquity of such seeming anomalies. He coined the standard term for them: “fractals”, from the Latin fractus, meaning broken or shattered, and saw his own name given to the best-known, the “Mandelbrot set”. Mandelbrot died in 2010; his wife, Aliette, completed the book. It contains little mathematics (the only formula is the one that generates the Mandelbrot set), focusing instead on the path of discovery and the people who helped along the way.
It was a winding and random-seeming path. Mandelbrot moved from Warsaw to Paris in 1936 when he was a boy, and then to rural France at the outbreak of war. Along the way he picked up a patchy but inspiring mathematical education thanks to clever relatives, old textbooks, some excellent schoolmasters, and a geometric intuition that helped him to pass notoriously difficult exams in France. Too restless to settle down as a university professor, he spent scattered years doing odd things like working on colour television for Philips Electronics and as mathematician-in-residence at an institute set up by a child psychologist, Jean Piaget.
Eventually Mandelbrot landed at IBM’s research facility in New York, where the different strands of the story come together, not least because the development of computer graphics allowed fractals to be plotted with ease for the first time. Brimming with examples, conjectures and ground-breaking images, his best-known book, “The Fractal Geometry of Nature”, published in 1982, launched an avalanche of colourful T-shirts, mugs and calendars, and made him famous, at least for a mathematician.
Mandelbrot made no secret of his belief that glory lay as much in coming up with a conjecture as in proving it, leading some to dismiss him as a mere “hand-waver”, as mathematicians call those who elide a proof’s tricky steps. Most of the mathematics used to analyse fractals was developed by others. But to his admirers Mandelbrot was a spell-worker who saw connections no one else did and united apparently disparate phenomena. The mathematics of fractals—and pictures of the Mandelbrot set—offered many budding mathematicians their first taste of “real” mathematics, in all its beauty, utility and sheer unexpectedness.
No hay comentarios:
Publicar un comentario
COMENTE SIN RESTRICCIONES PERO ATÉNGASE A SUS CONSECUENCIAS