[In the following review, Pippard discusses chaos theory and offers a favorable assessment of Gleick's treatment of the subject in Chaos.]

Haydn's Creation opens with a Representation of Chaos which, while adhering to the rules of musical grammar, confounds at every turn the listener's expectations. Chaos is for Haydn no illogical riot but a paradoxical coexistence of logic and unpredictability, and it is in this sense that the word was imported into science in 1975 to describe what has since been recognized as a pervasive mode of behaviour. The technical literature is voluminous—there is already a new journal dedicated to its study—and very heavy reading it makes, even when not presented in the impenetrable shorthand of pure mathematics.

To appreciate what is involved, consider a tennis-player repeatedly bouncing a ball into the air off his racket. Of course he times his movements to keep the ball in play, but let us replace him by a simple automaton that jiggles the racket up and down perfectly regularly; will the ball lock into synchronization with the racket so that it bounces to the same height every time? Yes, if the amplitude of the racket's oscillation is about right. But let the amplitude be increased steadily, and at a certain moment the even spacing of the bounces will give way to alternating longer and shorter intervals; the motion is still regular, but now with a repetition cycle of two bounces, one higher and one lower, over and over again. Further increase of the racket's amplitude leads to a close succession of period doublings; the cycle switches from two to four different heights of bounce, and then from four to eight, and so on until a critical amplitude is reached when one has to wait indefinitely for the pattern to repeat itself. This is the chaotic response of the ball to the perfectly regular motion of the racket.

The seemingly random response is not, as with many other random processes, the consequence of extraneous disturbance; exactly this sequence of period doubling occurs when the process is modelled mathematically and the rules for the ball's trajectory are laid down unambiguously. This is an example of an iterative program of the sort that computers thrive on. Given the position and speed of the ball at one moment, the computer will rapidly calculate its position and speed one period of the racket's oscillation later. And it will apply the same procedure to find position and speed one further period later, and so on until it's told to stop. The pair of values, position and speed at a given moment, can be represented by a point on a graph, and the whole history of the ball's progress from each sampling to the next shown as a series of points. In the simplest case the points may converge on to a single position, or they may jump between two, or four, or higher powers of two. In the chaotic state no point is touched twice, but discrete regions of the graph are spattered with points at a density only limited by the time the computer is allowed to run. Yet, however many points there are, their distribution always shows a pattern within the occupied regions, no matter how fine the scale on which they are examined. This is an interesting distinction between chaotic and random processes, for the latter always lead eventually to an even coverage of the available area.

To have carried through programs of this enormously repetitive character before the development of adequately powerful computers was not feasible, but now that they are readily available many different iterative programs have been studied, and have shown that period-doubling sequences leading to chaos are remarkably prevalent. Moreover they bear a great family likeness to one another, in spite of being generated by quite different sets of equations. It is the universality of their features that attracts mathematicians and physicists, who are too busy disentangling the underlying reasons to spare time explaining the basic ideas in an assimilable style.

James Gleick thinks it a pity that the fascination of chaos should be withheld from a wider audience [in Chaos], since no great command of mathematics is needed to appreciate the general ideas. One of the most entertaining aspects is still only imperfectly understood at a deep level, the connection between chaos and the rather new topic of fractal geometry. Everyone has seen a picture of a man carrying a picture of a man carrying a picture of. … This is an example of one aspect of fractals—structures possessing no inherent scale, so that a small portion when magnified appears identical to the initial structure. The patterned distribution of points described above has this characteristic fractal structure. Looked at from another point of view, the infinite regress of pictures is the outcome of an iterative process, and it is from the study of iterations that there have arisen computer-drawings of marvellous complexity, some of which Gleick reproduces. Among them is a mass of curlicues like sea-horse’ tails, but tails that are covered with spines, each of which when magnified is seen to be another sea-horse with a spiny tail, etc. The resulting picture is neither stiff nor trivial, but as entrancing as florid, yet perfectly disciplined, calligraphy.

There is, however, much more to chaos than computer programs which, besides their intrinsic interest, alert one to the possibility of similar behaviour occurring elsewhere and in real life. The motions of fluids are governed by seemingly innocent equations which nevertheless can generate those chaotically fluctuating flow patterns that have long been studied under the name of turbulence. Two particles in the fluid may begin close together, but after a while move progressively further apart until their paths bear no relation to one another. This habit of divergence, leading ultimately to uncorrelated behaviour of elements that were initially hardly distinguishable, is characteristic of all chaotic systems. However well you know the initial state there will come a time, usually a rather well-defined “horizon of predictability.” beyond which no useful statement can be made about details of behaviour. This is bad news for weather-forecasters, for the atmosphere is almost certainly a fluid in a state of chaos. To be sure, there are gross features of climate for which prediction may be possible over months or even years; but if one is interested in details of the weather (will rain spoil the next Test match?) there is no point in blaming the forecasters for their mistakes. Nor will a better knowledge of the physics, or more weather stations, or a larger computer do much to extend the range of reliability. Long-range forecasting is likely to remain in the province of sorcery rather than science.

This may also be bad news for those physicists who are persuaded that their prime task is to discover fundamental principles, the application of those principles to particular systems being a somewhat derivative activity. What the study of chaos shows is that there are countless processes which will never be brought to a state of computer-predictability, however well the basic laws are known, and which will have to be studied in their own terms if anything useful is to be said. The physicist can console himself, however, with the knowledge that there are also many systems, like the planetary orbits, which are not chaotic—indeed it is from examining these that physics sprang into being. It was possible to make progress by ignoring for the time being the awkward, apparently lawless, phenomena; returning to them, as now with chaos, in the confidence of great achievements. Is there a lesson here for the social scientist, or is the chaotic state of the social organism such that the only regularities discernible are those that no one really wants to know about, while all the answers to interesting questions lie beyond the horizon of predictability?

The study of chaos is surely more than a mathematical recreation, and may well prove an important aspect of all sciences. James Gleick has made a very praiseworthy job, in Chaos: Making a New Science, of assimilating what is known so far and presenting it in digestible doses. His achievement is the more remarkable in that he is by calling a writer rather than a scientist. But he has talked at length with many of the leaders, mostly still young, and has put together an admirably accurate-story. The journalistic tone of his character-sketches I find incongruous in an intellectual adventure. But if this is the way to attract readers, and he is probably a better judge of this than I, it's a small price to pay for a book that few would have been enterprising enough to attempt.