Chaos in the Clockwork
The heavens themselves, the planets, and this centre, Observe degree, priority, and place, Insisture, course, proportion, season, form, Office, and custom, in all line of order.
WILLIAM SHAKESPEARE (1564-1616), Troilus and Cressida
SWEEPING AROUND the sun along a grand loop, the flattened ball we know as Earth hurtles through space at a breath-defying pace. Held captive by the sun's gravity, it maintains an elliptical course, every second adding another 30 kilometers to the log of its perpetual voyage. Its traveling companions maintain a respectful distance, each one held to its own well-worn track around the sun.
These nine planets respond not only to the enduring attraction of the sun but also to that of their neighbors. The competing tugs of these celestial compatriots append a restless spectrum of minute wiggles to a basic, sun-dominated motion, causing a jangle of deviations from perfect geometry. In this intricate, discordant symphony of the planets, the giants Jupiter and Saturn call out most loudly. Mercury, Venus,Mars, Uranus, Neptune, and Pluto contribute quieter voices. But even the imperceptible, evanescent whispers of the lesser objects in the solar system--asteroids, satellites, and comets--add to the celestial chorus, as does the thrum of the fluttering solar wind of accelerated particles and radiation continually erupting from the sun.
At the same time, the globe on which we live and wonder spins on its own axis even as it orbits the sun. Like a gargantuan twirling top, it wobbles and tilts. It shudders with every earthquake and twists fitfully with every giant swirl within its atmosphere or seas. Any unevenness in its shape or in the distribution of materials making up its crust unbalances its movements and provides a lever by which the sun and other bodies can further wrench Earth from a pure and simple motion.
This modern, remarkably detailed picture of solar system dynamics represents an astonishing triumph of human reasoning. Tugged one way, then another while whirling through space, Earth serves at best as a rickety platform from which to observe and contemplate the heavens. Moreover, nothing in everyday experience tells us directly that Earth moves. What we know about the workings of the solar system we learned first from the movements across the night sky of nothing more than pinpricks of light.
Discovery starts with personal experience. It doesn't take long for a young child to accumulate sufficient empirical evidence to sense that we are somehow tied to the ground on which we first lie, then crawl, walk, and run. Objects fall. Rocks tumble down slopes. Water flows along channels to lower basins. An arrow pierces the air in a great arc, first rising but inevitably falling to Earth. These movements have a consistent, natural direction.
The unfailing, daily passages of the sun across the sky add another crucial element, as do the corresponding movements of stars at night. These cyclic, predictable motions powerfully suggest the existence of some sort of Earth-centered order that contrasts sharply with an unpredictable existence fraught with the vagaries of weather and the perplexities of human behavior. In an unruly environment, such evidence of regularity provides reassurance in the face of great unknowns.
It's hardly surprising that in days long gone, priests, scholars, poets, and astronomers focused on celestial cycles, on sequences of events in the skies that repeated themselves at soothingly regular intervals. Only in the stately progress of the stars, repeating their motions night after night with pristine precision, in the unflagging regularity of the sun's daytime trek across the sky, and in the periodic changes in the moon's appearance and position did our forebears find the clearest evidence of some sort of rational design for the bewildering universe in which they found themselves.
But closer observations over months and years reveal subtle shifts in these patterns and a variety of unexpected movements. The sun, for instance, doesn't rise at precisely the same point on the horizon every day. Instead, the location of sunrise drifts back and forth along the horizon. These recurring excursions define a longer cycle tied to the changing seasons. Similarly, particular stars rise at different locations along the horizon and, at certain times of the year, disappear entirely from the sky for lengthy periods. These movements also have a definite rhythm tuned to the seasons.
The moon has its own cycles and thus provides another natural unit of time. Like the sun, it rises and sets at different points along thehorizon, but its cycle is much shorter than the sun's. In fact, the moon completes nearly 13 such cycles in the year that it takes the sun to complete one. The moon also regularly changes its appearance in a separate cycle of phases, from the vanishingly thin crescent of the new moon to the rotund magnificence of the full moon.
Even more curious, a handful of stars appears to shift position relative to the other stars, while maintaining their nightly traverses of the sky. Their courses confined to a broad belt stretched across the sky, these wanderers follow their own peculiar cycles. Close inspection reveals a fine structure of inexplicable whirlpools embedded in their generally smooth and languid motions.
Faced with such hints of complexity, humans naturally began to monitor the movements of lights in the sky. Endeavoring to satisfy an innate curiosity and penchant for solving puzzles, they applied their ingenuity to observing, recording, and measuring celestial events. And they turned to a characteristically human means of expressing the relationships involved and imposing order on them. Beginning with counting and going on to arithmetic and geometry, they had developed mathematics as a practical way to deal with an array of preoccupations, from enumerating livestock, dividing land, and estimating crop yields to toting up wealth, measuring area and volume, and calculating taxes. This wondrous, painstakingly assembled tool kit offered precise techniques for extracting order from confusion and uncertainty--tech--niques that could also be applied to the cosmos.
Equipped with these mathematical tools, early symbol manipulators soon realized that they could employ geometry to depict celestial motions and arithmetic to express their cyclic nature. By imprisoningtheir descriptions of the cosmos within a mathematical cage, they believed they could predict and somehow capture what they viewed as the silent but harmonious rhythms of the heavens. There was enough regularity in the movements to suggest that universal laws might govern them, and sufficient complexity and irregularity to spur the development of mathematics and science.
Human needs drove much of this development. Whether it was a matter of defining the seasons, predicting an eclipse, establishing the start of spring planting, fixing the date of a religious festival, or casting a horoscope, astronomical computation supplied reassuring answers. Mathematical investigations into celestial mechanics, which concerns the motions of the solar system, benefited navigation. These studies provided sailors with increasingly precise means of determining their position when out of sight of land, and that capability set the stage for wide-ranging exploration, global commerce, and ceaseless wandering. Those who wished to rule the waves set their best mathematicians to work grinding out tables of future planetary positions and lunar phases. Indeed, that tradition continues in today's astronomical and nautical almanacs, which are produced annually by such national institutions as the United States Naval Observatory and the Royal Greenwich Observatory.
At the heart of all this calculation lies the deeply held conviction that natural phenomena are, in essence, the consequence of just a small number of physical laws, and that these laws are best expressed in the language of mathematics. The goal is to construct a working model of the universe out of commonplace notions: ideas of number and order and measures of time and distance. With such a working model, we can leap ahead in time and predict what the otherwise opaque future has in store for us.
However, mathematics often offers more than one scheme for describing nature. More than 2000 years ago, Greek scholars proposed, pondered, and vigorously debated a variety of ingenious schemes for describing and accounting for the movements of the "wandering stars," which we know as planets. Most put Earth firmly at the center of their theories, but a few took the considerably more daring position that Earth itself was in motion.
Around the year 530, one of the last members of the Academy that Plato had founded nearly a thousand years earlier enunciated thepowerful tradition that had been passed down from Plato's time. In his commentary on one of Aristotle's works, Simplicius of Cilicia wrote, "Plato lays down the principle that the heavenly bodies' motion is circular, uniform, and constantly regular. Thereupon he sets mathematicians the following problem: What circular motions, uniform and perfectly regular, are to be admitted as hypotheses so that it might be possible to save the appearances presented by the planets?"
The machinery built to meet Plato's criteria worked remarkably well, and astronomers over the centuries tinkered with its details to improve their predictions. Other than as a point of philosophical or theological dispute, it didn't really matter whether the planets revolved around Earth or around the sun. The question always came down to which set of mathematical formulas could most efficiently generate tables containing the most accurate predictions.
But more than accuracy was at stake. It's much easier to carry around a scrap of paper containing three or four simple formulas than it is to lug around a huge volume filled with facts and detailed observations. The object of a mathematical model, then, is not just to describe natural phenomena accurately, but also to do it in the most compact, economical way possible.
In the seventeenth century, first Johannes Kepler and Galileo Galilei and then Isaac Newton established a physical basis for the mathematical model that best describes natural phenomena. Newton was able not only to extract out of a heap of confusing observational data the underlying physical laws but also to invent a new, labor-saving mathematical tool--the calculus--with which to express those relationships. In essence, he demonstrated that problems in mechanics deal with objects moving in response to forces and that a surprisingly wide array of natural phenomena can be deduced, described, and explained by those forces and by the mathematically expressed laws of motion governing them. Central to Newton's ideas was the notion that the fundamental laws of nature are the same everywhere. A stone dropped from a tower in Italy will behave in the same way as a stone dropped from a tower in England. The moon is subject to the same kind of force as a falling apple.
In his magnificent, revolutionary treatise, Philosophiae naturalis principia mathematica (The mathematical principles of natural philosophy), Newton focused on gravity and its effects, but there are hintsthat he was prepared to go much farther. In the preface, he wrote, "I wish I could derive the rest of the phenomena of Nature by the same kind of reasoning from mechanical principles, for I am induced by many reasons to suspect that they may all depend upon certain forces by which the particles of bodies, by some causes hitherto unknown, are either mutually impelled toward one another and cohere in regular figures, or are repelled and recede from one another. These forces being unknown, philosophers have hitherto attempted the search of Nature in vain."
Newton's equations of motion and his law of universal gravitation were sufficient for handling any problem that involved the movements of bodies in the solar system. With such a fundamental, simple framework in place, it didn't take much of a leap for Newton's successors to imagine planets confined forever to totally predictable, orderly orbits around the sun. Such a perpetual clockwork would never need rewinding or adjusting. This mathematical machinery, by encapsulating completely the solar system's past, present, and future, in principle seemed to leave no room for the unforeseen. It therefore turned astronomers and mathematicians naturally to the question of the solar system's stability.
Concerns about stability arise in practically any dynamical system. Imagine a ball sitting at the bottom of a large, hemispherical bowl. A small nudge forces the ball aside, but it immediately tries to return to its original position. If there is no friction, the ball continues to oscillate back and forth or to trace a curved path along the bowl's contours. This represents a stable situation. In contrast, a ball precariously balanced on top of an overturned bowl represents an unstable situation. A small nudge moves the ball aside, and it continues to move from its original position, never to return.
In the case of the solar system, the question is whether the multifarious influences of every body on every other body shift the planets from their essentially unchanging orbits only slightly and temporarily, or whether these effects can eventually lead to radical, irreversible changes. Billions of years in the future, will the planets continue tracing roughly the same paths as they do now, or will a time come when Mars will catastrophically smash into Earth, or Pluto escape from the solar system? Could Earth itself drift close enough to the sun to become a twin of veiled, noxious Venus?
In one way or another, the problem of the solar system's stability has fascinated and tormented astronomers and mathematicians for more than 200 years. Somewhat to the embarrassment of contemporary experts, it remains one of the most perplexing, unsolved issues in celestial mechanics. Each step toward resolving this and related questions has only exposed additional uncertainties and even deeper mysteries.
The crux of the matter hinges on the fact that it is one thing to write down the equations expressing the laws of motion and a totally different thing to solve those equations. As Newton and his successors quickly discovered, computing the motions of the planets and other bodies in the solar system is no simple matter. In fact, the computations are often so complex that researchers now use supercomputers or specialized, custom-built, electronic machines to solve them.
At first glance, the need for high levels of computing power to describe celestial motions appears puzzling. Apart from asteroids, planetary satellites, and a variety of itinerant but minor bodies, the solar system comprises just nine planets and the sun. The tug of gravity is the only significant force affecting their motions, and the mathematical formula that expresses the relationship between this force and the masses and separations of the planets has been known for more than 300 years. It should be possible to calculate the positions of the planets at any time and to explore what the laws of physics have in store for the solar system. Why isn't it?
If the entire solar system consisted of only the sun and Earth, a straightforward pencil-and-paper exercise would explain all the movements. Just as Newton did three centuries ago, one can use a so-called differential equation to express the physical law governing the moment-by-moment relationships between the positions, velocities, and accelerations of the two bodies. Solving, or integrating, the equation means deducing from these relationships the actual trajectories followed by the bodies in question--where they have been and where they will go. The values of the variables at one instant completely determine those values at all subsequent and preceding times. The mathematics confirms that any orbit in such an idealized, two-body system would be stable. As Newton demonstrated, Earth, for instance, would whirl endlessly around the sun, keeping forever to its elliptical course.
But add another planet to the system, and now there are three bodies tugging on one another. Earth can no longer keep to its precisely elliptical path. It continues to orbit the sun, but, depending on its distance from the other planet, it is affected at different times by a different gravitational pull. Those perturbations distort its trajectory in space, just as Earth's influence in turn perturbs its companion's orbit.
In this case, the equations representing the movements of the three gravitationally interacting bodies spawn no simple mathematical formula that can describe and predict the paths of all three bodies with unlimited accuracy for all time. The problem becomes even worse when four or more bodies are involved. The best that anyone can do is to calculate first the major effects--such as the sun's preponderant influence--and then step by step take into account other, less significant perturbations. Such strings of approximations allow mathematicians and astronomers to close in on the answers they're after. When applied to an actual planetary arrangement, however, this procedure requires horrendous amounts of computation. Indeed, the sheer physical labor involved in applying so-called perturbation theory--the notion that a "real" problem can be solved by making minor modifications to a simple, ideal situation-limits its usefulness as a mathematical tool. This is especially true when one wants to peer ahead billions of years or look back to the origins of the solar system.
Before the advent of reliable mechanical calculators and high-speed electronic computers, the trick was to adopt plausible approximations that provided predictions of planetary positions for a given date with a prescribed level of precision. However, the more precisely one tried to pinpoint a given celestial event, the more calculations it took to make a sufficiently reliable prediction. And the computations were done by hand with the aid of interminable, eye-taxing tables of logarithms. Indeed, until the last half century, the word computer meant a person who performed the painstakingly tedious calculations required in astronomy and other number-intensive fields. Whereas present-day astronomers can turn to the electronic machines on their desks, few astronomers in the past could get by without the ranks of anonymous assistants and clerks who performed the necessary calculations.
Electronic digital computers now handle the monotonous calculations required to compile the astronomical tables used in predicting planetary positions and celestial events such as eclipses. But scientists continue to probe the details of those calculations to understand and correct the minuscule defects that mar the accuracy of their predictions. Measurements relayed from spacecraft, along with radar signals bounced off the moon and other objects, enable distance and mass to be determined with an accuracy never before achieved. With the help of this detailed information and the prodigious calculating capacity of computers, scientists can tackle dynamical subtleties that were out of reach just decades ago.
Such precision has many uses. For example, careful computation of both planetary and spacecraft positions, combined with judicious course corrections, steered Voyager 2 nearly 5 billion kilometers on a 12-year trek from Earth; the spacecraft arrived for its encounter with Neptune on schedule and within a few kilometers of its target. Similarly, historians count on calculations of the timing of eclipses to help date historical events that took place thousands of years ago. Conversely, comparisons between ancient astronomical records and contemporary computations yield discrepancies that shed light on minute changes in the rate at which Earth spins on its axis.
Whether seated beneath a planetarium's darkened dome or before a high-resolution monitor displaying celestial scenes created by sophisticated software, we too can wander freely through time and space, confident that the sky appears to us as it did to Ptolemy nearly 2000 years ago, and as it will to our descendants a hundred years hence.
The quality of observation and calculation is now sufficiently high that scientists can probe the solar system's near future and recent past with a considerable degree of confidence. They can watch orbits evolve over eons and check for signs of instability. They can look into Earth's orbital history for evidence of minute wobbles and changes in orbital shape that may have been strong enough to affect our planet's climate and geological history.
But computation has also clearly demonstrated what mathematician Henri Poincaré at the turn of the century understood but couldn't quite visualize--that mechanics and the laws of physics, as formulated by Isaac Newton, are really much richer than Newton and his disciplesdreamed possible. Newton's equations encompass not only the precisely predictable but also the erratic and chaotic. Moreover, this dual nature of the equations appears to mirror the behavior of physical systems that can readily shift from an apparently orderly, predictable type of motion to an irregular, unpredictable course.
The double pendulum is an example of this type of physical system. A single pendulum consisting of a rod pivoted at one end simply swings back and forth. But the addition of a second rod, pivoted from the bottom of the first, greatly increases the motion's complexity. Gyrating like a trapeze artist bent at the waist, the double pendulum oscillates capriciously, sometimes swinging in wide arcs, sometimes pausing for short periods before resuming its unpredictable course.Sometimes the lower rod does all the swinging, while sometimes the upper rod carries the bulk of the motion. At other times the double pendulum acts as if it were a single pendulum. These remarkable gyrations can have such a strong, hypnotic appeal that museum and airport gift shops regularly offer glitzy contraptions patterned on the double pendulum.
In both science and mathematics, chaos is the technical term now used to describe such erratic activity. First applied in 1975 by mathematician James Yorke, chaos refers to the apparently unpredictable behavior of a deterministic system governed by mathematically expressed rules. Although its everyday meaning suggests wild, confused behavior, chaos to its initiates merely signifies limited predictability. The key idea is that the behavior of a chaotic system tends to change drastically in response to a slight change in initial conditions. Two identical double pendulums starting at slightly different positions will very quickly go out of synch and embark on highly individual movement patterns. Similarly, certain mathematical equations will produce very different strings of results if the initial values differ by even a tiny fraction.
Spectacular advances in computing power have brought about a dramatic increase in the understanding of basic dynamics. Taken together, the solutions to the equations that describe a dynamicalsystem typically encompass a variety of coexisting states, some ordered and some disordered. For example, displacing a double pendulum by only a small angle from its vertical starting position produces a regular, predictable motion. But displacing the same pendulum by a large angle initiates an unpredictable, chaotic motion. Order and chaos represent twin manifestations of an underlying determinism. Neither exists in isolation.
Numerical investigations also suggest that the chaos and order of theoretical Newtonian mechanics have counterparts in the solar system. Several strikingly different methods of computing and tracking the evolution of planetary orbits all generate results confirming that chaos lurks in the planetary clockwork. Researchers see evidence of dynamical chaos in the gaps between the orbits of the minor planets of the asteroid belt, in the tumbling motion of Hyperion (one of Saturn's satellites), and in the perplexingly twisted rings that encircle the outer planets. The latest numerical evidencepoints to traces of chaos in Pluto's peregrinations and even in Earth's orbital motion, although the effects are so far too minute to send either planet careening out of its present orbit. The solar system has apparently survived for more than four billion years in some semblance of its present form, but it isn't quite as placid or predictable as its venerable clockwork image suggests. Nothing guarantees that the future holds no surprises.
However, the presence of chaos in the solar system remains a controversial idea. Not everyone accepts the verdicts emerging from computations that simulate planetary motions over billions of years. Critics argue that approximations are involved in the calculations and that numerical errors inevitably creep in. The fact that previous computational models produced quite different, sometimes contradictory, answers adds to their skepticism.
But computer models of the solar system have improved considerably. Present-day calculations extending millions of years into the past and future are sufficiently accurate that, in the words of modeler Scott Tremaine, "If I were an astrologer, I'm confident that I could tell you what your zodiacal sign was if you had been born a million years ago."
Vastly different methods of computing and tracking the evolution of planetary orbits seem to generate remarkably similar answers. Even more significant is the fact that mathematicians have proved that certain simplified mathematical models of the solar system have chaotic solutions. These achievements are important because, in principle, computation by itself can never prove that a particular system is truly chaotic. However strong the numerical evidence, it can only suggest the possibility of chaos.
None of this work provides evidence that the solar system is falling apart. The real issue is much more fundamental. It's difficult, if not impossible, to make accurate predictions more than a few million years into the future. The dynamics of the solar system contain an element that simply can't be captured by calculation.
If the Newtonian mathematical apparatus used to model the solar system is an appropriate stand-in for the real thing, what does its bizarre, two-faced behavior suggest about what we can determine of the solar system's long-term future? Is there a horizon beyond whichno computation can provide a reliable picture? Does this mean the end of our efforts to settle the question of the solar system's stability?
In 1773, at the age of 24, Pierre-Simon de Laplace became one of the first to make a serious attempt at a mathematical proof of the solar system's fundamental stability. Opinion at that time was decidedly mixed on the subject. Isaac Newton believed that divine intervention might occasionally be necessary to put the solar system back in order and prevent its dissolution. Leonhard Euler, impressed by the immense difficulty of accounting for even the moon's motion, despaired of being able to cope sufficiently with the innumerable forces and complicated interactions contained in any realistic model of the solar system. He thought it impossible to make fair predictions of its destiny.
"All the effects of nature are only the mathematical consequences of a small number of immutable laws," Laplace insisted, and he applied his formidable mathematical powers to deciphering the dynamics of the solar system. After completing his exhaustive analysis, he declared that the solar system is stable. The planets repeat their complicated cycles forever, never straying far from their mandated courses.
The notion of Earth and the other planets endlessly whirling around the sun, keeping forever to their nearly elliptical orbits, seems both reassuring and dreary. But Laplace's hard-won proof of such stability applied to a mathematical model of an idealized solar system, not to the real world. His model neglected a variety of subtle gravitational influences that might have changed the outcome of his investigations.
The stability question remains open, but now scientists can explore, to an unprecedented level of detail, related questions about changes in planetary orbits. High-speed computers make possible the creation of sophisticated models of nature, on which experiments can be performed. The result is a new kind of science that is neither true experimentation nor pure theory, and its success depends on the degree to which computational models can be made to behave like natural phenomena. Its capabilities open up areas of research that were inaccessible in the past.
Astronomy, in particular, has felt the impact of the computer revolution. For generations, astronomers working in celestial mechanics were stuck with just one solar system on which to test their theories.Today, using electronic time-and-space machines, they can in effect audition alternative solar systems and acquire a sense of the variability that nature allows. They can shift the position of a single planet or create a solar system with a unique geometry and then experiment with the system's behavior. And they can do it again and again.
Such experiments may have something to say about the uniqueness of our own solar system. Although its stability hasn't been proved in a mathematical sense, it's evident that the solar system has survived for billions of years in roughly its present configuration. Is this particular distribution of planets just one of many possible stable arrangements, or is it the only one that can produce a stable solar system? Would there be room for another planet, or would its influence disturb the system's apparent stability? Did the solar system in its very early history slough off excess material so that only a stable core remained? To what extent did the need for stability force the solar system into its present arrangement?
In one sense, such questions bring astronomers back to the kind of widely discredited numerology that fascinated scientists like Johannes Kepler. Kepler's first major astronomical work contained an explanation for the existence of precisely six planets--the number known in 1596 when the book was published--at particular distances from the sun. He connected the planetary orbits with the five regular polyhedra, or Platonic figures, by nesting the appropriate polyhedron between each successive pair of spheres defined by the orbits of the planets.
Kepler's ingenious structure wasn't the only one to attach significance to the solar system's numerical parameters. In 1766, Johann D. Titius, a German mathematician, physicist, and astronomer, noted an intriguing, amazingly simple numerical pattern that apparently captured planetary distance relationships. He started with a sequence of numbers: 0, 3, 6, 12, 24, 48, 96, 192, 384, 768, and so on, in which each new term beyond 3 is obtained by doubling the previous term. By adding 4 to each term and then dividing by 10, he obtained a new sequence of numbers, each of which represented the average distance of a known planet from the sun relative to the average distance of Earth from the sun. For the most part, the resulting numbers corresponded closely to the measured ratios.
No one paid much attention to the inconspicuous footnote (appended to his translation of a French science book) in which Titiusfirst announced this peculiar numerical relationship. Then Johann Eilert Bode of the Berlin Observatory dug it up and made it public in 1772. The Titius-Bode formula gained credibility when the planet Uranus was discovered in an orbit whose average distance from the sun was reasonably close to where the formula predicted it would be. The formula also led to the sighting of the first asteroid, which rendered significant a number in the sequence that previously had been without a corresponding celestial body. But it failed to produce accurate relative distances for the outlying planets, Neptune and Pluto, when those planets were discovered.
The fact that the Titius-Bode formula comes even as close as it does to predicting the positions of the planets continues to intrigue some planetary scientists. Is this numerical relationship really just a coincidence, or does it have something to do with the dynamics and stability of the solar system? Perhaps there was something in the early history of the solar system that inevitably led to such a pattern.
Because we can't experience the five billion or so years remaining before the sun inflates to a red giant and destroys the solar system as we know it, we must rely on mathematics as the most effective and trustworthy means that we have to understand what we see around us. That mysteriously powerful creation of the human mind, which allows us to compress the universe into arrays of symbols on a printed page, continually amazes with its applicability to natural phenomena and with its subtlety and depth.
The history of astronomy and of the perennial effort to predict the course of celestial bodies and to solve the puzzles of the solar system are intricately intertwined with the history of mathematics and computation. Astronomy was the first mathematical science. It was arithmetic and geometry, in conjunction with observation, that led to the all-encompassing theoretical structure that we have at our disposal today.
It seems in some ways fortunate that our solar system happens not to be so complex that we could never have entertained the notion that simple physical laws govern the universe and have developed a language with which to express those laws. Imagine the tangled motions that astronomers on a planet caught in the clutches of two orbiting stars would observe. Could they ever discern the simple mathematical formulas underlying those erratic movements? Could any form of life, let alone astronomers, even exist in a situation so unstable?
The exploration of our own solar system has taken us from clockwork precision into chaos and complexity. Human limits and computational triumphs marked the halting steps along the way, providing the ingredients of a marvelous mathematical detective story. The tale begins with our distant forebears, who first saw in the night sky a message to pore over and comprehend. It leads to the startling perspectives that the time-and-space machines of modern research have provided of our solar system and the universe in which it drifts.