Chemistry: Chaos Theory

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Chemistry: Chaos Theory


Chaos theory is the study of mathematical systems that exhibit certain characteristic properties, one of which is extraordinarily erratic behavior. Examples of such systems include population growth, turbulent fluids, and the motion of the planet's. Though chaotic systems had been recognized (but not defined) throughout human history, it was not until the 1970s that the mathematical tools existed to examine these sorts of complicated behaviors in a quantitative fashion.

Through intensive interdisciplinary work by an international set of researchers, chaos study has moved from a small group of interested practitioners into a worldwide phenomenon. Chaos theory has been applied to weather, populations, economics, turbulence, information theory, and neuroscience, among other topics. Chaos theory entered into public consciousness with the publication of colorful fractal pictures, popular science books, and public debates over its validity.

Although popular among select audiences, chaos was seen by others as an attack on the existing way of doing science, reductionism, and early practitioners met with a lot of resistance. Even today, some researchers are concerned about the use of chaos in both the natural and social sciences.

Historical Background and Scientific Foundations

Before the term “chaos” was used by mathematicians and scientists, chaotic phenomena had been observed in nature. In the middle of the seventeenth century, Isaac Newton (1642–1727) developed differential equations, which show how quantities change over time, and used them to describe the laws of planetary motion. Newton was able to solve the problem of determining the location of a single planet orbiting the sun at a particular time in the future, the so-called “two-body problem,” but extending this to a system of more than two bodies, such as the moon or other planet's, was problematic. The equations of the conservation laws (such as the conservation of energy) that governed such complex motion could not be solved with simple algebraic methods. The many-body problem was more than theoretical: knowledge of the moon's position was important for celestial navigation. However, existing analytic methods were not powerful enough to solve it, and it was not until 1885 that progress was made.

That year, a mathematics professor in Sweden held a mathematical competition to honor King Oscar II's sixtieth birthday, asking entrants to address any of four pressing questions. One of them, tackled by Henri Poincaré (1854–1912), was the problem that Newton could not solve: the many-body problem. The question read something like this: Given a number of masses that obey Newton's law of gravitation, find the equations describing the position of the masses at any time.

Poincaré's paper, which for simplicity analyzed a three-body problem, derived a result that demonstrated the stability of the solar system (a many-body problem). In the future, he posited, the planets would not fly off into deep space. When this paper was being prepared for publication, however, several mathematical errors became apparent that invalidated this result. (In his defense, though, the point that Poincaré was trying to prove was so complex that it took vast energies just to perform the calculations, energies that would later be made easier by the computer.)

In the years since Newton's first attempt, however, mathematicians had come to understand that not all differential equations could be solved exactly, and mathematical tools had been developed to explain the qualitative properties of the solution. The judges, under the time constraints of selecting a winner before the birthday celebrations, saw the importance of Poincaré's paper and awarded him the prize.

A greatly corrected, revised, and lengthened version of the paper published in 1890, titled Sur le probléme des trios corps et les equations de la dynamique,” (On the Three-body Problem and the Equations of Motions; in modern terms, Equations of “Dynamics”) found a different result from the first. Far from depicting a stable Newtonian universe, Poincaré's second analysis of the problem showed chaotic behavior, describing a figure of curves that formed an intricate mesh so complex that he refused to attempt to draw it. His work, because of the impossibility of simple solutions, was largely qualitative and based on geometric reasoning. He also noted that a small difference in the initial position of the planets resulted in very large differences in the position of the bodies in the long run. A meteorologist would come to similar conclusions in the 1960s.

The First Route to Chaos

At the Massachusetts Institute of Technology in 1960, American meteorologist Edward Lorenz (1917–) programmed 12 weather-simulating equations into his vacuum-tube computer. While the equations governing the motion of air and water—both treated as fluids mathematically—had long been known, using them to predict weather changes had proved extremely difficult. These equations, like those Poincaré and Newton had studied before, described a dynamical system—a system that evolves over time.

In a dynamical system, both initial conditions (such as the positions and velocities of the planets at a particular time) and a set of rules (the differential equations governing motion) are used to calculate the state of the system in the future. For most dynamical systems, the future state of the system cannot be computed immediately: to do so, the state of the system has to be calculated for each moment from the beginning to the end point. Without a computer, this task was virtually impossible, but with his computer, Lorenz was able to calculate a series of numbers that showed various features of the simulated meteorological system, like temperature.

In the winter of 1962 Lorenz wanted to examine a particular sequence of numbers from his weather simulation in closer detail. When he ran it a second time, however, he got radically different results. This was unexpected because the weather system was deterministic; with the same initial conditions and the same equations, the system should have behaved exactly the same. Investigating the discrepancy, Lorenz discovered the cause of the strange behavior: he had run his second simulation with the initial condition 0.506, but the first calculation had used six digits, 0.506127.

In a dynamical system, Lorenz found, two very slightly different initial conditions (0.506 instead of 0.506127) led to radically different behavior in the long run—a conclusion that harked back to Poincaré's conclusion. The technical term for this conclusion, “sensitive dependence on initial conditions,” was eventually condensed in popular shorthand to “the butterfly effect,” which is based on the idea that even the flap of a butterfly's wings could produce a small atmospheric change that would create a chain of events resulting in a drastic change in weather patterns. Its practical result suggests that perfect weather prediction might be impossible. If the equations governing weather were anything like the equations Lorenz used to simulate the weather in his computer, the smallest discrepancy in the initial data would yield radically different results in the future state of the weather system.

To examine this effect further, Lorenz created a simpler three-equation system to describe atmospheric heat flow, known as convection. Like his previous 12-equation model, the three convection equations were deterministic, and for the most part the system was periodic: it repeated its behavior after a set period of time. However, when the temperature difference between the top of the atmosphere and the bottom of the atmosphere was great enough, the system would show turbulence. (This is similar to a pot of water. As the bottom heats up, the temperature difference between the top of the water and bottom of the water grows ever larger, until eventually turbulence occurs and the water boils.)

The second model behaved similar to the first, displaying a similar sensitive dependence on initial conditions. Unlike Poincaré, however, Lorenz was able to use his computer to generate a graphical display of his system. Plotting the evolution of the convection system, he started with a particular initial condition, then traced a curve that looked like butterfly wings. Further investigation revealed that all initial conditions would trace a similar curve—they were “attracted” to the same general region. These curves defined a space that would come to be known as the Lorenz attractor, part of a larger class of mathematical objects known as strange attractors.

After Lorenz, work with physical systems led to the discovery of more strange attractors, including the Rössler attractor, which came out of the study of chemical reactions, and the Chua attractor, which resulted from the study of an electronic circuit. From Lorenz's pioneering work, it is clear that modern chaos theory would not have been possible without computers to perform millions of operations in a short period of time. But computers were more than simply calculating robots—they became electronic laboratories themselves, simulating natural phenomena.

Two More Routes to Chaos

Chaos theory did not develop with one person or in one place, but was the product of different scientists working in different places, solving different problems. In addition to Lorenz, two other figures in the 1960s established key sections of chaos theory's intellectual framework.

One of these was the American mathematician Stephen Smale (1930–) at the University of California at Berkeley, who became internationally famous for his study of topology (the study of the properties that remain unchanged as geometric figures are stretched and folded). In the 1960s, Smale turned his research from topology to dynamical systems. Early on, he theorized that stable systems—those in which a small perturbation would not change the overall outcome of the system—could not behave erratically. At this point, he was not aware of the Lorenz attractor, which was a structurally stable system that did behave erratically. (This is not surprising, since Lorenz's paper was published in a specialty journal, traditionally read only by meteorologists. Like many important scientific discoveries, the importance of his work was not immediately recognized.)

One of Smale's colleagues, however, told Smale that the Lorenz attractor refuted his conjecture. Forced to rethink his conclusion, Smale came up with a visual way to understand why his conjecture was incorrect. The horseshoe map was a topological version of sensitive dependence on initial conditions, a way to deform a system (say a square sheet of rubber) topologically through repeated squeezing and folding, so that any two points close to each other in the original system ended up arbitrarily far apart after enough folding and stretching.

Smale's work expanded the study of dynamical systems into the domain of topology and reshaped the disciplinary boundary between physics and mathematics. Smale himself said, “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 … By 1968 this had completely turned around.”

The last figure setting the stage for an explosion of research in the 1970s was Belgian-French mathematical physicist David Ruelle (1935–). Located at a prestigious institute outside of Paris in 1971, he and mathematician Floris Takens (1940–) published “On the Nature of Turbulence,” which suggested that the onset of turbulence in a fluid was caused by topological properties of the fluid equations themselves, known as the Navier-Stokes equations. Just as the equations Lorenz used to study convection gave rise to a strange attractor—the butterfly-shaped Lorenz attractor—the authors argued that the Navier-Stokes equations also gave rise to strange attractors, and the presence of these attractors was responsible for the onset of turbulence.

Since the early nineteenth century, the transition of a stable system into a turbulent one had been ill-understood. Some scientists have called it the greatest unsolved problem in classical physics. The prevailing theory before Ruelle and Takens was promoted by Russian physicist Lev Landau (1908–1968) in the 1940s.

His model argued that eddies formed in fluids will generate smaller eddies within them, and these smaller eddies will then generate even smaller eddies within them, ad infinitum. As more and more eddies are created, the fluid flow begins its transformation from being predictable and regular to exhibiting turbulence. The eddies, in Landau's theory, are created through small external disturbances to the fluid. For Landau, the transition to turbulence was based not on the fluid equations themselves, but on external noise influencing the system. Taking the opposite position, Ruelle and Takens argued that turbulence could be explained by the fluid equations alone.

At the beginning of the 1970s the study of fluid mechanics and the question of turbulence lay at the uneasy intersection between mathematics and engineering. Ruelle and Takens's work had two important consequences. First, it inaugurated the merger between the study of fluid mechanics and dynamical systems. Second, it suggested physical experiments that could test the validity of Ruelle's ideas. No longer would the study of these strange dynamical systems be confined to paper or computer simulations. With Lorenz, Smale, and Ruelle building the foundation, by the end of the 1960s the stage was set for chaos to enter the academic community.

An Era of Interdisciplinarity

Lorenz was a meteorologist, Smale a mathematician, and Ruelle a mathematical physicist, yet all three were working on similar things. The 1970s ushered in a period of interdisciplinarity, in which biologists, hydrodynamicists, meteorologists, mathematicians, and physicists, among others, would read each others' papers, attend the same conferences, and enter into work traditionally outside their own specialized fields. Divisions among scientific disciplines have never been so solid as to prevent interdisciplinary work, but chaos, acting as a common meeting ground, brought together scientific nomads. It was also during this decade that experimental work brought some credibility to chaos, but at the same time, practitioners were unable to pin down exactly what they were studying.

Scientists, when investigating natural phenomena, would often encounter phase transitions—points at which a system changed its character dramatically. One example was the onset of turbulence, but there are many more, such as the magnetization of a nonmagnet or the transformation of a conductor into a superconductor. A number of studies drew analogies between various types of phase transitions; mathematically their descriptions appeared similar. In 1973 American physicists Harry L. Swinney (1939–) and Jerry P. Gollub (1944–) were investigating phase transitions in fluids. They set up an inexpensive apparatus: two cylinders, one inside the other, with a fluid in between. As they began to rotate the inner cylinder (keeping the outer cylinder at rest), they studied the properties of fluid flow. Once they reached a certain threshold, they observed turbulence. The experiment was not new; it had been performed before in 1923. What was new was the data-collection method, using the deflection of a laser beam in the fluid to measure the scattering.

Swinney and Gollub expected to confirm the older theory of turbulence proposed by Laundau. What they found comported better with the ideas of Ruelle and Takens. A second similar and widely discussed experiment was performed by French physicist Albert J. Libchaber (1934–) in 1977. The study of turbulence and convection was beginning to become an interesting scientific topic again. Instead of being squarely in the purview of mathematics or engineering, it began to be investigated by hydrodynamicists, plasma physicists, statistical physicists, thermodynamicists, and chemists. In the years between 1973 and 1977, a good number of conferences were held on turbulence. These acted simultaneously as sites where disciplines collided and where collaborations were forged.

James Yorke (1941–), an American mathematician, had stumbled across Lorenz's 1963 paper almost a decade after its publication and was enchanted by its conclusions. He sent a copy to Smale and made and distributed a number of other copies for his colleagues. He also cowrote a 1973 article in the widely read American Mathematical Monthly that brought the ideas to a new generation of mathematicians. The title of this article—“Period Three Implies Chaos”—marked the first time “chaos” was used in a technical sense.

Yorke's article made an important claim using a simple formula that was first used in 1838. The logistic equation is a model that predicts the population of a species over time, given information about how fast it reproduces and the maximum population sustainable in an environment. Yorke used the logistic equation to illustrate a powerful point: For most situations, the population of a species would remain the same after a long enough time had passed. However, after tweaking the conditions, he found that the population could eventually oscillate between two values—one year the population would be x, the next year y, the year after x, and so forth. Mathematically, this is called a bifurcation. It turns out the conditions could be tweaked so that the population would oscillate between 4, 8, 16, 32, etc., values. At a certain point, however, the system would become what Yorke called chaotic: The population would not oscillate among a fixed number of values, but rather, would never repeat itself.

Yorke used the logistic equation to illustrate a powerful mathematical theorem. Technically, it said that in any one-dimensional system (like the logistic equation), if a cycle of period 3 appears, then there would have to be cycles of every other period, as well as chaotic cycles. In a general sense this meant that even a simple equation can demonstrate some peculiar and unexpected behavior—chaos. More than that, in the midst of chaotic behavior, there can be pockets in which the system behaves nicely. In this way, chaos began to take on the meaning of its opposite: order. Even though the term chaos implied something erratic, it arose out of very simple systems, and even when chaos was observed, there seemed to be some kind of mathematical order within the chaos.

Universality in Chaos

In 1975 mathematical physicist Mitchell Feigenbaum (1944–) was a researcher at Los Alamos National Laboratory. After listening to a lecture by Smale, he began his work on the simple logistic map. He found that the bifurcations came at somewhat predictable intervals. In mathematical terms, they were converging geometrically. The convergent rate for the logistic map was calculated to be about 4.669. Feigenbaum discovered that the critical value of 4.669 occurred not just in the logistic map, but in a large class of equations—just as Yorke found that chaotic behavior occurred in a large class of equations.

He recalled in 1980 that “I spent a part of a day trying to fit the convergent rate value, 4.669, to the mathematical constants I knew. The task was fruitless, save for the fact that it made the number memorable.” That indicated that this number was a new universal constant. Just as the ratio of the circumference to the diameter of any circle will always be π, the convergent rate for any of a large class of equations will always be about 4.669.

Feigenbaum believed he had discovered a new law of nature. If a natural phenomenon could be described mathematically by one of the large class of equations, then Feigenbaum's work was a quantitative way to discuss the route to phenomena exhibiting chaotic behavior. Through his work, the analogy made between hydrodynamics (the transition to turbulence) and phase transitions in physics was reinforced. The equations were all part of the same class.

As often occurs with historically important works, Feigenbaum had difficulty publishing—it was not quite mathematics to the mathematicians because it did not have a grand level of abstraction, nor was his claim rigorously proved enough for physicists. In spite of this, his ideas spread through lectures and conversation and generated excitement in an assortment of intellectual circles. In retrospect, scientists look back on this work as a watershed, bestowing a sense of legitimacy to the idea that chaos was present in many natural systems, and that this chaotic behavior could be explored quantitatively instead of qualitatively and geometrically.

Moreover, in subsequent analyses, he used mathematical tools (renormalization group methods) commonly used by physicists, thus making his chaos relevant for that community. As a result, a number of physicists in the late 1970s and early 1980s began to look at hydrodynamics and turbulence, when previously the problem had belonged in the domain of mathematicians and engineers; a disciplinary reorientation took place. Libchaber's 1977 experiment on turbulence, for example, was concerned with calculating Feigenbaum's constant. His experimental value and the theoretical value were dissimilar, but not incompatible, and later experiments conducted by others in the early 1980s brought a closer correspondence between the theoretical and experimental values. These experiments showed that Feigenbaum's constant, and chaos theory in general, had a role to play in the physical world.

Two important chaos meetings were held in the late 1970s. In 1977, the New York Academy of Sciences held a conference on bifurcation theory and applications in scientific disciplines. This meeting, the first of such a large scale, brought dozens of researchers together: economists, physicists, chemists, biologists, and others. A second meeting was held two years later, this time attended by hundreds of researchers. Chaos was becoming a more popular topic of study in a variety of fields. By 1990, one bibliography of chaos—in Chaos II—listed 117 books and 2,244 articles on the subject from a variety of disciplines, and, significantly, from many different countries too.


The Lorenz attractor is a graphical representation of the behavior of the convection equations. The logistic map is a discrete version of the logistic equation and is mathematically written: xn1 = rx (1 蜢 xn). We need not go into details about the mathematics here, but what should be clear is that this simple one-dimensional equation demonstrates some very complicated behavior.

As the value of r increases, the long-term behavior of the system changes. For small values of r, the system settles down to a single value. As r increases, something special happens: the system settles down not to one, but two values: It bifurcates.

As r increases further, the system settles down to four values, then eight, then sixteen, and so on. Eventually, when r reaches a critical value, the system never repeats itself, never settles down to a finite set of numbers. Note that the behavior of the equation becomes more and more erratic as r increases, but even in the chaotic regime, there are pockets of nonerratic behavior.

While chaos theory's audience was increasing, there was—and remains—no single accepted definition. As late as 1994 American mathematician Steven Strogatz (1959–) wrote that: “No definition of the term chaos is universally accepted yet, but almost everyone would agree on the three ingredients used in the following working definition: Chaos is aperiodic long-term behavior in a deterministic system that exhibits sensitive dependence on initial conditions.”

“Aperiodic long-term behavior” simply means that the system does not settle down to a state in which nothing moves, or a state in which the system repeats itself over and over. Instead, the system should eventually lead to erratic behavior, like that in a Lorenz attractor.

Popularization: Chaos in Popular Consciousness

As chaos became a popular topic in the sciences, ideas about chaos theory also entered into the public domain. People learned about chaos through various channels, and one of these channels was fractals. Benoit Mandelbrot (1924–), a Polish-born, French-educated American mathematician, saw a universe full not of ordinary Euclidean geometry, in which figures were smooth, but full of “fractals,” a term he coined in 1975.


By the mid-1980s a wide-ranging set of systems had been shown to have exhibited chaotic behavior. More than mathematics and physics, even animals and humans fell under its purview as noted in an article in Scientific American in 1986:

In the past few years a growing number of systems have been shown to exhibit randomness due to a simple chaotic attractor. Among them are the convection pattern of fluid heated in a small box, oscillating concentration levels in a stirred-chemical reaction, the beating of chicken-heart cells and a large number of chemical and mechanical oscillators. In addition, computer models of phenomena ranging from epidemics to the electricity of a nerve cell to stellar oscillations have been shown to possess this simple type of randomness. There are even experiments now under way that are searching for chaos in areas as disparate as brain waves and economics.

SOURCE: Crutchfield, James P., et al. “Chaos.” Scientific American 54, no. 12 (December 1986): 46–57.

Fractals are objects that display self-similarity at all scales: If a fractal is magnified, the resulting image will have properties similar to the original fractal. Because fractals are so intricate—they can be magnified over and over again—they are often described as beautiful. As a result of their intricacy, however, everyday concepts like distance and area are difficult to apply. When studying these figures, Mandelbrot devised a new way to measure an object's dimensions. With his definition, a line still has one dimension, a plane still has two, but fractals could possess a dimension in-between—a nonintegral dimension, such as a 3/2 dimension.

Strange attractors such as the Lorenz attractor, it turns out, are not points, lines, or surfaces, but rather fractals with a dimension between 2 and 3. (A trajectory in the Lorenz attractor is infinitely long, but it is bounded by a finite volume—the butterfly shape.) Thus, understanding the properties of fractals and uncovering ways to study them could shed light on strange attractors—and consequently, on chaos.

Also important, however, is the explosion of fractals into popular culture. Mandelbrot's popular books Fractals: Form, Chance, and Dimension and The Fractal Geometry of Nature, transported fractals to a wider audience, including a number of professionals. Ruelle exclaimed, “I have not spoken of the esthetic appeal of strange attractors. These systems of curves, these clouds of points suggest sometimes fireworks or galaxies, sometimes strange and disquieting vegetal proliferations. A realm lies there of forms to explore, and harmonies to discover.”

The 1970s and 1980s fractal geometry was used to create art—illustrations of nature in addition to more abstract entities, known as Julia sets. Artists too found fodder in a new concept of space.

In addition to Mandelbrot's popular books and the widely reproduced pictures of fractals, American author James Gleick's (1954–) popular science book Chaos: Making a New Science also helped propel chaos theory into popular consciousness. Published in 1987, Chaos was an immediate bestseller.

Narrating the history of chaos as a paradigm shift, Gleick introduced chaos theory not only to the wider public, but also to numerous scientists. The impact of this book can be seen in the high number of citations it receives in both scientific and popular literature. (The Science Citation Index, which counts the number of times a particular book or article is cited in various research journals, puts the number at well over 1,000 articles.)

If Gleick introduced the concept of chaos, the publication of American novelist Michael Crichton's (1942–) book Jurassic Park in 1990 and the release of Steven Spielberg's (1946–) motion picture of the same name three years later put chaos theory center stage.

In the movie, the scientist Malcolm saw the dinosaur park as a physical system governed by chaos. His interpretation of chaos, however, was a theory of inherent unpredictability and disorder, an interpretation that is not quite accurate. Chaotic systems are often defined by deterministic equations—meaning that, in theory, at least, they are perfectly predictable. And, as noted before, one of the key features of chaos is the concept of order—not disorder. Regardless, Jurassic Park, Gleick's book, and captivating pictures of fractals, generated by many on their home computers, helped bring an interest in chaos to the wider public.

Chaos Outside the Traditional Sciences

Social scientists have long used statistics and mathematical models to better understand social phenomena; some hold the hope expressed in the introduction to Chaos Theory in the Social Sciences: Foundations and Applications, edited by L. Douglas Kiel and Euel Elliott, that chaos theory will be “a promising means for a convergence of the sciences that will serve to enhance understanding of both natural and social phenomena.” Chaos theory, these scientists believe, may finally explain the complex phenomena observed in social systems.

Beginning in the 1990s, political science, economics, sociology, and even in literary theory began to use chaos theory. Some of these fields rely heavily on its mathematical tools to analyze social data, while others capitalize on the metaphorical power of chaos. Political scientists, for example, have studied shifting public opinions and international conflict using some of the same mathematical tools used to study chaos. Some economists posit that because many simple economic systems show a sensitive dependence on initial conditions, the standard form of economics should be reevaluated, neoclassical economics. And literary theorist N. Katherine Hayles makes the case that since both science and the humanities are rooted in the same culture, when scientists were countenancing the implications of disorder, so too were fiction authors and literary theorists.

Modern Cultural Connections

In recent years, chaos has lost some of the excitement that fueled it earlier. Per Bak (1948–2002), a Danish theoretical physicist at Brookhaven National Laboratory, believed that chaos theory had run its course as early as 1985. Physicist J. Doyne Farmer (1952–) said, “After a while, though, I got pretty bored with chaos … [I] felt ‘So what?’ The basic theory had already been fleshed out. So there wasn't that excitement of being on the frontier, where things aren't understood.” Chaos today can be seen not as a highly active area of scientific research, but instead as a conglomeration of mathematical tools used in a wide variety of disciplines.

Chaos, however, has taken on a new incarnation in recent years: complexity, which, like chaos, defies a universal definition. (One scientist compiled 31 distinct definitions.) However, most often it is described as the state between order and chaos. It is in this regime that complexity researchers believe self-organization occurs. In highly ordered systems, nothing novel can emerge. Chaotic systems are too erratic to have something structured emerge. It is at the intersection, some believe, that interesting and complex behavior emerges. Complexity theorists work in a number of different fields, including artificial intelligence, information theory, linguistics, chemistry, physiology, evolutionary biology, computer science, archeology, and network theory. In fact, in 1999, the prestigious journal Science devoted an entire issue to examining research done on complex systems in a variety of fields.


Though the face of Benoit Mandelbrot (1924–) is not well known, the colorful fractal figures he pioneered are familiar to almost everyone. He is one of the most famous living mathematicians, having developed a geometry that would infiltrate a number of different arenas, including the study of turbulence, the stock market, physiology, and art.

Mandelbrot was born in Warsaw and raised in France. His father was a buyer and seller of clothes, his mother was a medical doctor, and his uncle, Szolem Mandelbrojt (1899–1983), was a Polish-born French mathematician at the Collège de France. While his uncle taught the younger Mandelbrot that mathematics was an honorable profession, he was also a purist who believed that mathematics and beauty were mutually exclusive.

Benoit Mandelbrot's mathematics education took place mainly in France, punctuated by brief stints at the California Institute of Technology and Princeton's Institute for Advanced Study. He was seriously affected by the existing political conditions, noting in an interview that “when I look back I see a pattern. For a long time that pattern was imposed by catastrophes, namely the fall of Poland and the occupation of France during the second world war. Those events dictated everything … Being raised under such hair-raising conditions can have a strong effect on someone's personality.” In 1952 he obtained a doctorate in mathematical sciences at the University of Paris.

Mandelbrot joined the research team at the International Business Machines (IBM) Thomas J. Watson Research Center in New York in 1958, remaining there until 1987, when he joined the mathematics department at Yale University. At IBM, Mandelbrot reveled in the freedom afforded in his research, allowing him to move in directions that a university position would not encourage; his research included the flooding of the Nile River, cotton prices, and the geometric shape of coastlines. It was in these studies that the idea of fractals began to form, a concept Mandelbrot continued to expand and develop in the following years. One of the most famous fractals, called the Mandelbrot Set, is based on work done by French mathematicians Gaston Julia (1893–1978) and Pierre Fatou (1878–1929).

The complex Mandelbrot set is created through a simple mathematical transformation. Every point in the plane undergoes this transformation, and if the result is infinity, the point is shaded a particular color. If the result is not infinite, the point is shaded black. Interestingly, his uncle Szolem introduced him to this work in 1945, but Mandelbrot would not return to it until the 1970s. Mandelbrot retired from Yale in 2005.

The major institution promoting the study of complex systems is the Santa Fe Institute (SFI) in New Mexico. George Cowan, founder of the SFI, and once the head of research at the national laboratory at Los Alamos, also advised President Ronald Reagan on the White House Science Council. During this time, he was made further aware of the interconnections between science and morality, economics, the environment, and other topics. Looking back, he said:

The royal road to a Nobel Prize has generally been through the reductionist approach…. You look for the solution of some more or less idealized set of problems, somewhat divorced from the real world, and constrained sufficiently so that you can find a solution … [a]nd that leads to more and more fragmentation of science. Whereas the real world demands—though I hate the word—a more holistic approach.

A reductionist view, he believed, also leads to studying simple systems, instead of whole, messy, complicated systems.

With the rise of computer use and numerical simulation, complex behavior could be investigated. By the 1980s the complex behaviors Cowan was interested in were being investigated. These complex systems were not linear, in the sense that they could be broken down into smaller parts; what made them interesting was their global behavior created by the nonlinear interactions between individual parts. He took this belief and transformed it into the now world-famous interdisciplinary SFI, founded in 1984, eventually adding three Nobel laureates to its staff. This institute is now synonymous with the study of complexity. At the institute and elsewhere, the study of chaos theory was incorporated into a broader framework of “complexity.”

Chaos and complexity: These terms hint at a new structure for science. Like Cowan, many see the sciences moving away from reductionism and linearity into something completely different. The term “paradigm shift” is often invoked by chaos researchers and popular writers to describe this transformation. In 1962, historian Thomas Kuhn (1922–1996) argued that science did not accumulate facts progressively. Instead, science is a cyclical process, punctuated by large-scale transformations known as “paradigm shifts.” (Paradigms, loosely defined, are ways of understanding the world. Paradigm shifts are brought about by finding anomalies that cannot be explained in the current paradigm.) Resolving these anomalies led to a new way of understanding the world, a way in which the anomaly makes sense. It should be noted that historians of science have found much to praise and criticize with Kuhn's theory about how science operates.


In his review of a book by Stephen Wolfram titled A New Kind of Science, Nobel Prize-winning physicist Steven Weinberg noted that the increasing study of complexity in physics had led to a disciplinary division—those who believe that reductionism is the best way to study nature, and those who argue that complexity is better. Weinberg levies a common criticism at those favoring complexity: there is no set paradigm that complexity has produced.

There is a low-intensity culture war going on between scientists who specialize in free-floating theories [like chaos] and those (mostly particle physicists) who pursue the old reductionist dream of finding laws of nature that are not explained by anything else, but that lie at the roots of all chains of explanation. The conflict usually comes to public attention only when particle physicists are trying to get funding for a large new accelerator. Their opponents are exasperated when they hear talk about particle physicists searching for the fundamental laws of nature. They argue that the theories of heat or chaos or complexity or broken symmetry are equally fundamental, because the general principles of these theories do not depend on what kind of particles make up the systems to which they are applied. In return, particle physicists like me point out that, although these free-floating theories are interesting and important, they are not truly fundamental, because they may or may not apply to a given system; to justify applying one of these theories in a given context you have to be able to deduce the axioms of the theory in that context from the really fundamental laws of nature….

Lately particle physicists have been having trouble holding up their end of this debate. Progress toward a fundamental theory has been painfully slow for decades, largely because the great success of the “Standard Model” developed in the 1960s and 1970s has left us with fewer puzzles that could point to our next step. Scientists studying chaos and complexity also like to emphasize that their work is applicable to the rich variety of everyday life, where elementary particle physics has no direct relevance.

Scientists studying complexity are particularly exuberant these days. Some of them discover surprising similarities in the properties of very different complex phenomena, including stock market fluctuations, collapsing sand piles, and earthquakes … But all this work has not come together in a general theory of complexity. No one knows how to judge which complex systems share the properties of other systems, or how in general to characterize what kinds of complexity make it extremely difficult to calculate the behavior of some large systems and not others. The scientists who work on these two different types of problem don't even seem to communicate very well with each other. Particle physicists like to say that the theory of complexity is the most exciting new thing in science in a generation, except that it has the one disadvantage of not existing.

SOURCE: Weinberg, Steven. “Is the Universe a Computer?” New York Review of Books 49, no. 16 (October 24, 2002).

Researchers interested in chaos and complexity tout the beginnings of a paradigm shift. The old paradigm, reductionism, was on its way to being replaced, or at least supplanted by, nonreductionism, which sees the world as a system of many parts that interact in nonlinear ways. Many scientists are skeptical of this claim. They do not believe complexity is a paradigm shift in the Kuhnian sense, nor do they believe that the reductionist program has yielded all its secrets. Whether a paradigm shift, a fad, or something else, science has of recent been changing its flavor, and chaos and complexity have played a role.

Primary Source Connection

This patent describes an unusual application of chaos theory to a real-world problem: wrinkly clothes. The patent was issued October 1, 1996.


Description of the Prior Arts

Washing machines presently employed use a pulsator or drum. The washing machines using the pulsator increase the washing power by irregular flow of washing water in the washing tank by repeatedly rotating the pulsator disposed in the bottom of the washing tank clockwise and counter-clockwise.

The washing machine using the drum increases the washing power by a head of laundry derived by rotating the drum itself in which the laundry and washing water are contained.

However, such washing machines using the pulsators have the disadvantage in that the laundry is wrinkled while being rotated together with the washing water, and further, it is difficult to obtain the higher washing effect due to limitations of the washing power that is dependent on the rotation power of water current.

On the other hand, washing machines using the drum have the disadvantage due to the difficulty of obtaining the higher washing effect because of the limited washing power that is dependent on the head of the laundry, as well as the laundry being wrinkled by the regular and reverse rotation of the drum.

Further, a problem exists in that the manufacturing cost increases because an additional program or hardware, must be installed in order to prevent the wrinkling of the laundry.

Summary of the Invention

It is an object of the present invention to provide a chaos washing machine which can improve the washing effect by using the random generation of water current in order to overcome the aforementioned defects.

It is another object of the present invention to provide a chaos washing machine which can reduce the wrinkling of the laundry by creating a random stream of water for short periods, random generation of the water stream and then producing a strong turbulent flow in the washing tank.

It is a further object of the present invention to provide a method of washing of a chaos washing machine which can improve a washing power and prevent the wrinkling of clothes by using the random generation of the water current.

These and other objects of the present invention are accomplished by means of a chaos washing machine which is composed of a first washing tank having a plurality of inducing holes for inducing the water current in which a detergent dissolved into the space where the laundry is contained, a second washing tank for enclosing the washing tank to be filled with water and a detergent, a water current fan and a fan motor creating a turbulent flow to water in which a detergent is dissolved and pushing the turbulent flow from the second washing tank through the inducing holes into the first washing tank and a washing tank motor for rotating the first washing tank. […]

As stated above, according to the chaos washing machine of using a convection flow of the present invention, the laundry is not wrinkled. Therefore, the present invention eliminates the need for mechanically and electrically means for sensing the wrinkle of the laundry or for proceeding the twist preventing pattern of the laundry such as the prior pulsator washing tank thereby easily accomplishing the structure of the washing machine and the design and the operation of the control program.

Further, according to the present invention, the laundry will be not rotated with the washing tank. Therefore, the present invention can reduce the damage created to the laundry that is created by being regular and reverse rotation of the laundry such as the prior pulsator or drum washing machines thereby extending the life of cloth and maintaining the cleanliness of clothes after washing.

Bo Wang

wang, bo. “chaos washing machine and a method of washing thereof.” patent storm. (accessed november 21, 2007).

See Also Physics: Aristotelian Physics; Physics: Articulation of Classical Physical Law; Physics: Newtonian Physics.



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Sameer Shah

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