## Metadata - Author: [[James Gleick]] - Full Title: Chaos - Category: #books ## Highlights - Lorenz saw that there must be a link between the unwillingness of the weather to repeat itself and the inability of forecasters to predict it—a link between aperiodicity and unpredictability. ([Location 359](https://readwise.io/to_kindle?action=open&asin=B004Q3RRPI&location=359)) - Smale conceived of the entire range of possibilities in the oscillator, the entire phase space, as physicists called it. Any state of the system at a moment frozen in time was represented as a point in phase space; all the information about its position or velocity was contained in the coordinates of that point. As the system changed in some way, the point would move to a new position in phase space. As the system changed continuously, the point would trace a trajectory. For a simple system like a pendulum, the phase space might just be a rectangle: the pendulum’s angle at a given instant would determine the east-west position of a point and the pendulum’s speed would determine the north-south position. For a pendulum swinging regularly back and forth, the trajectory through phase space would be a loop, around and around as the system lived through the same sequence of positions over and over again. Smale, instead of looking at any one trajectory, concentrated on the behavior of the entire space as the system changed—as more driving energy was added, for example. His intuition leapt from the physical essence of the system to a new kind of geometrical essence. His tools were topological transformations of shapes in phase space—transformations like stretching and squeezing. Sometimes these transformations had clear physical meaning. Dissipation in a system, the loss of energy to friction, meant that the system’s shape in phase space would contract like a balloon losing air—finally shrinking to a point at the moment the system comes to a complete halt. To represent the full complexity of the van der Pol oscillator, he realized that the phase space would have to suffer a complex new kind of combination of transformations. He quickly turned his idea about visualizing global behavior into a new kind of model. His innovation—an enduring image of chaos in the years that followed—was a structure that became known as the horseshoe. ([Location 759](https://readwise.io/to_kindle?action=open&asin=B004Q3RRPI&location=759)) - “When I started my professional work in mathematics in 1960, which is not so long ago, modern mathematics in its entirety—in its entirety—was rejected by physicists, including the most avant-garde mathematical physicists. So differentiable dynamics, global analysis, manifolds of mappings, differential geometry—everything just a year or two beyond what Einstein had used—was all rejected. The romance between mathematicians and physicists had ended in divorce in the 1930s. These people were no longer speaking. They simply despised each other. Mathematical physicists refused their graduate students permission to take math courses from mathematicians: Take mathematics from us. We will teach you what you need to know. The mathematicians are on some kind of terrible ego trip and they will destroy your mind. That was 1960. By 1968 this had completely turned around.” ([Location 800](https://readwise.io/to_kindle?action=open&asin=B004Q3RRPI&location=800)) - One helpful simplification was to model the world in terms of discrete time intervals, like a watch hand that jerks forward second by second instead of gliding continuously. Differential equations describe processes that change smoothly over time, but differential equations are hard to compute. Simpler equations—“difference equations”—can be used for processes that jump from state to state. Fortunately, many animal populations do what they do in neat one-year intervals. Changes year to year are often more important than changes on a continuum. Unlike people, many insects, for example, stick to a single breeding season, so their generations do not overlap. To guess next spring’s gypsy moth population or next winter’s measles epidemic, an ecologist might only need to know the corresponding figure for this year. A year-by–year facsimile produces no more than a shadow of a system’s intricacies, but in many real applications the shadow gives all the information a scientist needs. ([Location 903](https://readwise.io/to_kindle?action=open&asin=B004Q3RRPI&location=903)) - LATER, PEOPLE WOULD SAY that James Yorke had discovered Lorenz and given the science of chaos its name. The second part was actually true. ([Location 970](https://readwise.io/to_kindle?action=open&asin=B004Q3RRPI&location=970)) - Yorke enjoyed an unusual freedom to work on problems outside traditional domains, and he enjoyed frequent contact with experts in a wide range of disciplines. One of these experts, a fluid dynamicist, had come across Lorenz’s 1963 paper “Deterministic Nonperiodic Flow” in 1972 and had fallen in love with it, handing out copies to anyone who would take one. He handed one to Yorke. ([Location 981](https://readwise.io/to_kindle?action=open&asin=B004Q3RRPI&location=981)) - Lorenz’s paper was a piece of magic that Yorke had been looking for without even knowing it. It was a mathematical shock, to begin with—a chaotic system that violated Smale’s original optimistic classification scheme. But it was not just mathematics; it was a vivid physical model, a picture of a fluid in motion, and Yorke knew instantly that it was a thing he wanted physicists to see. Smale had steered mathematics in the direction of such physical problems, but, as Yorke well understood, the language of mathematics remained a serious barrier to communication. If only the academic world had room for hybrid mathematician/physicists—but it did not. Even though Smale’s work on dynamical systems had begun to close the gap, mathematicians continued to speak one language, physicists another. ([Location 984](https://readwise.io/to_kindle?action=open&asin=B004Q3RRPI&location=984)) - Pedagogically speaking, a good share of physics and mathematics was—and is—writing differential equations on a blackboard and showing students how to solve them. Differential equations represent reality as a continuum, changing smoothly from place to place and from time to time, not broken in discrete grid points or time steps. As every science student knows, solving differential equations is hard. But in two and a half centuries, scientists have built up a tremendous body of knowledge about them: handbooks and catalogues of differential equations, along with various methods for solving them, or “finding a closed-form integral,” as a scientist will say. It is no exaggeration to say that the vast business of calculus made possible most of the practical triumphs of post-medieval science; nor to say that it stands as one of the most ingenious creations of humans trying to model the changeable world around them. So by the time a scientist masters this way of thinking about nature, becoming comfortable with the theory and the hard, hard practice, he is likely to have lost sight of one fact. Most differential equations cannot be solved at all. ([Location 1009](https://readwise.io/to_kindle?action=open&asin=B004Q3RRPI&location=1009)) - Enrico Fermi once exclaimed, “It does not say in the Bible that all laws of nature are expressible linearly!” The mathematician Stanislaw Ulam remarked that to call the study of chaos “nonlinear science” was like calling zoology “the study of non elephant animals.” ([Location 1025](https://readwise.io/to_kindle?action=open&asin=B004Q3RRPI&location=1025)) - In the history of chaos, Mandelbrot made his own way. Yet the picture of reality that was forming in his mind in 1960 evolved from an oddity into a full-fledged geometry. To the physicists expanding on the work of people like Lorenz, Smale, Yorke, and May, this prickly mathematician remained a sideshow—but his techniques and his language became an inseparable part of their new science. ([Location 1279](https://readwise.io/to_kindle?action=open&asin=B004Q3RRPI&location=1279)) - In part, Bourbaki began in reaction to Poincaré, the great man of the late nineteenth century, a phenomenally prolific thinker and writer who cared less than some for rigor. Poincaré would say, I know it must be right, so why should I prove it? Bourbaki believed that Poincaré had left a shaky basis for mathematics, and the group began to write an enormous treatise, more and more fanatical in style, meant to set the discipline straight. Logical analysis was central. A mathematician had to begin with solid first principles and deduce all the rest from them. The group stressed the primacy of mathematics among sciences, and also insisted upon a detachment from other sciences. Mathematics was mathematics—it could not be valued in terms of its application to real physical phenomena. And above all, Bourbaki rejected the use of pictures. A mathematician could always be fooled by his visual apparatus. Geometry was untrustworthy. Mathematics should be pure, formal, and austere. Nor was this strictly a French development. In the United States, too, mathematicians were pulling away from the demands of the physical sciences as firmly as artists and writers were pulling away from the demands of popular taste. A hermetic sensibility prevailed. Mathematicians’ subjects became self-contained; their method became formally axiomatic. A mathematician could take pride in saying that his work explained nothing in the world or in science. Much good came of this attitude, and mathematicians treasured it. Steve Smale, even while he was working to reunite mathematics and natural science, believed, as deeply as he believed anything, that mathematics should be something all by itself. With self-containment came clarity. And clarity, too, went hand in hand with the rigor of the axiomatic method. Every serious mathematician understands that rigor is the defining strength of the discipline, the steel skeleton without which all would collapse. Rigor is what allows mathematicians to pick up a line of thought that extends over centuries and continue it, with a firm guarantee. ([Location 1309](https://readwise.io/to_kindle?action=open&asin=B004Q3RRPI&location=1309)) - ALWAYS A BELIEVER in creating his own mythology, Mandelbrot appended this statement to his entry in Who’s Who: “Science would be ruined if (like sports) it were to put competition above everything else, and if it were to clarify the rules of competition by withdrawing entirely into narrowly defined specialties. The rare scholars who are nomads-by–choice are essential to the intellectual welfare of the settled disciplines.” ([Location 1334](https://readwise.io/to_kindle?action=open&asin=B004Q3RRPI&location=1334)) - Discontinuity, bursts of noise, Cantor dusts—phenomena like these had no place in the geometries of the past two thousand years. The shapes of classical geometry are lines and planes, circles and spheres, triangles and cones. They represent a powerful abstraction of reality, and they inspired a powerful philosophy of Platonic harmony. Euclid made of them a geometry that lasted two millennia, the only geometry still that most people ever learn. Artists found an ideal beauty in them, Ptolemaic astronomers built a theory of the universe out of them. But for understanding complexity, they turn out to be the wrong kind of abstraction. Clouds are not spheres, Mandelbrot is fond of saying. Mountains are not cones. Lightning does not travel in a straight line. The new geometry mirrors a universe that is rough, not rounded, scabrous, not smooth. It is a geometry of the pitted, pocked, and broken up, the twisted, tangled, and intertwined. The understanding of nature’s complexity awaited a suspicion that the complexity was not just random, not just accident. It required a faith that the interesting feature of a lightning bolt’s path, for example, was not its direction, but rather the distribution of zigs and zags. Mandelbrot’s work made a claim about the world, and the claim was that such odd shapes carry meaning. The pits and tangles are more than blemishes distorting the classic shapes of Euclidian geometry. They are often the keys to the essence of a thing. ([Location 1397](https://readwise.io/to_kindle?action=open&asin=B004Q3RRPI&location=1397)) - HOW BIG IS IT? How long does it last? These are the most basic questions a scientist can ask about a thing. They are so basic to the way people conceptualize the world that it is not easy to see that they imply a certain bias. They suggest that size and duration, qualities that depend on scale, are qualities with meaning, qualities that can help describe an object or classify it. ([Location 1598](https://readwise.io/to_kindle?action=open&asin=B004Q3RRPI&location=1598)) - If one scientist announces that a thing is probably true, and another demonstrates it with rigor, which one has done more to advance science? Is the making of a conjecture an act of discovery? Or is it just a cold-blooded staking of a claim? Mathematicians have always faced such issues, but the debate became more intense as computers began to play their new role. Those who used computers to conduct experiments became more like laboratory scientists, playing by rules that allowed discovery without the usual theorem-proof, theorem-proof of the standard mathematics paper. ([Location 1688](https://readwise.io/to_kindle?action=open&asin=B004Q3RRPI&location=1688)) - In theory the World War II atomic bomb project was a problem in nuclear physics. In reality the nuclear physics had been mostly solved before the project began, and the business that occupied the scientists assembled at Los Alamos was a problem in fluid dynamics. ([Location 1842](https://readwise.io/to_kindle?action=open&asin=B004Q3RRPI&location=1842)) - Any liquid or gas is a collection of individual bits, so many that they may as well be infinite. If each piece moved independently, then the fluid would have infinitely many possibilities, infinitely many “degrees of freedom” in the jargon, and the equations describing the motion would have to deal with infinitely many variables. But each particle does not move independently—its motion depends very much on the motion of its neighbors—and in a smooth flow, the degrees of freedom can be few. ([Location 1871](https://readwise.io/to_kindle?action=open&asin=B004Q3RRPI&location=1871)) - Like so much of chaos itself, phase transitions involve a kind of macroscopic behavior that seems hard to predict by looking at the microscopic details. ([Location 1921](https://readwise.io/to_kindle?action=open&asin=B004Q3RRPI&location=1921)) - To study Couette flow, Swinney and Gollub built an apparatus that fit on a desktop, an outer glass cylinder the size of a skinny can of tennis balls, about a foot high and two inches across. An inner cylinder of steel slid neatly inside, leaving just one-eighth of an inch between for water. “It was a string-and–sealing-wax affair,” said Freeman Dyson, one of an unexpected series of prominent sightseers in the months that followed. “You had these two gentlemen in a poky little lab with essentially no money doing an absolutely beautiful experiment. It was the beginning of good quantitative work on turbulence.” ([Location 1960](https://readwise.io/to_kindle?action=open&asin=B004Q3RRPI&location=1960)) - The strange attractor lives in phase space, one of the most powerful inventions of modern science. Phase space gives a way of turning numbers into pictures, abstracting every bit of essential information from a system of moving parts, mechanical or fluid, and making a flexible road map to all its possibilities. Physicists already worked with two simpler kinds of “attractors”: fixed points and limit cycles, representing behavior that reached a steady state or repeated itself continuously. In phase space the complete state of knowledge about a dynamical system at a single instant in time collapses to a point. That point is the dynamical system—at that instant. At the next instant, though, the system will have changed, ever so slightly, and so the point moves. The history of the system time can be charted by the moving point, tracing its orbit through phase space with the passage of time. ([Location 2029](https://readwise.io/to_kindle?action=open&asin=B004Q3RRPI&location=2029)) - Every piece of a dynamical system that can move independently is another variable, another degree of freedom. Every degree of freedom requires another dimension in phase space, to make sure that a single point contains enough information to determine the state of the system uniquely. ([Location 2057](https://readwise.io/to_kindle?action=open&asin=B004Q3RRPI&location=2057)) - The great quantum theorist Richard P. Feynman expressed this feeling. “It always bothers me that, according to the laws as we understand them today, it takes a computing machine an infinite number of logical operations to figure out what goes on in no matter how tiny a region of space, and no matter how tiny a region of time. How can all that be going on in that tiny space? Why should it take an infinite amount of logic to figure out what one tiny piece of space/time is going to do?” ([Location 2085](https://readwise.io/to_kindle?action=open&asin=B004Q3RRPI&location=2085)) - Rössler felt that these shapes embodied a self-organizing principle in the world. He would imagine something like a wind sock on an airfield, “an open hose with a hole in the end, and the wind forces its way in,” he said. “Then the wind is trapped. Against its will, energy is doing something productive, like the devil in medieval history. The principle is that nature does something against its own will and, by self-entanglement, produces beauty.” ([Location 2153](https://readwise.io/to_kindle?action=open&asin=B004Q3RRPI&location=2153)) - “In a way, art is a theory about the way the world looks to human beings. It’s abundantly obvious that one doesn’t know the world around us in detail. What artists have accomplished is realizing that there’s only a small amount of stuff that’s important, and then seeing what it was. So they can do some of my research for me. When you look at early stuff of Van Gogh there are zillions of details that are put into it, there’s always an immense amount of information in his paintings. It obviously occurred to him, what is the irreducible amount of this stuff that you have to put in. Or you can study the horizons in Dutch ink drawings from around 1600, with tiny trees and cows that look very real. If you look closely, the trees have sort of leafy boundaries, but it doesn’t work if that’s all it is—there are also, sticking in it, little pieces of twiglike stuff. There’s a definite interplay between the softer textures and the things with more definite lines. Somehow the combination gives the correct perception. With Ruysdael and Turner, if you look at the way they construct complicated water, it is clearly done in an iterative way. There’s some level of stuff, and then stuff painted on top of that, and then corrections to that. Turbulent fluids for those painters is always something with a scale idea in it. ([Location 2810](https://readwise.io/to_kindle?action=open&asin=B004Q3RRPI&location=2810)) - But physiologists have also begun to see chaos as health. It has long been understood that nonlinearity in feedback processes serves to regulate and control. Simply put, a linear process, given a slight nudge, tends to remain slightly off track. A nonlinear process, given the same nudge, tends to return to its starting point. ([Location 4319](https://readwise.io/to_kindle?action=open&asin=B004Q3RRPI&location=4319)) - With all such control phenomena, a critical issue is robustness: how well can a system withstand small jolts. Equally critical in biological systems is flexibility: how well can a system function over a range of frequencies. A locking-in to a single mode can be enslavement, preventing a system from adapting to change. Organisms must respond to circumstances that vary rapidly and unpredictably; no heartbeat or respiratory rhythm can be locked into the strict periodicities of the simplest physical models, and the same is true of the subtler rhythms of the rest of the body. Some researchers, among them Ary Goldberger of Harvard Medical School, proposed that healthy dynamics were marked by fractal physical structures, like the branching networks of bronchial tubes in the lung and conducting fibers in the heart, that allow a wide range of rhythms. Thinking of Robert Shaw’s arguments, Goldberger noted: “Fractal processes associated with scaled, broadband spectra are ‘information-rich.’ Periodic states, in contrast, reflect narrow-band spectra and are defined by monotonous, repetitive sequences, depleted of information content.” Treating such disorders, he and other physiologists suggested, may depend on broadening a system’s spectral reserve, its ability to range over many different frequencies without falling into a locked periodic channel. ([Location 4332](https://readwise.io/to_kindle?action=open&asin=B004Q3RRPI&location=4332)) - “Is it possible that mathematical pathology, i.e. chaos, is health? And that mathematical health, which is the predictability and differentiability of this kind of a structure, is disease?” ([Location 4343](https://readwise.io/to_kindle?action=open&asin=B004Q3RRPI&location=4343)) - “When you reach an equilibrium in biology you’re dead,” he said. “If I ask you whether your brain is an equilibrium system, all I have to do is ask you not to think of elephants for a few minutes, and you know it isn’t an equilibrium system.” ([Location 4349](https://readwise.io/to_kindle?action=open&asin=B004Q3RRPI&location=4349)) - Many other scientists began to apply the formalisms of chaos to research in artificial intelligence. The dynamics of systems wandering between basins of attraction, for example, appealed to those looking for a way to model symbols and memories. A physicist thinking of ideas as regions with fuzzy boundaries, separate yet overlapping, pulling like magnets and yet letting go, would naturally turn to the image of a phase space with “basins of attraction.” Such models seemed to have the right features: points of stability mixed with instability, and regions with changeable boundaries. Their fractal structure offered the kind of infinitely self-referential quality that seems so central to the mind’s ability to bloom with ideas, decisions, emotions, and all the other artifacts of consciousness. With or without chaos, serious cognitive scientists can no longer model the mind as a static structure. They recognize a hierarchy of scales, from neuron upward, providing an opportunity for the interplay of microscale and macroscale so characteristic of fluid turbulence and other complex dynamical processes. Pattern born amid formlessness: that is biology’s basic beauty and its basic mystery. Life sucks order from a sea of disorder. Erwin Schrödinger, the quantum pioneer and one of several physicists who made a nonspecialist’s foray into biological speculation, put it this way forty years ago: A living organism has the “astonishing gift of concentrating a ‘stream of order’ on itself and thus escaping the decay into atomic chaos.” ([Location 4371](https://readwise.io/to_kindle?action=open&asin=B004Q3RRPI&location=4371)) - Now all that has changed. In the intervening twenty years, physicists, mathematicians, biologists, and astronomers have created an alternative set of ideas. Simple systems give rise to complex behavior. Complex systems give rise to simple behavior. And most important, the laws of complexity hold universally, caring not at all for the details of a system’s constituent atoms. ([Location 4409](https://readwise.io/to_kindle?action=open&asin=B004Q3RRPI&location=4409)) - More and more of them realized that chaos offered a fresh way to proceed with old data, forgotten in desk drawers because they had proved too erratic. More and more felt the compartmentalization of science as an impediment to their work. More and more felt the futility of studying parts in isolation from the whole. For them, chaos was the end of the reductionist program in science. ([Location 4418](https://readwise.io/to_kindle?action=open&asin=B004Q3RRPI&location=4418)) - Dynamics freed at last from the shackles of order and predictability…. Systems liberated to randomly explore their every dynamical possibility…. Exciting variety, richness of choice, a cornucopia of opportunity. ([Location 4456](https://readwise.io/to_kindle?action=open&asin=B004Q3RRPI&location=4456)) - John Hubbard, exploring iterated functions and the infinite fractal wildness of the Mandelbrot set, considered chaos a poor name for his work, because it implied randomness. To him, the overriding message was that simple processes in nature could produce magnificent edifices of complexity without randomness. In nonlinearity and feedback lay all the necessary tools for encoding and then unfolding structures as rich as the human brain. ([Location 4458](https://readwise.io/to_kindle?action=open&asin=B004Q3RRPI&location=4458)) - The journal Nature carried a running debate about whether the earth’s climate followed a strange attractor. Economists looked for recognizable strange attractors in stock market trends but so far had not found them. Dynamicists hoped to use the tools of chaos to explain fully developed turbulence. Albert Libchaber, now at the University of Chicago, was turning his elegant experimental style to the service of turbulence, creating a liquid-helium box thousands of times larger than his tiny cell of 1977. Whether such experiments, liberating fluid disorder in both space and time, would find simple attractors, no one knew. As the physicist Bernardo Huberman said, “If you had a turbulent river and put a probe in it and said, ‘Look, here’s a low-dimensional strange attractor,’ we would all take off our hats and look.” ([Location 4465](https://readwise.io/to_kindle?action=open&asin=B004Q3RRPI&location=4465)) - However expressed, the Second Law is a rule from which there seems no appeal. In thermodynamics that is true. But the Second Law has had a life of its own in intellectual realms far removed from science, taking the blame for disintegration of societies, economic decay, the breakdown of manners, and many other variations on the decadent theme. These secondary, metaphorical incarnations of the Second Law now seem especially misguided. In our world, complexity flourishes, and those looking to science for a general understanding of nature’s habits will be better served by the laws of chaos. Somehow, after all, as the universe ebbs toward its final equilibrium in the featureless heat bath of maximum entropy, it manages to create interesting structures. Thoughtful physicists concerned with the workings of thermodynamics realize how disturbing is the question of, as one put it, “how a purposeless flow of energy can wash life and consciousness into the world.” ([Location 4480](https://readwise.io/to_kindle?action=open&asin=B004Q3RRPI&location=4480)) - Compounding the trouble is the slippery notion of entropy, reasonably well-defined for thermodynamic purposes in terms of heat and temperature, but devilishly hard to pin down as a measure of disorder. Physicists have trouble enough measuring the degree of order in water, forming crystalline structures in the transition to ice, energy bleeding away all the while. But thermodynamic entropy fails miserably as a measure of the changing degree of form and formlessness in the creation of amino acids, of microorganisms, of self-reproducing plants and animals, of complex information systems like the brain. Certainly these evolving islands of order must obey the Second Law. The important laws, the creative laws, lie elsewhere. Nature forms patterns. Some are orderly in space but disorderly in time, others orderly in time but disorderly in space. Some patterns are fractal, exhibiting structures self-similar in scale. Others give rise to steady states or oscillating ones. Pattern formation has become a branch of physics and of materials science, allowing scientists to model the aggregation of particles into clusters, the fractured spread of electrical discharges, and the growth of crystals in ice and metal alloys. The dynamics seem so basic—shapes changing in space and time—yet only now are the tools available to understand them. It is a fair question now to ask a physicist, “Why are all snowflakes different?” ([Location 4487](https://readwise.io/to_kindle?action=open&asin=B004Q3RRPI&location=4487)) - Snowflakes are nonequilibrium phenomena, physicists like to say. They are products of imbalance in the flow of energy from one piece of nature to another. The flow turns a boundary into a tip, the tip into an array of branches, the array into a complex structure never before seen. As scientists have discovered such instability obeying the universal laws of chaos, they have succeeded in applying the same methods to a host of physical and chemical problems, and, inevitably, they suspect that biology is next. In the back of their minds, as they look at computer simulations of dendrite growth, they see algae, cell walls, organisms budding and dividing. ([Location 4535](https://readwise.io/to_kindle?action=open&asin=B004Q3RRPI&location=4535)) - For any one scientist the ideas of chaos could not prevail until the method of chaos became a necessity. Every field had its own examples. In ecology, there was William M. Schaffer, who trained as the last student of Robert MacArthur, the dean of the field in the fifties and sixties. MacArthur built a conception of nature that gave a firm footing to the idea of natural balance. His models supposed that equilibriums would exist and that populations of plants and animals would remain close to them. To MacArthur, balance in nature had what could almost be called a moral quality—states of equilibrium in his models entailed the most efficient use of food resources, the least waste. Nature, if left alone, would be good. Two decades later MacArthur’s last student found himself realizing that ecology based on a sense of equilibrium seems doomed to fail. The traditional models are betrayed by their linear bias. Nature is more complicated. Instead he sees chaos, “both exhilarating and a bit threatening.” Chaos may undermine ecology’s most enduring assumptions, he tells his colleagues. “What passes for fundamental concepts in ecology is as mist before the fury of the storm—in this case, a full, nonlinear storm.” Schaffer is using strange attractors to explore the epidemiology of childhood diseases such as measles and chicken pox. He has collected data, first from New York City and Baltimore, then from Aberdeen, Scotland, and all England and Wales. He has made a dynamical model, resembling a damped, driven pendulum. The diseases are driven each year by the infectious spread among children returning to school, and damped by natural resistance. Schaffer’s model predicts strikingly different behavior for these diseases. Chicken pox should vary periodically. Measles should vary chaotically. As it happens, the data show exactly what Schaffer predicts. To a traditional epidemiologist the yearly variations in measles seemed inexplicable—random and noisy. Schaffer, using the techniques of phase-space reconstruction, shows that measles follow a strange attractor, with a fractal dimension of about 2.5. ([Location 4556](https://readwise.io/to_kindle?action=open&asin=B004Q3RRPI&location=4556)) - The ordinary-sized stuff which is our lives, the things people write poetry about—clouds—daffodils—waterfalls—and what happens in a cup of coffee when the cream goes in—these things are full of mystery, as mysterious to us as the heavens were to the Greeks…. The future is disorder. A door like this has cracked open five or six times since we got up on our hind legs. It’s the best possible time to be alive, when almost everything you thought you knew was wrong. ([Location 4626](https://readwise.io/to_kindle?action=open&asin=B004Q3RRPI&location=4626)) - IN THE HEADY early days, researchers described chaos as the century’s third revolution in the physical sciences, after relativity and quantum mechanics. What has become clear now is that chaos is inextricable from relativity and quantum mechanics. There is only one physics. ([Location 4669](https://readwise.io/to_kindle?action=open&asin=B004Q3RRPI&location=4669))