The answer to life, the universe and everything

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The Independent Online
Does God play dice with the universe? Or is everything that happens the consequence of rigid "laws of nature"? Albert Einstein was convinced that the universe is law-abiding. In a letter to Max Born he wrote: "You believe in a god who plays dice, and I in complete law and order." Born was a pioneer of quantum physics, in which events on sub-atomic scales are held to be governed by pure chance. Einstein disagreed profoundly.

Today's physicists mostly side with Born - though a few mavericks still wonder if Einstein was right all along. Outside the world of the quantum, though, it has become clear that drawing a sharp line between law and chance is a mistake. Statistics teaches us that random events have their own laws; the new mathematics of "chaos theory" and the intricate shapes known as "fractals" - made up from tiny copies of themselves such as a cloud or a tree - have revealed that rigid laws often give rise to entirely random behaviour. So the question for today's science is no longer whether God plays dice, but how and to what effect.

At a recent conference in Italy the geologist Donald Turcotte announced that by marrying chaos theory to hard data he had discovered a new universal pattern in seemingly random events. The idea first arose when he was studying the severity of floods in the US. The same pattern has since shown up in magnets, aircraft turbulence, species in an ecosystem, avalanches, earthquakes, and forest fires, and these are just the tip of the iceberg. In fact, the same pattern probably occurs in icebergs, too.

Turcotte's discovery threatens the statistical supremacy of the "bell curve", which describes how measurements fluctuate around their average. The new pattern could be important in engineering, insurance, weather prediction, ecology - any area where rare events are important. Insurers who use the bell curve to work out the chances of a major hurricane are underestimating the likelihood of such disasters and so have to pay out a lot more money than they'd expected. The new pattern gives a far more accurate prediction.

The bell curve originated in 18th-century astronomy, as a universal pattern in observational errors. Here, observed data cluster around some central value in a simple, universal manner. An overestimate is just as likely as an underestimate, and big errors are less likely than small ones. If you represent the distribution of the data by a curve, you get a picture that vaguely resembles a bell: a hump in the middle, tailing off symmetrically to left and right.

If the bell curve had remained the property of astronomers, a lot of trouble might have been avoided. But the Frenchman Adolphe Quetelet discovered that the bell curve also shows up in social data: people's heights, criminal convictions, drunkenness. The bell curve came to occupy a central role in statistics, and a scientific urban myth came into being that it is the "natural" way to describe variable data. Real data, however, do not always follow the bell curve. The bell curve predicts that Bill Gates cannot exist, because the chance of any individual owning so much wealth is (so it says) so incredibly small. Ecosystems contain more rare species than the bell curve indicates. Turcotte's experience with real data led him to replace the bell curve by one that extends further in one direction. This lopsided pattern increases the probability of rare events. Events that the bell curve forbids become possible, though still rare, and Bill Gates can exist after all.

Why does the same new curve keep turning up all over science? The systems involved are all fractal - they have detailed structure on many scales. Turbulent air flow involves vortices of all sizes; large ecosystems have much in common with small ones. This "self-similarity" seems to be why the same lopsided curve is showing up in so many different areas of science.

How novel is this idea? It depends who you talk to. Research statisticians already know that the bell curve is often misapplied. Observational data, for example exchange rates on the money markets or the sizes of sand grains on beaches, often follow radically different patterns. Unfortunately, this message has not filtered down to many users of statistics. Turcotte's curve is just the latest in a growing pile of alternatives - but this one comes with a pedigree: the fractal connection. In some quarters any such connection is a cause for scepticism, on the curious grounds that fractal pictures are pretty. Yes, they are: this is because nature's mathematics is beautiful. Chaos and fractals are central features of the mathematics that underlies much of today's science. It is silly to pretend otherwise.

Be that as it may, Turcotte's discovery tells us that if God does play dice, then those dice are biased in favour of rarities, which may explain why this particular universe is such an interesting one.

Ian Stewart is professor of mathematics at Warwick University.