## The geometry of Mandelbrot

### The founding father of fractal theory warns that another Great Crash is a real threat. Martin Baker meets him

Preposterous though it seems, there is something of the lone gunslinger about Benoit Mandelbrot. He is Sterling Professor of Mathematical Sciences at Yale University. And, yes, he is turning 80 next month. But if ever a man fixed the establishment in the eye and shot from the hip, it is Mandelbrot.Next week is the 75th anniversary of the Wall Street Crash of 1929 - now consider this extract from the book Mandelbrot is publishing next month: "The financiers and investors of the world are, at the moment, like mariners who heed no weather warnings."

No, no, no. Surely it couldn't happen again? Conventional wisdom tells us that we understand risk so much better now. Those clever hedge fund chappies have got it all sewn up, surely. There has been the occasional disaster, such as the collapse of the huge US hedge fund Long Term Capital Management in the late 1990s. But we all survived. The investment industry has a sophisticated understanding of the riskiness of the market, and by using analytical tools such as chaos theory, the risks are managed and controlled.

Unfortunately not, according to Mandelbrot. And he should know. As the founder of fractal geometry and the discoverer of the Mandelbrot set (pictorially represented as beautiful, complex swirls of coral) Mandelbrot is acknowledged as the father of chaos theory. Here are his views of the current state of play: "A few fund managers have experimented with these concepts [of price dependence, whatever that is, and volatility]. They often call it chaos theory - though strictly speaking that is marketing language riding on the coat-tails of a popular scientific trend. In reality, the mathematics is still young, the research barely begun, and reliable applications still distant."

Mandelbrot has tweaked a few tails in the academic world, too. James Gleik, author of one of the many books on chaos theory, acknowledges Mandelbrot's contribution to the doctrine, while calling him "exasperating and indispensable".

So what is the principal theory of this near-octogenarian, so often described as a maverick? It was clearly a good idea to find out before having lunch with him.

Fractal geometry is a way of describing complex, irregular shapes that repeat themselves in nature. Take a leaf on a fir tree, for example. The leaf itself is a mini-me version of the whole tree. And if you look at the individual bits of the leaf, they look like the leaf that looks like the tree. So you have a complex mathematical formula that describes a pattern that keeps repeating itself. Thus a single formula describes lots and lots of data. This is the kind of thing that makes mathematicians happy.

The practical applications of the theory are that it can be used to model and describe, though not predict, a huge number of complex phenomena such as coastlines, water and air turbulence, galaxy clusters, and the fluctuations of stock markets and commodity prices (there is an hilarious passage in the book describing early research on cotton price fluctuations that reads like a comedy thriller). By describing such phenomena, fractal geometry moves on from Euclidian geometry, which is confined to smooth shapes and planes.

Armed with this very basic understanding of his work, I meet the man himself in a fantastically noisy French restaurant. Mandelbrot is tall, fairly robust, and has the thick-lensed spectacles of academic cliche. He has a Clouseau-esque accent, despite having worked at IBM in the US since 1958. But there is little of Peter Sellers about this man. His mind is like a steel trap.

It turns out the publication of The (Mis)Behaviour of Markets - A Fractal View of Risk, Ruin and Reward is not entirely designed to upset the financial community. Mandelbrot, after all, is a teacher with a didactic urge: "Part of my business may be to return mathematics and geometry to its role as an instrument to organise and understand the patterns of nature."

At this point readers may be crying out that markets are supposed to be rational phenomena, not natural ones. According to the efficient market hypothesis (a phrase coined by one of Mandelbrot's students, apparently) only rationally relevant information is priced into an asset or market. So what use is a formula that describes the natural world?

Mandelbrot's point is that, whatever the causal factors that go into price movements, markets and prices behave as if they are natural phenomena. He says: "My purpose is always to observe the symptoms and have a model of what is being seen. In the case of markets, it is frightening because there are so many people of great brilliance and extraordinary greed who work there. They don't understand the market, but they understand the numbers."

It is easy to see why Mandelbrot has a reputation for arrogance. He is, simply, very clever indeed, and is impatient with those who aren't. His fractal theory identifies three states of randomness - mild, slow and wild - and he believes that this model describes market behaviour far better than any other theories of randomness.

If he is hard on the financial community, it is because he believes investment managers and advisers are failing investors: "A stockbroker wrote me a very plaintive letter asking why I was giving stockbrokers such a hard time. His argument was that what he did was right 98 percent of the time. Why bother about the events that occur in the rest of the time? The answer is that those events are the ones that really count."

It is said that no one hurries like an old man, and Mandelbrot knows that at 79, time is precious. This only exacerbates his impatience with the financial community: "It is quite clear that some portfolios that were declared to be free of risk turned out not to be. They are very good for 90 percent or more of the time, but at the critical moment, they fail. They are just dreadful. Given the inter-connectedness of things, they may lead to very, very embarrassing complications for the whole world."

His best attempt to save the world, or at least make society aware of its incomprehension of the riskiness of the markets it depends on so much, is probably contained in a book that is surprisingly entertaining (much credit must go to Richard Hudson, the former European bureau chief of The Wall Street Journal, and co-author).

Now Mandelbrot is writing his memoirs - "purely what I remember. I won't research and check dates unless I absolutely have to". They should make compelling reading. Born in Warsaw in 1924, his family moved to Paris in 1936. As a Jew he was lucky to survive the Nazis, and had to move constantly. One possible benefit was the lack of a conventional education (he was tutored by an artist uncle who was also a professor of mathematics).

Mandelbrot says he inherited his independent tendencies from a father who saved his own life by refusing to stay on the road with fellow Jews who had been liberated from an internment camp during the war. Mandelbrot senior set out on his own and took refuge in a wood; those who stayed on the road were strafed by Stuka fighter planes. Following his father's untimely death, the youthful Mandelbrot also learned about the market, selling the crude clothing his father had made to eke out a living at distressed seller prices. "It put food in our mouths," he says.

Mandelbrot was a brilliant student and held a variety of academic positions, before resigning a post in France in 1958 in order to work in IBM's famous ideas factory (a group of oddball intellectuals paid to come up with great innovations). He was already seen as a cross-disciplinarian and a maverick and had created for himself "a very hostile intellectual environment. France does not like people not to belong".

For the last five years he has been at Yale, feted for the long-delayed publication of his paper on fractal geometry. And now, in his memoirs, he's turning those fearsome analytical powers on himself: "There are many things I begin to understand better now."

*The*by Benoit B Mandelbrot and Richard L Hudson is published by Profile Books, £18.99

*(*Mis*)*behaviour of Markets: A Fractal View of Risk, Ruin and RewardSunday Telegraph 17/10/2004