It is rare that the passing of a mathematician makes headlines in the mass media. It is rarer still for one to achieve celebrity for work on problems we can all understand. But then, the Franco-American mathematician Benoit Mandelbrot, who died this month at the age of 85, was as extraordinary as the phenomena he studied.
At the age of 20, Mandelbrot declared that his ambition was: "To find a corner of science, not necessarily very extensive or even significant, of which I would know enough to be its Kepler or even its Newton." As his recent obituary in this newspaper shows, he certainly achieved the latter: it carried the headline "the father of fractals". Yet the significance of his work on these strange geometrical shapes remains deeply controversial, for reasons that should concern us all.
While Mandelbrot gave them their name, from the Latin for "broken", in the 1970s, fractals had been studied decades earlier by the English mathematician Lewis Fry Richardson, who identified many of their key properties. Put simply, fractals are shapes which reveal ever more detail as they are magnified.
An everyday example of a fractal, and the one studied extensively by both Richardson and Mandelbrot, is the coastline of a country. Take the case of the UAE: according to textbooks, the Emirates have a coastline of 1,318km, a figure that one could at least roughly confirm by getting a map and measuring it with a ruler. But as Richardson noted, the precise answer depends critically on the scale of the map. If it's large scale, it won't capture the ins and outs around, say, Al Mirfa. As a result, it will give a figure for the coastline much shorter than that found using a finer-scale map.
So what is the correct answer ? As Richardson and Mandelbrot showed, there really isn't one. The best one can do is state the length at a given scale, along with a number which captures the "jaggedness" of the coastline. Plugged into a simple mathematical formula, this so-called fractal dimension shows how the length of the coastline changes with scale.
All this might sound like just the sort of esoteric curiosity only mathematicians get excited by. The significance of fractals lies in their ability to cram a lot of detail into a tiny mathematical "package". That gives them a host of applications in everything from data compression to creating computer-generated images.
In the early 1980s, Mandelbrot, by then working for the computer giant IBM, made his first foray into the limelight with his lavishly illustrated book The Fractal Geometry of Nature, packed with many startling images of fractals, among them the bizarre snowman-like shape known as the Mandelbrot Set. But in his latter years he achieved notoriety for his attempts to use fractals to understand financial markets.
The idea is simple enough: when plotted against time, the prices of everything from stocks to currencies to commodities give graphs that look like jagged coastlines. That suggests they might be described by simple fractal formulas, which in turn hints at the possibility of predicting their behaviour.
Mandelbrot's work on this fascinating possibility began in the 1960s, even before he understood the power of fractals to capture the notion of "jaggedness". After analysing real-life market data, he found evidence that fractals might be useful in capturing the key concept of market volatility, that is, just how "jagged" the prices are over time.
For example, the price of cotton appeared to suffer especially violent swings, making it a riskier investment than, say, wheat. Similarly, the stock prices of companies such as Westinghouse and American Tobacco proved more jumpy than those of, say, Standard Oil.
Mandelbrot suggested that, in principle at least, fractals could capture these different levels of volatility in a single number. Yet this innocuous-sounding problem made him deeply unpopular with most financial market analysts. That was because they believed they already had a perfectly good way of understanding volatility, based around something much more familiar than fractals: the so-called bell curve.
Mandelbrot, however, insisted that the bell curve could seriously under-estimate the chances of a huge market swing, making a mockery of the standard risk analysis performed by financial institutions.
He became convinced that the financial markets were heading for an almighty calamity, and voiced his concerns in 2004 with the publication of The (Mis)Behaviour of Markets. Despite stirring up considerable controversy, his calls for more research into the application of fractals to market behaviour went largely unheeded.
The crash of 2007 seemed to vindicate Mandelbrot's claims. Yet even now there seems no huge drive to adopt Mandelbrot's methods. Why is this? The history of science holds a clue. In general, it is not enough to be right: one also has to be able to offer practical techniques for going beyond what is already possible.
The plain fact is that, as Mandelbrot himself conceded, his ideas were barely half-baked. He had impressive evidence that fractals could be useful, but could not offer financial analysts a toolkit allowing them to move seamlessly from the bell curve to fractals.
In a world where, as the sequel to the movie Wall Street puts it, "money never sleeps", that is unacceptable. With the passing of Mandelbrot, it falls to others to pick up the torch he lit and waved aloft. Whether the financial world will like what is revealed by its fractal flame is quite another matter.
Robert Matthews is Visiting Reader in Science at Aston University, Birmingham, England