Read The Faber Book of Science Online
Authors: John Carey
Benoit Mandelbrot (b. 1924), a maverick mathematician of Lithuanian extraction, coined the word ‘fractal’ (from the Latin
fractus,
broken) in 1975, to describe irregular geometrical shapes that repeat themselves endlessly on smaller and smaller scales – a feature known as ‘self-similarity’. Fractals, he found, were everywhere in nature – in coastlines, trees, mountains, forked lightning, the cardio-vascular system, or cauliflowers (where each floret of the edible head resembles the whole). Fractal pictures, produced by repeatedly feeding equations into computers, became popular in T-shirt and poster design in the 1980s. The connection between fractals and chaos theory is explained here by Caroline Series, reader in mathematics at Warwick University.
Most people have become familiar in recent years with pictures of fractals, those elusive shapes that, no matter how you magnify them, still look infinitely crinkled. The pictures you saw were probably drawn by computer, but examples abound in nature – the edge of a leaf, the outline of a tree, or the course of a river. Fractal curves differ from those studied in normal geometry. The curve of a circle, for instance, if magnified sufficiently, just about becomes a straight line. A fractal curve, on the other hand, when viewed on many different scales, from macroscopic to microscopic, reveals the same intricate pattern of convolutions.
In recent years, there has been a revolution of interest in fractals. Previously, only a few people had appreciated the significance and beauty of these strange shapes. Benoit Mandelbrot drew much attention to their potential use in describing the natural world. At the same time, the development of high-speed computing and computer graphics has made them easily accessible and this has drawn many people to study them more closely.
Another reason for the interest in fractals is that they are connected with chaos. In mathematics, chaos has a specialized meaning. The easiest way to understand chaos is by some examples.
Suppose a particle is moving in a confined region of space according to a definite deterministic law. Following the path traced out by the particle, we are likely to observe that it settles down to one of three possible behaviours – the geometrical description of which is called an attractor. The particle may be attracted to a final resting position (like, for example, the bob on a pendulum as it gradually settles down to rest). In this case, the attractor is just a point – the final resting position of the bob. Or the particle may settle down in a periodic cycle (like the planets in their orbits around the Sun). Here the attractor is an ellipse and the future motion can be predicted with astonishingly high accuracy as far ahead as we want. The last possibility is that the particle may continue to move wildly and erratically while, nevertheless, remaining in some bounded region of space. The motion of some of the asteroids, for example, appears to exhibit exactly this phenomenon. Tiny inaccuracies in measuring the position and speed of the asteroid quickly lead to enormous errors in predicting its future path. This phenomenon is the signal of chaotic motion. The regions of space traced out by such motions are called strange attractors.
Once a particle is attracted to a strange attractor there is no escaping. Almost anywhere you start inside the attractor, the point moves, on the average, in the same way, just as no matter how you start off a pendulum, it always eventually comes to rest at the same point. Although the motion is specified by precise laws, for all practical purposes, the particle behaves as if it were moving randomly. The interesting point here is that strange attractors are very frequently fractals.
The meteorologist Edward Lorenz pioneered chaos theory in a 1963 paper (though he did not call it chaos theory – the name was invented in 1972 by the mathematician James Yorke). Studying weather systems, Lorenz attributed their unpredictability to the fact that a very small initial difference could enormously change the future state of the system. This became known as ‘the butterfly effect’ from the title of Lorenz’s 1979 paper ‘Predictability: Does the Flap of a Butterfly’s Wings in Brazil Set Off a Tornado in Texas?’ Lorenz at first used a seagull as his example, but the butterfly was more dramatic.
Fractal pictures, and their link with chaos, have inspired science-writers to make claims about the underlying ‘beauty’ of nature or mathematics. Ian Stewart’s rhapsody is typical.
Chaos is beautiful. This is no accident. It is visible evidence of the beauty of mathematics, a beauty normally confined within the inner eye of the mathematician but which here spills over into the everyday world of human senses. The striking computer graphics of chaos have resonated with the global consciousness; the walls of the planet are papered with the famous Mandelbrot sets.
This seems to assume that beauty is an absolute, rather than a subjective value. But of course it is not. In reality any of us might protest that fractal pictures are not beautiful, but as kitsch as a luminous cauliflower, and no one could prove us wrong. To any arts undergraduate this would be an obvious point, and Stewart’s neglect of it illustrates the gap that still exists between the two cultures. So (in the reverse direction) does the confusion among arts students about chaos theory, which is widely interpreted as meaning both that causation does not exist (‘nature is chaotic’) and, simultaneously, that chains of causation are so rigid that a butterfly’s wing can cause a tornado. Paul Davies’s is one of the most lucid attempts to resolve these difficulties in terms intelligible to the semi-numerate.
All science is founded on the assumption that the physical world is ordered. The most powerful expression of this order is found in the laws of physics. Nobody knows where these laws come from, nor why they apparently operate universally and unfailingly, but we see them at work all around us: in the rhythm of night and day, the pattern of planetary motions, the regular ticking of a clock.
The ordered dependability of nature is not, however, ubiquitous. The vagaries of the weather, the devastation of an earthquake, or the fall of a meteorite seem to be arbitrary and fortuitous. Small wonder that our ancestors attributed these events to the moodiness of the gods. But how are we to reconcile these apparently random ‘acts of God’ with the supposed underlying lawfulness of the Universe?
The ancient Greek philosophers regarded the world as a
battleground
between the forces of order, producing cosmos, and those of disorder, which led to chaos. They believed that random or
disordering
processes were negative, evil influences. Today, we don’t regard the role of chance in nature as malicious, merely as blind. A chance event may act constructively, as in biological evolution, or destructively, such as when an aircraft fails from metal fatigue.
Though individual chance events may give the impression of lawlessness, disorderly processes, as a whole, may still display
statistical regularities. Indeed, casino managers put as much faith in the laws of chance as engineers put in the laws of physics. But this raises something of a paradox. How can the same physical processes obey both the laws of physics and the laws of chance?
Following the formulation of the laws of mechanics by Isaac Newton in the 17th century, scientists became accustomed to thinking of the Universe as a gigantic mechanism. The most extreme form of this doctrine was strikingly expounded by Pierre Simon de Laplace in the 19th century. He envisaged every particle of matter as
unswervingly
locked in the embrace of strict mathematical laws of motion. These laws dictated the behaviour of even the smallest atom in the most minute detail. Laplace argued that, given the state of the Universe at any one instant, the entire cosmic future would be uniquely fixed, to infinite precision, by Newton’s laws.
The concept of the Universe as a strictly deterministic machine governed by eternal laws profoundly influenced the scientific world view, standing as it did in stark contrast to the old Aristotelian picture of the cosmos as a living organism. A machine can have no ‘free will’; its future is rigidly determined from the beginning of time. Indeed, time ceases to have much physical significance in this picture, for the future is already contained in the present. Time merely turns the pages of a cosmic history book that is already written.
Implicit in this somewhat bleak mechanistic picture was the belief that there are actually no truly chance processes in nature. Events may appear to us to be random but, it was reasoned, this could be attributed to human ignorance about the details of the processes concerned. Take, for example, Brownian motion. A tiny particle suspended in a fluid can be observed to execute a haphazard zigzag movement as a result of the slightly uneven buffeting it suffers at the hands of the fluid molecules that bombard it. Brownian motion is the archetypical random, unpredictable process. Yet, so the argument ran, if we could follow in detail the activities of all the individual molecules involved, Brownian motion would be every bit as predictable and deterministic as clockwork. The apparently random motion of the Brownian particle is attributed solely to the lack of information about the myriads of participating molecules, arising from the fact that our senses are too coarse to permit detailed observation at the molecular level.
For a while, it was commonly believed that apparently ‘chance’
events were always the result of our ignoring, or effectively averaging over, vast numbers of hidden variables, or degrees of freedom. The toss of a coin or a die, the spin of a roulette wheel – these would no longer appear random if we could observe the world at the molecular level. The slavish conformity of the cosmic machine ensured that lawfulness was folded up in even the most haphazard events, albeit in an awesomely convoluted tangle.
Two major developments of the 20th century have, however, put paid to the idea of a clockwork universe. First there was quantum mechanics. At the heart of quantum physics lies Heisenberg’s uncertainty principle, which states that everything we can measure is subject to truly random fluctuations. Quantum fluctuations are not the result of human limitations or hidden degrees of freedom; they are inherent in the workings of nature on an atomic scale. For example, the exact moment of decay of a particular radioactive nucleus is intrinsically uncertain. An element of genuine unpredictability is thus injected into nature.
Despite the uncertainty principle, there remains a sense in which quantum mechanics is still a deterministic theory. Although the outcome of a particular quantum process might be undetermined, the relative probabilities of different outcomes evolve in a deterministic manner. What this means is that you cannot know in any particular case what will be the outcome of the ‘throw of the quantum dice’, but you can know completely accurately how the betting odds vary from moment to moment. As a statistical theory, quantum mechanics remains deterministic. Quantum physics thus builds chance into the very fabric of reality, but a vestige of the Newtonian-Laplacian world view remains.
Then along came chaos. The essential ideas of chaos were already present in the work of the mathematician Henri Poincaré at the turn of the century, but it is only in recent years, especially with the advent of fast electronic computers, that people have appreciated the full significance of chaos theory.
The key feature of a chaotic process concerns the way that predictive errors evolve with time. Let me first give an example of a non-chaotic system: the motion of a simple pendulum. Imagine two identical pendulums swinging in exact synchronism. Suppose that one pendulum is slightly disturbed so that its motion gets a little out of step with the other pendulum. This discrepancy, or phase shift, remains small as the pendulums go on swinging.
Faced with the task of predicting the motion of a simple pendulum, one could measure the position and velocity of the bob at some instant, and use Newton’s laws to compute the subsequent behaviour. Any error in the initial measurement propagates through the calculation and appears as an error in the prediction. For the simple pendulum, a small input error implies a small output error in the predictive computation. In a typical non-chaotic system, errors accumulate with time. Crucially, though, the errors grow only in proportion to the time (or perhaps a small power thereof), so they remain relatively manageable.
Now let me contrast this property with that of a chaotic system. Here a small starting difference between two identical systems will rapidly grow. In fact, the hallmark of chaos is that the motions diverge exponentially fast. Translated into a prediction problem, this means that any input error multiples itself at an escalating rate as a function of prediction time, so that before long it engulfs the calculation, and all predictive power is lost. Small input errors thus swell to
calculation-wrecking
size in very short order.
The distinction between chaotic and non-chaotic behaviour is well illustrated by the case of the spherical pendulum, this being a pendulum free to swing in two directions. In practice, this could be a ball suspended on the end of a string. If the system is driven in a plane by a periodic motion applied at the pivot, it will start to swing about. After a while, it may settle into a stable and entirely predictable pattern of motion, in which the bob traces out an elliptical path with the driving frequency. However, if you alter the driving frequency slightly, this regular motion may give way to chaos, with the bob swinging first this way and then that, doing a few clockwise turns, then a few anticlockwise turns in an apparently random manner.
The randomness of this system does not arise from the effect of myriads of hidden degrees of freedom. Indeed, by modelling mathematically only the three observed degrees of freedom (the three possible directions of motion), one may show that the behaviour of the pendulum is nonetheless random. And this is in spite of the fact that the mathematical model concerned is strictly deterministic …
Chaos evidently provides us with a bridge between the laws of physics and the laws of chance. In a sense, chance or random events can indeed always be traced to ignorance about details, but whereas Brownian motion appears random because of the enormous number of
degrees of freedom we are voluntarily overlooking, deterministic chaos appears random because we are necessarily ignorant of the ultra-fine detail of just a few degrees of freedom. And whereas Brownian chaos is complicated because the molecular bombardment is itself a complicated process, the motion of, say, the spherical pendulum is complicated even though the system itself is very simple. Thus, complicated behaviour does not necessarily imply complicated forces or laws. So the study of chaos has revealed how it is possible to reconcile the complexity of a physical world displaying haphazard and capricious behaviour with the order and simplicity of underlying laws of nature.