[In the following review of Chaos, Coyne finds Gleick's book adequate for lay readers, but notes shortcomings in Gleick's incomplete grasp of the topic and in his newspaper-style prose.]

Odd how the vocabulary of a newly credulous age seems to be invading even the best guarded of territories. Who would have thought that the words “catastrophe” and “chaos” could have become part of the common currency of that most self-consciously rational of disciplines, mathematics? Perhaps there is more to come? Could other current American predelictions, like the musings of Nostradamus, be quantified, codified and find themselves in the textbooks?

Don't rule it out. James Gleick's book [Chaos] is an object lesson in how, beneath the best explored of surfaces, there can lurk not the occult but—much worse, as far as science is concerned—the unpredictable. Moreover we are not talking of some obscure branch of algebra. Gleick's claims for the mathematics of chaos resemble those of Douglas Adams for the number 42. “Life, the Universe and Everything” is there as Gleick asserts that chaos sheds light on stock markets, heart attacks, earthquakes, the rise and fall of animal populations, the mechanics of star clusters and much, much more.

From where does chaos spring? In a phrase: from non-linear equations. Gleick rightly identifies the tendency among scientists to try and reduce everything to the linear—i.e. formulae without awkward exponents such as x² or x³—which even a computer can solve. They share too an almost religious belief that in the real universe the natural state is equilibrium. Left to themselves things will more or less stay where they are.

On the whole, science has done pretty well by both tenets, but, as ever, complacency sets in and there is a tendency to think that they are universally applicable, Just as a (very) little thing like the precession of the orbit of Mercury (43 seconds of arc a century for anyone interested) prompted Einstein to formulate the theory of General Relativity, so seemingly trivial problems in apparently unrelated disciplines have stimulated the development of revolutionary techniques.

Take, for example, weather forecasting. It is commonplace that forecasts are accurate for, at best, no more than a few days, but until recently the general view has been that this is due to the complexity of the problem and the huge number of variables involved. Given a big enough computer and sufficient good data, ran the argument, accurate longer-range forecasts should be possible. Maybe not. As long ago as 1963, Edward Lorenz, a meteorologist at Massachusetts Institute of Technology, had developed a very simple model of the atmosphere comprising no more than a dozen non-linear equations (compared to hundreds of the linear variety in conventional models), which seemed to mimic in essential details the behaviour of the weather. But the most intriguing aspect was the model's extraordinary sensitivity to its initial conditions. A difference in the fourth or fifth decimal point, of say a temperature fed into it, would lead to wildly different predictions for weather only days ahead. Mocking the scientific faith in equilibrium, it seemed that the tiniest cause could have massive effects. “It was as if,” said one mathematician, “the flap of a butterfly's wing in Brazil could set off a tornado in Texas.”

Little notice was taken of Lorenz for 15 years until suddenly, in the late 1970s, a whole host of problems started yielding to the techniques he had pioneered. The seemingly random fluctuations of fish numbers in a pond, the shape of snowflakes, the spread of epidemics, the movement of cotton prices over a century, the deadly propensity of the human heart to go into fibrillation rather than maintain a steady beat: all turned out to have aspects in common. That common factor was chaos; the potential to go suddenly from seeming equilibrium to wild fluctuation. Yet that chaos contained a curious sort of order.

Pictured graphically, the solutions to the chaotic equations resembled whorls, never repeating themselves but seemingly centred around a point or points, dubbed finally “strange attractors.” In solving them, the mathematicians had to resort to techniques which had been regarded as the purest of pure mathematics. Fractional dimensions, for example, turned out to have physical meaning. Fractional dimensions? They lie between the point (1) and the line (2) or between the line and the three-dimensional solid. What, for example, are the dimensions of a ball of string? The two dimensions of the (linear) string or the three of the ball? Mathematically, the answer turns out to be somewhere in between.

One result of these researches has been the generation of the Mandelbrot set. Arguably the most complex as well as the most beautiful object in mathematics, its infinitely detailed whorling shapes contain forms of size down to the sub-sub-microscopic, never repeating, yet revealing ever more variations on the basic shape. The Mandelbrot set has become a public emblem of chaos, featuring on book covers, brochures and even spawning a travelling exhibition of its own.

So does Gleick bring order to chaos? Up to a point. A certain breathlessness infects his prose:

A picture of reality built up over the years in Benoit Mandelbrot's mind. In 1960 it was a ghost of an idea, a faint unfocused image. But Mandelbrot recognised it when he saw it and there it was on the blackboard of Hendrik Houthakker's office.

One puts it down to the baleful influence of the New York Times, for which he works. He gives, too, the impression of not quite understanding a lot of what he describes, though, to be fair, the same could be said of a good few scientists in the field. On the plus side he has talked to virtually everybody of note on the subject and makes a very creditable attempt at rendering it intelligible to the lay person. One thing missing is a sense of irony. Just 60 years ago the search for the causes of atomic spectra ended with quantum mechanics and the abandoning of causality, so now the quest for more predictable weather forecasts leads to chaos. Disorder and unpredictability, it seems, lie at the heart of things.