Chaos: Making a New Science
By James Gleick
4/5
()
About this ebook
The “highly entertaining” New York Times bestseller, which explains chaos theory and the butterfly effect, from the author of The Information (Chicago Tribune).
For centuries, scientific thought was focused on bringing order to the natural world. But even as relativity and quantum mechanics undermined that rigid certainty in the first half of the twentieth century, the scientific community clung to the idea that any system, no matter how complex, could be reduced to a simple pattern. In the 1960s, a small group of radical thinkers began to take that notion apart, placing new importance on the tiny experimental irregularities that scientists had long learned to ignore. Miniscule differences in data, they said, would eventually produce massive ones—and complex systems like the weather, economics, and human behavior suddenly became clearer and more beautiful than they had ever been before.
James Gleick
James Gleick was born in New York in 1954. He worked for ten years as an editor and reporter for The New York Times. He is the bestselling author of Chaos, Genius, Faster, What Just Happened and a biography of Isaac Newton.
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Reviews for Chaos
1,123 ratings35 reviews
- Rating: 4 out of 5 stars4/5Well-Written; could have Employed a Little More MathFew writers write clearly and concisely about science and Mathematics. James Gleick, a former science writer for the New York Times, writes about the first years of the study of chaos. Focusing on scientists rather than science, Gleick explains the thought processes and investigative techniques researchers applied to chaos problems. Rather than attempt to explain Julia sets, Lorenz attractors, and the Mandelbrot Set with complicated equations, Chaos employs sketches, photographs, and descriptive prose.There are not many writers who have the ability to write on two planes. One is understandable by the general public. The other is appreciated by experts who grasp the subject matter and appreciate the author’s depth of understanding. I am not one of the latter. While reading the book, I found myself long for math that would connect the prose to the science.Nevertheless, this book is a history of a new science. Limited as it is, it inspired me to further study. It is probably asking too much to expect more from a book about science’s frontiers.
- Rating: 3 out of 5 stars3/5Read some time ago as part of my "physics book of the year" goal. It was interesting at the time, but I didn't invest a lot of brain-power in remembering details. Need to get an up-to-date book on the subject.
- Rating: 4 out of 5 stars4/5A very influential book, which brought the ideas of deterministic chaos to public notice. A best-seller for awhile.
- Rating: 4 out of 5 stars4/5James Gleick's early history of the science of chaos is a thorough and personal account compiled from hours of interviews, articles and lectures. Chaos is perhaps a somewhat controversial term in science and perhaps is better described as complexity forming out of simplicity or self-organizion emerging from apparent randomness. The simple, mechanistic way of viewing the world as deterministic, static and linear no longer holds water. In other words, systems are not clocklike machines destined to run down into a lifeless eternity, but rather evolve through time into more beautiful and complex patterns. At what point does a chemical feedback loop cease to become "mere chemicals" and become alive? Time can be viewed as a process, rather than a series of intervals. Gleick, for the most part, stays away from couching philosophical questions and rather lets the reader ask for themselves. This book is a fantastic introduction for those with the patience for scientific terms and interest in scientific history. For the less scientifically inclined a more general, great introduction on the subject is a book called the Turbulent Mirror, by John Briggs and F. David Peat.
- Rating: 5 out of 5 stars5/5Chaos studied here. The author makes the new way of understanding, well, everything, remarkably transparent. In the sense that I can see it, but still, I don't understand it. (!) This is not like Hofstadter's "strange loops". This is order of an even eerier take. He sort of starts with Lorenz' "butterfly effect" underlying the weather, jumps into the box of broken glass produced by Feigenbaum's nonlinear number calculations drawn from art and Nature, and then falls upon the sword of Mandelbrot's fractals. Birth pangs of a new science. New. Author is an editor/reporter for the New York Times.The failure to mention Hilbert's "Entscheidungs", posed as building blocks of mathematics, is nothing more than a personal disappointment.. I am also keening over the failure to limn the shadowing "strange loops" of Hofstadter and the ramble Bertrand Russell made of Godel, but that probably just dates me. Highest marks for taking on Nature and our Understandings of It, with sympathy, clarity and grace. Filled with snappy, even snarky biographical material. Science, made plummy.
- Rating: 5 out of 5 stars5/5My favorite chaos book. Interesting and informative. Perfectly readable for a layperson or a scientist.
- Rating: 5 out of 5 stars5/5An exhilarating read. Understanding variation, adaptability, and the overall beauty of our physical systems has never been this much fun to read about. Literally walked me through my own thesis. Absolutely brilliant!
- Rating: 4 out of 5 stars4/5As I recall (it's been decades since I read this, and I'm planning on reading "The Information"), it's a pretty good book. Lots of fractal stuff in it of course.
- Rating: 5 out of 5 stars5/5Although the book slowed down in spots it still was very exciting. I loved reading about how non linear systems or Chaos slowly emerged to change the face of many disciplines today. Being a consultant it makes me take a fresh look at the way I view and interpret data looking for strange attractors and other things. It was very interesting to see how many of the pioneers of this new science had no mentors or support within their communities. As a matter of fact many of them were warned that studying a new discipline would not bode well for their careers.
- Rating: 5 out of 5 stars5/5Absolute must-read for most anyone, but especially anyone who is involved in the sciences (professionally or otherwise) and would benefit from an alternative to standard reductionist thinking that pervades the common perceptions.
- Rating: 4 out of 5 stars4/5Achieves its goal - even after 18 years: When I first picked up Gleick's "Chaos" I was a little skeptical - could a book written in 1987 still work as an introduction to chaos and nonlinear dynamics, a field that has been evolving rapidly for the past eighteen years? Well, in a certain sense, it turns out it can.The truth is that the focus of Gleick's book is not so much chaos itself as it is the people who first explored chaos theory and eventually managed to make it respectable and bring it into the mainstream. As the book's subtitle hints, Gleick is concerned mainly with how a 'new science' is 'made', not necessarily with the actual science or math involved. This was not quite what I was expecting from "Chaos", but it is actually an advantage for the book, since its age becomes somewhat irrelevant: although chaos theory itself has been growing and evolving dramatically in recent decades, "Chaos" deals only with its roots in the '60s, '70s and early '80s. On the other hand, I was hoping for more discussion of the science itself, rather than the personalities involved in its early development.I was also not that taken with the style of Gleick's writing. His narrative tends to jump around rapidly, often spending only a few pages on some person or event before moving on to another, commonly with little in the way of connection or logical transition. This is fine for short articles in newspapers and magazines, but it doesn't work so well in a 300+ page book. The vast cast of characters (meteorologists, physicists, mathematicians, computer scientists, biologists, ecologists and many others) spins in and out of view, and it can be very difficult to get more than a general impression how the little pieces all fit together in the big picture.However, even though I'm complaining about the content and presentation, I'm still giving "Chaos" four stars. This is because "Chaos" managed to get me interested in and excited about nonlinear dynamics. Gleick was able to convey the sense of wonder and excitement that comes from looking at nature in a new way, through the lens of nonlinearity. He successfully presented the making of this new science as the greatest and most exciting scientific revolution since the development of quantum mechanics - with the difference that chaos is more accessible, more understandable, and applicable in a far wider range of fields.In short, "Chaos" still achieves its goal 18 years after it was written. It gets the reader (this reader, at least) interested in and excited about nonlinear dynamics and eager to explore the topic in greater depth. Reading Gleick's book inspired me to pick up a copy of Robert Hilborn's "Chaos and Nonlinear Dynamics" from the library and take a more serious look at the science itself. "Chaos" should make a good read for anyone who knows little or nothing about chaos or nonlinear dynamics but is curious about the topic and interested in learning a bit about its early development.
- Rating: 4 out of 5 stars4/5Very interesting, but not as well-written or as accessible as "Faster." Personally, I could have done with more concept and less profiling of the individual personalities involved.
- Rating: 5 out of 5 stars5/5I'm not a scientist, but I found this book both readable and fascinating. Gleick delves deep into some pretty abstruse subjects, but he keeps his finger on the human stories behind them and drives his history forward. Recommended.
- Rating: 4 out of 5 stars4/5I recommended this book to my son, the incipient neurobiologist, this fall and he had the same reaction to it I did: is it too late to become a mathematician? In my case, alas, the answer is obviously yes, but there is still hope for him. Anyone who can write this well and compellingly about a subject I don't even think I am interested in all that much is to be enormously commended.
- Rating: 5 out of 5 stars5/5By now a classic. This book got me into Chaos theory big time. I was even on the way to making it my thesis topic, but got seduced by the dark side... (experimental physics). Great read; what a science book should be.
- Rating: 5 out of 5 stars5/5Great Book. I read it to get a base on the 'new science' and how it was affecting thought and organization development. It's written in an easy yet engaging way, and I felt like I was reading a novel. Push my thinking through worm holes I didn't know existed and continues to challenge me. Is smeinal in my understanding of God, Church, and relationships.
- Rating: 4 out of 5 stars4/5The fundamental book on chaos science for the popular audience.
- Rating: 4 out of 5 stars4/5Chaos: Making a New Science is about a variety of topics: the sensitivity of some systems to their initial conditions, the weather being a prime example, which makes detailed long-term forecasting impossible; nonlinear systems; fractals; strange attractors; dynamical systems; etc. It is also about the people who discovered and studied these phenomena. It describes their difficulties in introducing these ideas into the scientific community. That's not an unusual situation in science. Einstein's special theory of relativity, for example, despite it's mathematical simplicity and fit with evidence, was not readily accepted.Gleick sometimes strays a bit from his topic, as when he briefly talks about Darwinian thinking in biology. He writes, "In biology, however, Darwin firmly established teleology as the central mode of thinking about cause. [...] Natural selection operates not on genes or embryos, but on the final product. [...] Final cause survives in science wherever Darwinian thinking has become habitual." (se p. 201 in the original hardback edition) I don't know where he got his information, but he got it wrong. Darwinian evolution through natural selection is not teleological. In What Evolution Is, Ernst Mayr writes, "... those who adopt teleological thinking will argue that progress is due to a built-in drive or striving toward perfection. Darwin rejected such a causation and so do modern Darwinians ..." In Darwin's Dangerous Idea, Daniel Dennett writes, "The theory of natural selection shows how ever feature of the natural world can be the product of a blind, unforesightful, nonteleological, ultimately mechanical process of differential reproduction over long periods of time." The nonteleological nature of Darwinian evolution is one of the principle themes of Dennett's book.Chaos is a long book about somewhat difficult ideas, mostly of a mathematical nature, but the mathematics is largely suppressed. One important point that I think he makes very clear is that very simple equations when iterated in real space can exhibit surprising behavior.The topics of this book are mostly outside my areas of even limited expertise, but I was wondering as I read it how many of the phenomena it describes depend on the use of real numbers, i.e., numbers that in general require infinite precision, e.g. π. If physical theories were to be developed on the basis of discrete mathematics, would some of these problems of chaos disappear? Consider the very first topic in the book: the sensitivity of weather models to initial conditions. With limited precision measuring instruments there are infinitely many states of the weather, if described by real numbers, that cannot be distinguished. So, if small differences, below the precision of measurement, can make a big difference as the weather develops, we have a problem that limits predictability. But, if the physics of weather were described by a mathematics with finite precision, then we might be able to make completely accurate measurements of initial conditions—in principle.I found Chaos interesting to read, but I am always skeptical about reading explanations of science written by journalists, just as I am skeptical of explanations of science written by philosophers.
- Rating: 5 out of 5 stars5/5I won't pretend I understood all of it. But it's written in a way so that anyone can understand much of it -- and every step on the journey is fascinating.
- Rating: 5 out of 5 stars5/5One of my all-time favorite books -- and the one that got me interested in dynamical systems. This would make the short list of books I would want along if stranded on the proverbial desert island. Gleick not only does an outstanding job of describing chaos (in the sense of sensitivity to initial conditions), but inspires readers to experiment. Reading this book while playing around with the ideas it inspires using your favorite programming language is a fun way to pass the time. (Try programming a model of the waterwheel and graphing its velocity!) Very, very highly recommended if you've ever had the slightest interest in science, math, or computers.
- Rating: 4 out of 5 stars4/5First, a couple of caveats: "Chaos: Making a New Science" was written almost twenty-five years ago, so I don't know how up-to-date it is. Also, sadly enough, I can't say I know too much about any sort of science, so I'm the "general reader" that this book is probably aimed at. Even so, I think that "Chaos" is high-quality science writing and very recommendable. Gleick approaches his subject from a number of angles: he discusses nonlinear equations and mathematical oddities, chance discoveries, computer simulations and real-world applications like turbulence and the rise and fall of animal populations, and his multifaceted approach gives the reader a remarkably complete portrait of what chaos is and how it might change our thinking about a number of well-established scientific imponderables. I'm pretty much a math illiterate and barely managed a B- in pre-calc in senior year, but Gleick's a top-shelf nonfiction writer: he's able to lay out his ideas in a way that makes them comprehensible while also preserving their uncanny beauty. The author basically squares the popular science writing circle by making the head-bendingly complex seem both important and eminently graspable. Chaos is also a new enough science that Gleick can also examine the more human side of chaos research: he seems to have personally interviewed many of this new science's major figures, describes their academic backgrounds and mindsets in detail, and is adept at capturing the excitement they must have felt when making their discoveries and transmitting it to his readers. I'm sure real-deal science students will want to start elsewhere, but I'd wager that this book has provided a good deal of inspiration and reflection among even non-scientists. From the way that scientists in widely differing fields came to ask the same sort of questions about dissimilar phenomena to the very idea that there might be some underlying structure to disorder itself, there's a lot to consider here. I'm looking forward to picking up more titles from this author.
- Rating: 4 out of 5 stars4/5Chaos was all the rage when I read Gleick's book. It's just part of the zeitgeist now.
- Rating: 5 out of 5 stars5/5Amazing book!! It is easy to read and a perfect introduction to ~Chaos.
- Rating: 4 out of 5 stars4/5I think I would have enjoyed this book more if I had read it 30 years ago, closer to when it was written. For one thing, I am 30 years older and my brain that much slower; for another, Gleick's book functions best as a history and it is disappointing that the story ended 33 years ago. Actually, i know the story didn't end, but I read the 1987 edition and I know there have been more developments in all the areas Gleick covered back then.
- Rating: 5 out of 5 stars5/5This book truly changed the way I view the world. However at times the explanations were too vague/simple and sometimes the book seemed boring.
- Rating: 5 out of 5 stars5/5A very intuitive and expressive way of explaining "The Butterfly Effect" - is what impressed me first about this book, while the detailing on the Logistic Map is the close second.If you miss understanding what a "fractal" is, you might find the rest of the book quite boring - happened to me over a couple of chapters, so I had to track back where I "lost it" and reread from there.Chaos Theory is mind-numbing and yet charmingly elegant. It'd be very frustrating to NOT know what actually cause something OR how some outcome can't be reproducible - but the world is filled with such events.
- Rating: 5 out of 5 stars5/5Like Entangled Life, it took me a while to finish this because it encompasses so many different fields that I'd stop all the time to check references. This is not a criticism of the book, though: James Gleick describes how scientists in areas as disparate as meteorology and the study of cardiac arrhythmia were able to make an impact by showing simple chaotic models that reproduce features of very real systems which were previously thought to be too random to understand. I find that pretty amazing.
While I'd have preferred a slightly more technical book, I've found that Gleick is really good at making layman explanations without being condescending. He also interviewed a lot of people and out of it he made a really compelling story of how the field developed. I particularly liked how he highlights the massive resistance the first people who studied chaos encountered. Nonlinear systems have historically been (and still are to this day) something of a footnote because analytic treatment of them is impossible, which makes theoretical scientists unenthusiastic. And the more practical scientists are also unlikely to be receptive because incorporating a dynamic systems approach to their study requires interdisciplinary work. - Rating: 3 out of 5 stars3/5Great
- Rating: 3 out of 5 stars3/5Not enough science in this history of science book.
- Rating: 4 out of 5 stars4/5Indeholder "Forord", "Sommerfugleeffekten", "Revolution", "Livets lyse og mørke sider", "Naturgeometri", "Fraktale attraktorer", "Universalitet", "Eksperimentatoren", "Billeder af kaos", "De dynamiske systemers kollektiv", "Indre rytmer", "Kaos og hinsides", "Litteratur", "Tak til", "Ordforklaring", "Indeks".Behandler Lorenz vejrsystem, Lorenz vandhjul, Penduler, Poincaré system, Smales hestesko, Populationsmodeller, periodefordoblinger, Mandelbrot, bomuldspriser, Cantor-støv, Koch-kurve, Menger-svamp, turbulens, Hénon-attraktor, Feigenbaum, Landau, D'Arcy Wentworth Thompson, Barnsleys bregne, Julia-mængder og Mandelbrot-mængder.Omkring side 250 er der en beskrivelse af forkammerflimren og hjerteflimmer og hjertedefibrillatorer og hjertestartere. Siden er der sket en masse på området, men det er dybt interessant at læse om starten af forskningen på området. Min far fik atrieflimmer og det var formentlig også det, han døde af, så det er jo et godt gæt at det også bliver min skæbne en dag!Glimrende beskrivelse af kaos-teoriens start men uden ret meget matematik. Det er mest en appetitvækker.
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Chaos - James Gleick
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MAKING A NEW SCIENCE
JAMES GLEICK
To Cynthia
human was the music,
natural was the static…
—JOHN UPDIKE
CONTENTS
Prologue
The Butterfly Effect
Edward Lorenz and his toy weather. The computer misbehaves. Long-range forecasting is doomed. Order masquerading as randomness. A world of nonlinearity. We completely missed the point.
Revolution
A revolution in seeing. Pendulum clocks, space balls, and playground swings. The invention of the horseshoe. A mystery solved: Jupiter’s Great Red Spot.
Life’s Ups and Downs
Modeling wildlife populations. Nonlinear science, the study of non-elephant animals.
Pitchfork bifurcations and a ride on the Spree. A movie of chaos and a messianic appeal.
A Geometry of Nature
A discovery about cotton prices. A refugee from Bourbaki. Transmission errors and jagged shores. New dimensions. The monsters of fractal geometry. Quakes in the schizosphere. From clouds to blood vessels. The trash cans of science. To see the world in a grain of sand.
Strange Attractors
A problem for God. Transitions in the laboratory. Rotating cylinders and a turning point. David Ruelle’s idea for turbulence. Loops in phase space. Mille-feuilles and sausage. An astronomer’s mapping. Fireworks or galaxies.
Universality
A new start at Los Alamos. The renormalization group. Decoding color. The rise of numerical experimentation. Mitchell Feigenbaum’s breakthrough. A universal theory. The rejection letters. Meeting in Como. Clouds and paintings.
The Experimenter
Helium in a Small Box. Insolid billowing of the solid.
Flow and form in nature. Albert Libchaber’s delicate triumph. Experiment joins theory. From one dimension to many.
Images of Chaos
The complex plane. Surprise in Newton’s method. The Mandelbrot set: sprouts and tendrils. Art and commerce meet science. Fractal basin boundaries. The chaos game.
The Dynamical Systems Collective
Santa Cruz and the sixties. The analog computer. Was this science? A long-range vision.
Measuring unpredictability. Information theory. From microscale to macroscale. The dripping faucet. Audiovisual aids. An era ends.
Inner Rhythms
A misunderstanding about models. The complex body. The dynamical heart. Resetting the biological clock. Fatal arrhythmia. Chick embryos and abnormal beats. Chaos as health.
Chaos and Beyond
New beliefs, new definitions. The Second Law, the snowflake puzzle, and loaded dice. Opportunity and necessity.
Afterword
Notes on Sources and Further Reading
Acknowledgments
Index
CHAOS
PROLOGUE
THE POLICE IN THE SMALL TOWN of Los Alamos, New Mexico, worried briefly in 1974 about a man seen prowling in the dark, night after night, the red glow of his cigarette floating along the back streets. He would pace for hours, heading nowhere in the starlight that hammers down through the thin air of the mesas. The police were not the only ones to wonder. At the national laboratory some physicists had learned that their newest colleague was experimenting with twenty-six–hour days, which meant that his waking schedule would slowly roll in and out of phase with theirs. This bordered on strange, even for the Theoretical Division.
In the three decades since J. Robert Oppenheimer chose this unworldly New Mexico landscape for the atomic bomb project, Los Alamos National Laboratory had spread across an expanse of desolate plateau, bringing particle accelerators and gas lasers and chemical plants, thousands of scientists and administrators and technicians, as well as one of the world’s greatest concentrations of supercomputers. Some of the older scientists remembered the wooden buildings rising hastily out of the rimrock in the 1940s, but to most of the Los Alamos staff, young men and women in college-style corduroys and work shirts, the first bombmakers were just ghosts. The laboratory’s locus of purest thought was the Theoretical Division, known as T division, just as computing was C division and weapons was X division. More than a hundred physicists and mathematicians worked in T division, well paid and free of academic pressures to teach and publish. These scientists had experience with brilliance and with eccentricity. They were hard to surprise.
But Mitchell Feigenbaum was an unusual case. He had exactly one published article to his name, and he was working on nothing that seemed to have any particular promise. His hair was a ragged mane, sweeping back from his wide brow in the style of busts of German composers. His eyes were sudden and passionate. When he spoke, always rapidly, he tended to drop articles and pronouns in a vaguely middle European way, even though he was a native of Brooklyn. When he worked, he worked obsessively. When he could not work, he walked and thought, day or night, and night was best of all. The twenty-four–hour day seemed too constraining. Nevertheless, his experiment in personal quasiperiodicity came to an end when he decided he could no longer bear waking to the setting sun, as had to happen every few days.
At the age of twenty-nine he had already become a savant among the savants, an ad hoc consultant whom scientists would go to see about any especially intractable problem, when they could find him. One evening he arrived at work just as the director of the laboratory, Harold Agnew, was leaving. Agnew was a powerful figure, one of the original Oppenheimer apprentices. He had flown over Hiroshima on an instrument plane that accompanied the Enola Gay, photographing the delivery of the laboratory’s first product.
"I understand you’re real smart, Agnew said to Feigenbaum.
If you’re so smart, why don’t you just solve laser fusion?"
Even Feigenbaum’s friends were wondering whether he was ever going to produce any work of his own. As willing as he was to do impromptu magic with their questions, he did not seem interested in devoting his own research to any problem that might pay off. He thought about turbulence in liquids and gases. He thought about time—did it glide smoothly forward or hop discretely like a sequence of cosmic motion-picture frames? He thought about the eye’s ability to see consistent colors and forms in a universe that physicists knew to be a shifting quantum kaleidoscope. He thought about clouds, watching them from airplane windows (until, in 1975, his scientific travel privileges were officially suspended on grounds of overuse) or from the hiking trails above the laboratory.
In the mountain towns of the West, clouds barely resemble the sooty indeterminate low-flying hazes that fill the Eastern air. At Los Alamos, in the lee of a great volcanic caldera, the clouds spill across the sky, in random formation, yes, but also not-random, standing in uniform spikes or rolling in regularly furrowed patterns like brain matter. On a stormy afternoon, when the sky shimmers and trembles with the electricity to come, the clouds stand out from thirty miles away, filtering the light and reflecting it, until the whole sky starts to seem like a spectacle staged as a subtle reproach to physicists. Clouds represented a side of nature that the mainstream of physics had passed by, a side that was at once, fuzzy and detailed, structured and unpredictable. Feigenbaum thought about such things, quietly and unproductively.
To a physicist, creating laser fusion was a legitimate problem; puzzling out the spin and color and flavor of small particles was a legitimate problem; dating the origin of the universe was a legitimate problem. Understanding clouds was a problem for a meteorologist. Like other physicists, Feigenbaum used an understated, tough-guy vocabulary to rate such problems. Such a thing is obvious, he might say, meaning that a result could be understood by any skilled physicist after appropriate contemplation and calculation. Not obvious described work that commanded respect and Nobel prizes. For the hardest problems, the problems that would not give way without long looks into the universe’s bowels, physicists reserved words like deep. In 1974, though few of his colleagues knew it, Feigenbaum was working on a problem that was deep: chaos.
WHERE CHAOS BEGINS, classical science stops. For as long as the world has had physicists inquiring into the laws of nature, it has suffered a special ignorance about disorder in the atmosphere, in the turbulent sea, in the fluctuations of wildlife populations, in the oscillations of the heart and the brain. The irregular side of nature, the discontinuous and erratic side—these have been puzzles to science, or worse, monstrosities.
But in the 1970s a few scientists in the United States and Europe began to find a way through disorder. They were mathematicians, physicists, biologists, chemists, all seeking connections between different kinds of irregularity. Physiologists found a surprising order in the chaos that develops in the human heart, the prime cause of sudden, unexplained death. Ecologists explored the rise and fall of gypsy moth populations. Economists dug out old stock price data and tried a new kind of analysis. The insights that emerged led directly into the natural world—the shapes of clouds, the paths of lightning, the microscopic intertwining of blood vessels, the galactic clustering of stars.
When Mitchell Feigenbaum began thinking about chaos at Los Alamos, he was one of a handful of scattered scientists, mostly unknown to one another. A mathematician in Berkeley, California, had formed a small group dedicated to creating a new study of dynamical systems.
A population biologist at Princeton University was about to publish an impassioned plea that all scientists should look at the surprisingly complex behavior lurking in some simple models. A geometer working for IBM was looking for a new word to describe a family of shapes—jagged, tangled, splintered, twisted, fractured—that he considered an organizing principle in nature. A French mathematical physicist had just made the disputatious claim that turbulence in fluids might have something to do with a bizarre, infinitely tangled abstraction that he called a strange attractor.
A decade later, chaos has become a shorthand name for a fast-growing movement that is reshaping the fabric of the scientific establishment. Chaos conferences and chaos journals abound. Government program managers in charge of research money for the military, the Central Intelligence Agency, and the Department of Energy have put ever greater sums into chaos research and set up special bureaucracies to handle the financing. At every major university and every major corporate research center, some theorists ally themselves first with chaos and only second with their nominal specialties. At Los Alamos, a Center for Nonlinear Studies was established to coordinate work on chaos and related problems; similar institutions have appeared on university campuses across the country.
Chaos has created special techniques of using computers and special kinds of graphic images, pictures that capture a fantastic and delicate structure underlying complexity. The new science has spawned its own language, an elegant shop talk of fractals and bifurcations, intermittencies and periodicities, folded-towel diffeomorphisms and smooth noodle maps. These are the new elements of motion, just as, in traditional physics, quarks and gluons are the new elements of matter. To some physicists chaos is a science of process rather than state, of becoming rather than being.
Now that science is looking, chaos seems to be everywhere. A rising column of cigarette smoke breaks into wild swirls. A flag snaps back and forth in the wind. A dripping faucet goes from a steady pattern to a random one. Chaos appears in the behavior of the weather, the behavior of an airplane in flight, the behavior of cars clustering on an expressway, the behavior of oil flowing in underground pipes. No matter what the medium, the behavior obeys the same newly discovered laws. That realization has begun to change the way business executives make decisions about insurance, the way astronomers look at the solar system, the way political theorists talk about the stresses leading to armed conflict.
Chaos breaks across the lines that separate scientific disciplines. Because it is a science of the global nature of systems, it has brought together thinkers from fields that had been widely separated. "Fifteen years ago, science was heading for a crisis of increasing specialization, a Navy official in charge of scientific financing remarked to an audience of mathematicians, biologists, physicists, and medical doctors.
Dramatically, that specialization has reversed because of chaos." Chaos poses problems that defy accepted ways of working in science. It makes strong claims about the universal behavior of complexity. The first chaos theorists, the scientists who set the discipline in motion, shared certain sensibilities. They had an eye for pattern, especially pattern that appeared on different scales at the same time. They had a taste for randomness and complexity, for jagged edges and sudden leaps. Believers in chaos—and they sometimes call themselves believers, or converts, or evangelists—speculate about determinism and free will, about evolution, about the nature of conscious intelligence. They feel that they are turning back a trend in science toward reductionism, the analysis of systems in terms of their constituent parts: quarks, chromosomes, or neurons. They believe that they are looking for the whole.
The most passionate advocates of the new science go so far as to say that twentieth-century science will be remembered for just three things: relativity, quantum mechanics, and chaos. Chaos, they contend, has become the century’s third great revolution in the physical sciences. Like the first two revolutions, chaos cuts away at the tenets of Newton’s physics. As one physicist put it: Relativity eliminated the Newtonian illusion of absolute space and time; quantum theory eliminated the Newtonian dream of a controllable measurement process; and chaos eliminates the Laplacian fantasy of deterministic predictability.
Of the three, the revolution in chaos applies to the universe we see and touch, to objects at human scale. Everyday experience and real pictures of the world have become legitimate targets for inquiry. There has long been a feeling, not always expressed openly, that theoretical physics has strayed far from human intuition about the world. Whether this will prove to be fruitful heresy or just plain heresy, no one knows. But some of those who thought physics might be working its way into a corner now look to chaos as a way out.
Within physics itself, the study of chaos emerged from a backwater. The mainstream for most of the twentieth century has been particle physics, exploring the building blocks of matter at higher and higher energies, smaller and smaller scales, shorter and shorter times. Out of particle physics have come theories about the fundamental forces of nature and about the origin of the universe. Yet some young physicists have grown dissatisfied with the direction of the most prestigious of sciences. Progress has begun to seem slow, the naming of new particles futile, the body of theory cluttered. With the coming of chaos, younger scientists believed they were seeing the beginnings of a course change for all of physics. The field had been dominated long enough, they felt, by the glittering abstractions of high-energy particles and quantum mechanics.
The cosmologist Stephen Hawking, occupant of Newton’s chair at Cambridge University, spoke for most of physics when he took stock of his science in a 1980 lecture titled Is the End in Sight for Theoretical Physics?
We already know the physical laws that govern everything we experience in everyday life…. It is a tribute to how far we have come in theoretical physics that it now takes enormous machines and a great deal of money to perform an experiment whose results we cannot predict.
Yet Hawking recognized that understanding nature’s laws on the terms of particle physics left unanswered the question of how to apply those laws to any but the simplest of systems. Predictability is one thing in a cloud chamber where two particles collide at the end of a race around an accelerator. It is something else altogether in the simplest tub of roiling fluid, or in the earth’s weather, or in the human brain.
Hawking’s physics, efficiently gathering up Nobel Prizes and big money for experiments, has often been called a revolution. At times it seemed within reach of that grail of science, the Grand Unified Theory or theory of everything.
Physics had traced the development of energy and matter in all but the first eyeblink of the universe’s history. But was postwar particle physics a revolution? Or was it just the fleshing out of the framework laid down by Einstein, Bohr, and the other fathers of relativity and quantum mechanics? Certainly, the achievements of physics, from the atomic bomb to the transistor, changed the twentieth-century landscape. Yet if anything, the scope of particle physics seemed to have narrowed. Two generations had passed since the field produced a new theoretical idea that changed the way nonspecialists understand the world.
The physics described by Hawking could complete its mission without answering some of the most fundamental questions about nature. How does life begin? What is turbulence? Above all, in a universe ruled by entropy, drawing inexorably toward greater and greater disorder, how does order arise? At the same time, objects of everyday experience like fluids and mechanical systems came to seem so basic and so ordinary that physicists had a natural tendency to assume they were well understood. It was not so.
As the revolution in chaos runs its course, the best physicists find themselves returning without embarrassment to phenomena on a human scale. They study not just galaxies but clouds. They carry out profitable computer research not just on Crays but on Macintoshes. The premier journals print articles on the strange dynamics of a ball bouncing on a table side by side with articles on quantum physics. The simplest systems are now seen to create extraordinarily difficult problems of predictability. Yet order arises spontaneously in those systems—chaos and order together. Only a new kind of science could begin to cross the great gulf between knowledge of what one thing does—one water molecule, one cell of heart tissue, one neuron—and what millions of them do.
Watch two bits of foam flowing side by side at the bottom of a waterfall. What can you guess about how close they were at the top? Nothing. As far as standard physics was concerned, God might just as well have taken all those water molecules under the table and shuffled them personally. Traditionally, when physicists saw complex results, they looked for complex causes. When they saw a random relationship between what goes into a system and what comes out, they assumed that they would have to build randomness into any realistic theory, by artificially adding noise or error. The modern study of chaos began with the creeping realization in the 1960s that quite simple mathematical equations could model systems every bit as violent as a waterfall. Tiny differences in input could quickly become overwhelming differences in output—a phenomenon given the name sensitive dependence on initial conditions.
In weather, for example, this translates into what is only half-jokingly known as the Butterfly Effect—the notion that a butterfly stirring the air today in Peking can transform storm systems next month in New York.
When the explorers of chaos began to think back on the genealogy of their new science, they found many intellectual trails from the past. But one stood out clearly. For the young physicists and mathematicians leading the revolution, a starting point was the Butterfly Effect.
THE BUTTERFLY EFFECT
Physicists like to think that all you have to do is say, these are the conditions, now what happens next?
—RICHARD P. FEYNMAN
THE SUN BEAT DOWN through a sky that had never seen clouds. The winds swept across an earth as smooth as glass. Night never came, and autumn never gave way to winter. It never rained. The simulated weather in Edward Lorenz’s new electronic computer changed slowly but certainly, drifting through a permanent dry midday midseason, as if the world had turned into Camelot, or some particularly bland version of southern California.
Outside his window Lorenz could watch real weather, the early-morning fog creeping along the Massachusetts Institute of Technology campus or the low clouds slipping over the rooftops from the Atlantic. Fog and clouds never arose in the model running on his computer. The machine, a Royal McBee, was a thicket of wiring and vacuum tubes that occupied an ungainly portion of Lorenz’s office, made a surprising and irritating noise, and broke down every week or so. It had neither the speed nor the memory to manage a realistic simulation of the earth’s atmosphere and oceans. Yet Lorenz created a toy weather in 1960 that succeeded in mesmerizing his colleagues. Every minute the machine marked the passing of a day by printing a row of numbers across a page. If you knew how to read the printouts, you would see a prevailing westerly wind swing now to the north, now to the south, now back to the north. Digitized cyclones spun slowly around an idealized globe. As word spread through the department, the other meteorologists would gather around with the graduate students, making bets on what Lorenz’s weather would do next. Somehow, nothing ever happened the same way twice.
Lorenz enjoyed weather—by no means a prerequisite for a research meteorologist. He savored its changeability. He appreciated the patterns that come and go in the atmosphere, families of eddies and cyclones, always obeying mathematical rules, yet never repeating themselves. When he looked at clouds, he thought he saw a kind of structure in them. Once he had feared that studying the science of weather would be like prying a jack-in–the-box apart with a screwdriver. Now he wondered whether science would be able to penetrate the magic at all. Weather had a flavor that could not be expressed by talking about averages. The daily high temperature in Cambridge, Massachusetts, averages 75 degrees in June. The number of rainy days in Riyadh, Saudi Arabia, averages ten a year. Those were statistics. The essence was the way patterns in the atmosphere changed over time, and that was what Lorenz captured on the Royal McBee.
He was the god of this machine universe, free to choose the laws of nature as he pleased. After a certain amount of undivine trial and error, he chose twelve. They were numerical rules—equations that expressed the relationships between temperature and pressure, between pressure and wind speed. Lorenz understood that he was putting into practice the laws of Newton, appropriate tools for a clockmaker deity who could create a world and set it running for eternity. Thanks to the determinism of physical law, further intervention would then be unnecessary. Those who made such models took for granted that, from present to future, the laws of motion provide a bridge of mathematical certainty. Understand the laws and you understand the universe. That was the philosophy behind modeling weather on a computer.
Indeed, if the eighteenth-century philosophers imagined their creator as a benevolent noninterventionist, content to remain behind the scenes, they might have imagined someone like Lorenz. He was an odd sort of meteorologist. He had the worn face of a Yankee farmer, with surprising bright eyes that made him seem to be laughing whether he was or not. He seldom spoke about himself or his work, but he listened. He often lost himself in a realm of calculation or dreaming that his colleagues found inaccessible. His closest friends felt that Lorenz spent a good deal of his time off in a remote outer space.
As a boy he had been a weather bug, at least to the extent of keeping close tabs on the max-min thermometer recording the days’ highs and lows outside his parents’ house in West Hartford, Connecticut. But he spent more time inside playing with mathematical puzzle books than watching the thermometer. Sometimes he and his father would work out puzzles together. Once they came upon a particularly difficult problem that turned out to be insoluble. That was acceptable, his father told him: you can always try to solve a problem by proving that no solution exists. Lorenz liked that, as he always liked the purity of mathematics, and when he graduated from Dartmouth College, in 1938, he thought that mathematics was his calling. Circumstance interfered, however, in the form of World War II, which put him to work as a weather forecaster for the Army Air Corps. After the war Lorenz decided to stay with meteorology, investigating the theory of it, pushing the mathematics a little further forward. He made a name for himself by publishing work on orthodox problems, such as the general circulation of the atmosphere. And in the meantime he continued to think about forecasting.
To most serious meteorologists, forecasting was less than science. It was a seat-of–the-pants business performed by technicians who needed some intuitive ability to read the next day’s weather in the instruments and the clouds. It was guesswork. At centers like M.I.T., meteorology favored problems that had solutions. Lorenz understood the messiness of weather prediction as well as anyone, having tried it firsthand for the benefit of military pilots, but he harbored an interest in the problem—a mathematical interest.
Not only did meteorologists scorn forecasting, but in the 1960s virtually all serious scientists mistrusted computers. These souped-up calculators hardly seemed like tools for theoretical science. So numerical weather modeling was something of a bastard problem. Yet the time was right for it. Weather forecasting had been waiting two centuries for a machine that could repeat thousands of calculations over and over again by brute force. Only a computer could cash in the Newtonian promise that the world unfolded along a deterministic path, rule-bound like the planets, predictable like eclipses and tides. In theory a computer could let meteorologists do what astronomers had been able to do with pencil and slide rule: reckon the future of their universe from its initial conditions and the physical laws that guide its evolution. The equations describing the motion of air and water were as well known as those describing the motion of planets. Astronomers did not achieve perfection and never would, not in a solar system tugged by the gravities of nine planets, scores of moons and thousands of asteroids, but calculations of planetary motion were so accurate that people forgot they were forecasts. When an astronomer said, Comet Halley will be back this way in seventy-six years,
it seemed like fact, not prophecy. Deterministic numerical forecasting figured accurate courses for spacecraft and missiles. Why not winds and clouds?
Weather was vastly more complicated, but it was governed by the same laws. Perhaps a powerful enough computer could be the supreme intelligence imagined by Laplace, the eighteenth-century philosopher-mathematician who caught the Newtonian fever like no one else: Such an intelligence,
Laplace wrote, "would embrace in the same formula the movements of the greatest bodies of the universe and those of the lightest atom; for it, nothing would be uncertain and the future, as the past, would be present to its eyes." In these days of Einstein’s relativity and Heisenberg’s uncertainty, Laplace seems almost buffoon-like in his optimism, but much of modern science has pursued his dream. Implicitly, the mission of many twentieth-century scientists—biologists, neurologists, economists—has been to break their universes down into the simplest atoms that will obey scientific rules. In all these sciences, a kind of Newtonian determinism has been brought to bear. The fathers of modern computing always had Laplace in mind, and the history of computing and the history of forecasting were intermingled ever since John von Neumann designed his first machines at the Institute for Advanced Study in Princeton, New Jersey, in the 1950s. Von Neumann recognized that weather modeling could be an ideal task for a computer.
There was always one small compromise, so small that working scientists usually forgot it was there, lurking in a corner of their philosophies like an unpaid bill. Measurements could never be perfect. Scientists marching under Newton’s banner actually waved another flag that said something like this: Given an approximate knowledge of a system’s initial conditions and an understanding of natural law, one can calculate the approximate behavior of the system. This assumption lay at the philosophical heart of science. As one theoretician liked to tell his students: "The basic idea of Western science is that you don’t have to take into account the falling of a leaf on some planet in another galaxy when you’re trying to account for the motion of a billiard ball on a pool table on earth. Very small influences can be neglected. There’s a convergence in the way things work, and arbitrarily small influences don’t blow up to have arbitrarily large effects." Classically, the belief in approximation and convergence was well justified. It worked. A tiny error in fixing the position of Comet Halley in 1910 would only cause a tiny error in predicting its arrival in 1986, and the error would stay small for millions of years to come. Computers rely on the same assumption in guiding spacecraft: approximately accurate input gives approximately accurate output. Economic forecasters rely on this assumption, though their success is less apparent. So did the pioneers in global weather forecasting.
With his primitive computer, Lorenz had boiled weather down to the barest skeleton. Yet, line by line, the winds and temperatures in Lorenz’s printouts seemed to behave in a recognizable earthly way. They matched his cherished intuition about the weather, his sense that it repeated itself, displaying familiar patterns over time, pressure rising and falling, the airstream swinging north and south. He discovered that when a line went from high to low without a bump, a double bump would come next, and he said, "That’s the kind of rule a forecaster could use." But the repetitions were never quite exact. There was pattern, with disturbances. An orderly disorder.
To make the patterns plain to see, Lorenz created a primitive kind of graphics. Instead of just printing out the usual lines of digits, he would have the machine print a certain number of blank spaces followed by the letter a. He would pick one variable—perhaps the direction of the airstream. Gradually the a’s marched down the roll of paper, swinging back and forth in a wavy line, making a long series of hills and valleys that represented the way the west wind would swing north and south across the continent. The orderliness of it, the recognizable cycles coming around again and again but never twice the same way, had a hypnotic fascination. The system seemed slowly to be revealing its secrets to the forecaster’s eye.
One day in the winter of 1961, wanting to examine one sequence at greater length, Lorenz took a shortcut. Instead of starting the whole run over, he started midway through. To give the machine its initial conditions, he typed the numbers straight from the earlier printout. Then he walked down the hall to get away from the noise and drink a cup of coffee. When he returned an hour later, he saw something unexpected, something that planted a seed for a new science.
THIS NEW RUN should have exactly duplicated the old. Lorenz had copied the numbers into the machine himself. The program had not changed. Yet as he stared at the new printout, Lorenz saw his weather diverging so rapidly from the pattern of the last run that, within just a few months, all resemblance had disappeared. He looked at one set of numbers, then back at the other. He might as well have chosen two random weathers out of a hat. His first thought was that another vacuum tube had gone bad.
Suddenly he realized the truth. There had been no malfunction. The problem lay in the numbers he had typed. In the computer’s memory, six decimal places were stored: .506127. On the printout, to save space, just three appeared: .506. Lorenz had entered the shorter, rounded-off numbers, assuming that the difference—one part in a thousand—was inconsequential.
It was a reasonable assumption. If a weather satellite can read ocean-surface temperature to within one part in a thousand, its operators consider themselves lucky. Lorenz’s Royal McBee was implementing the classical program. It used a purely deterministic system of equations. Given a particular starting point, the weather would unfold exactly the same way each time. Given a slightly different starting point, the weather should unfold in a slightly different way. A small numerical error was like a small puff of wind—surely the small puffs faded or canceled each other out before they could change important, large-scale features of the weather. Yet in Lorenz’s particular system of equations, small errors proved catastrophic.
HOW TWO WEATHER PATTERNS DIVERGE. From nearly the same starting point, Edward Lorenz saw his computer weather produce patterns that grew farther and farther apart until all resemblance disappeared. (From Lorenz’s 1961 printouts.)
He decided to look more closely at the way two nearly identical runs of weather flowed apart. He copied one of the wavy lines of output onto a transparency and laid it over the other, to inspect the way it diverged. First, two humps matched detail for detail. Then one line began to lag a hairsbreadth behind. By the time the two runs reached the next hump, they were distinctly out of phase. By the third or fourth hump, all similarity had vanished.
It was only a wobble from a clumsy computer. Lorenz could have assumed something was wrong with his particular machine or his particular model—probably should have assumed. It was not as though he had mixed sodium and chlorine and got gold. But for reasons of mathematical intuition that his colleagues would begin to understand only later, Lorenz felt a jolt: something was philosophically out of joint. The practical import could be staggering. Although his equations were gross parodies of the earth’s weather, he had a faith that they captured the essence of the real