Summary of the book "Chaos" - By James Gleick

Key Concepts in this book:

1. After identifying the unpredictability of weather, meteorologist Edward Lorenz became the conceptual father of chaos theory.
2. Nonlinear systems with simple inputs can yield immensely complicated behaviour.
3. Nonlinear systems were first studied seriously by physicists and mathematicians in the 1970s.
4. The behaviour of animal populations is similar to that of nonlinear dynamical systems.
5. The indefinitely intricate patterns of complex dynamical systems were unveiled by Mandelbrot's fractal geometry.
6. Strange attractors aided scientists in their understanding of turbulence's complex dynamics.
7. Mitchell Feigenbaum raised chaos theory to new heights of legitimacy by discovering the universal principles of nonlinear systems.
8. To popularize chaos theory, a group of young mathematicians at UC Santa Cruz employed computer graphics and ordinary phenomena.
9. Nonlinear dynamical systems can be found all across nature, and they're especially vital to our biology.

Who can benefit the most from this book:

• Curious minds interested in unraveling the mysteries of the universe.
• People interested in the history of modern science.
• Anyone looking to understand and appreciate the chaos of life.

What am I getting out of it? Discover the order that lies beneath life's chaos.

The world would run like clockwork if physicists had their way: routinely and predictably, according to a few simple rules. And they studied the globe as though it did for a long time. Any evidence of randomness or disorder in their data was brushed aside as a fluke.

However, in the 1970s, a small group of scientists decided to investigate these anomalies further. They discovered chaotic behaviour everywhere using modern computer technology: in weather patterns, the irregular drip of a faucet, even the beat of our hearts. Then they noticed something fascinating: behind the turmoil, there was a curious order.

This summary illustrates why chaos might be the ordering basis of life and how the emerging discipline of chaos theory changed science.

• You'll learn why we might blame a butterfly in Peking for a storm in New York.
• Why the British coast is eternally long.
• And how to cause a mosquito jet lag in this summary.

1. After identifying the unpredictability of weather, meteorologist Edward Lorenz became the conceptual father of chaos theory.

What percentage of the time do you believe the weather forecast?

Scientists in the 1950s were very confident about their ability to forecast – and even manipulate – the weather. New computer technology was the source of this promise.

Of course, they were well aware that accurate measures of anything as complex as the weather were difficult to come by. However, they believed that with enough data and computing capacity, it would be possible to forecast the weather for months in advance — at least roughly.

They have no clue how delicate, unstable, and unpredictable physical systems such as the Earth's weather are. This was demonstrated by a meteorologist with a mathematical bent.

Here's the main point: After identifying the unpredictability of weather, meteorologist Edward Lorenz became the conceptual father of chaos theory.

Edward Lorenz started a weather simulation on his spanking new computer in 1960. He was interested in learning more about how weather patterns evolve over time. And he came across something really disturbing.

Lorenz's weather simulation was quite basic, with no clouds at all. Numbers were used to representing conditions such as temperature and airstream. Lorenz would pick one of the variables and print out a graph that displayed its variations over time to investigate how they behaved over time.

He wanted to redo a simulation from the day before in 1961. However, he chose to begin the simulation in the middle, manually putting in the values from the previous output.

At first, the second simulation was identical to the first. The variables' behaviour began to deviate after that. They became increasingly out of sync as the simulation progressed. Finally, the second graph's motion appeared to be completely different from the first.

What caused such a huge discrepancy? Only up to the third decimal point had Lorenz filled in the values from the previous simulation. He'd typed in.506. for airstream, for example. However, the computer's calculations were accurate to the sixth decimal place:.506127. This seemingly insignificant adjustment was enough to knock the weather forecast entirely off course.

Lorenz was taken aback. He, like other scientists at the time, felt that tiny oscillations in large-scale processes like the weather had little impact. Instead, his blunder demonstrated how unstable, unpredictable, and chaotic complex systems can be.

The butterfly effect was coined by Lorenz. This means that weather systems are so sensitive to minor perturbations that a butterfly flapping its wings in Beijing today could cause a devastating storm in New York next month. This is referred to as "sensitive dependency on initial conditions" in science, and it constituted the foundation of the new area of chaos theory.

2. Nonlinear systems with simple inputs can yield immensely complicated behaviour.

There's a lot of sensitivity to beginning conditions. If you've ever missed a bus, which caused you to miss a flight, which ruined a whole business trip, you know how small mistakes can quickly spiral out of control.

One of the reasons Lorenz was attracted by the weather was because of the butterfly effect. It means that even if we have strewn weather sensors a foot apart throughout the globe, we wouldn't be able to forecast the weather for a few weeks. Another remarkable aspect of weather is that it is aperiodic — that is, it is almost cyclical but never quite repeats itself.

In truth, Lorenz' talent was in displaying the almost-orderly patterns in the chaos, not in illuminating the chaos itself.

This is the most important message: Nonlinear systems with simple inputs can yield immensely complicated behaviour.

Following his discovery of meteorological chaos, Lorenz set out to identify physical systems that behaved similarly. A simple waterwheel, turning as the flow of water fills its buckets, was one of the most renowned chaotic systems he discovered. Lorenz discovered that if the water flow is fast enough, the buckets cease filling fully and the wheel's motion slows or reverses. The motion gets chaotic at high speeds.

The waterwheel, like our weather, is a nonlinear dynamical system. But what exactly does that imply?

When Lorenz looked into the algebra behind such systems, he discovered that chaotic behaviour could be produced using just three simple nonlinear equations. The output value of a nonlinear equation is not proportional to the input value. As a result, a nonlinear dynamical system is one in which small variations can have arbitrarily large consequences.

In the actual world, many nonlinear dynamical systems are both damped and driven. Consider a playground swing that you can speed up by giving it the same push each time, but that is also slowed down by friction. The swing should rapidly acquire its equilibrium, swinging at the same height and speed every time, according to common sense. That, however, is not the case. Most damped-and-driven systems, in fact, never reach equilibrium.

When Lorenz plotted his three equations on a graph, he discovered a distinctive shape: a weird three-dimensional double spiral that resembles a pair of butterfly wings. Like the weather, the waterwheel, or a playground swing, its motions were virtually cyclical but never exactly repeat themselves.

The discovery by Lorenz that a few simple equations can yield complicated chaos patterns was revolutionary. And, as with all revolutions, it was faced with resistance by those who clung to the existing quo.

3. Nonlinear systems were first studied seriously by physicists and mathematicians in the 1970s.

Scientists, like the rest of us, enjoy having their expectations dashed. They certainly weren't expecting some of our world's most fundamental physical systems to act in such a chaotic and unexpected manner. As a result, the majority of them were not enthusiastic about the new chaos theory, which was welcomed by younger, more freethinking scientists beginning in the 1970s.

It sounded unscientific, outlandish, and, most crucially, it seemed to confuse – if not contradict – what they thought they knew about the cosmos. Until Lorenz, most scientists had stuck to linear descriptions of the world.

Galileo, for example, was so sure of a linear theory of motion that he perceived a regularity that didn't exist when studying pendulums.

What is the most important message? Nonlinear systems were first studied seriously by physicists and mathematicians in the 1970s.

Galileo believed that a pendulum, no matter how wildly it swings, always preserves the same time. It swings slowly if it swings narrowly. It swings that much faster if it swings that wide. However, friction, air resistance, and the changing angle of a swinging pendulum transform it into a nonlinear dynamical system with chaotic motion.

Pendulums become one of the most popular objects to study for scientists interested in chaos.

Stephen Smale, a mathematician at UC Berkeley, was one of the first to take chaos seriously, even though he had never heard of Lorenz's work. Smale studied topology, a branch of mathematics that investigates which properties of geometric objects remain constant when they are distorted, twisted, or stretched.

He was able to visualize chaotic systems thanks to his geometric method. Smale researched oscillating electronic circuits, specifically the Van der Pol oscillator. Using his topology knowledge, he devised a striking visual comparison for the behaviour of this nonlinear system. He pictured a horseshoe-shaped rectangle in three-dimensional space that was squeezed, stretched, and folded. After that, you can wrap another rectangle around the horseshoe and repeat the process as needed. You can never estimate where two close points of the rectangle will end up on the horseshoe map, no matter which two you choose.

Smale discovered, to his astonishment, that chaos and instability are not the same things. Nonlinear systems, he discovered, can have substantially more stable average behaviour than linear systems. Even when exposed to external noise and perturbations, a nonlinear system quickly reverts to its original chaotic structure.

Smale didn't learn about Lorenz's work until later, and he was shocked that a meteorologist had predicted chaotic mathematics ten years before him. When individuals started integrating Lorenz's and Smale's work, it ushered in a new generation of chaos experts who were enthralled by the richness and complexity that simple, predictable systems may produce.

4. The behaviour of animal populations is similar to that of nonlinear dynamical systems.

Mathematicians and physicists aren't the only ones who are interested in nonlinear dynamical systems. Nonlinear systems are fundamental to nature, as Lorenz demonstrated when studying weather.

For example, animal populations change in a nonlinear, dynamical manner. Ecology is a branch of biology that analyzes how organisms behave over time, and it was one of the first fields to link its findings to chaos theory.

The math behind population growth is straightforward. The greater the number of animals you have, the more offspring they can have. However, animal populations do not continue to increase indefinitely for a variety of reasons. When you include limited food resources, for example, the calculation becomes considerably more difficult. A tiny population may increase swiftly and exponentially at first. However, the larger it becomes, the slower it grows. It occasionally, and inexplicably, collapses totally. This is referred to as a "boom-and-bust cycle" in ecology and economics.

What is the main point here? The behaviour of animal populations is similar to that of nonlinear dynamical systems.

Consider how an ecologist may track changes in the gipsy moth population over time. This happens naturally in the actual world, one month at a time. Differential equations are mathematical equations that can describe such a smooth, nonlinear shift. However, differential equations are difficult to solve, and most biologists dislike math, to begin with. As a result, they employ difference equations, which assess changes in small increments – such as year to year.

After a certain point, a realistic difference equation that represents how the gipsy moth population evolves year by year must control expansion. A logistic differential equation is the simplest equation that meets this requirement. For a long time, scientists assumed that this form of the equation, like the animal population, would always attain equilibrium.

Robert May, an ecologist, was experimenting with a logistic differential equation when he made an unexpected discovery. May discovered that if he increased the "boom-and-bustiness" of his fictitious animal population, it began to act abnormally. The population's periodic cycles would first double in length, then double again, looping into so-called period-doubling bifurcations. The entire system would eventually become chaotic.

May sought an explanation from his mathematical buddy James Yorke. Yorke established in his important study "Period Three Implies Chaos" that when a system starts breaking up into period-doubling bifurcations, chaos is just a matter of degrees away.

Scientists, he said, tended to ignore such bifurcations because they didn't want to see the chaos that lurked in the systems they investigated. May was one of the first scientists to take epidemics seriously after applying his interest in chaos theory to the study of them.

5. The indefinitely intricate patterns of complex dynamical systems were unveiled by Mandelbrot's fractal geometry.

Benoit Mandelbrot, a mathematician and polymath, has spent most of his life in settings where he was not wanted. He first fled Poland with his family to France in the 1930s when he was a boy. He felt suffocated by the intellectual milieu at the École Polytechnique, where he studied mathematics, after the war. His colleagues didn't appreciate his pictorial approach to mathematical issues because pure arithmetic was all the rage at the time. As a result, Mandelbrot fled to the United States, where he obtained work at IBM's New York research division.

Mandelbrot chose unfavourable regions in his research as well. Studying economic trends, such as income distribution and price variations, was one of his earliest interests at IBM. When he was studying cotton price swings in the nineteenth century, he got a glimpse of the finding that would make him famous: our universe's densely nested nature.

Here's the main point: The indefinitely intricate patterns of complex dynamical systems were unveiled by Mandelbrot's fractal geometry.

Prices, according to economists of the time, fluctuated randomly in the near term but responded to actual economic causes over time, such as economic policy and new technology. They also believed that most prices should converge on average. The cotton prices of the previous century, on the other hand, had definitely not done so. Mandelbrot examined using the most up-to-date computers IBM had to offer. And he discovered something intriguing: daily price variations matched monthly price changes, with little trends nested inside larger trends, and so on.

This symmetry of scale captivated Mandelbrot, and he soon recognized it in numerous structures, both abstract mathematical structures and real-world phenomena. He began to see, for example, how many natural structures, such as mountains and clouds, can be broken down into smaller and smaller versions of themselves. Mandelbrot coined the term "fractal" to describe structures that are self-similar.

Mandelbrot likes to use a simple question to demonstrate his discovery: How long is the British coast?

You may simply answer this question by looking at a map, measuring the shoreline with a ruler, and then bringing the measurement up to scale. But did you actually take into account all of the coastline's nooks and crannies? Most likely not. To do so, you'd have to walk the length of the shore and measure each twist and turn. But hold on a second, can you make out the outline of each rock and stone that makes up the coastline? The shoreline becomes longer as the units of measurement become smaller. The length of Britain's coastline approaches infinity as your units of measurement shrink - say, to an atom.

This infinite — the rugged, scattered, and fragmented nature of our reality – is accounted for by Mandelbrot's novel fractal geometry. Mandelbrot became something of a celebrity in academia because fractal geometry was so elegant and beautiful. His inexhaustibly complex geometrical constructions became the visual embodiment of chaos theory.

6. Strange attractors aided scientists in their understanding of turbulence's complex dynamics.

Werner Heisenberg, a quantum scientist, pledged on his deathbed to ask God two questions regarding physics: What is the point of relativity? What is the significance of turbulence? Heisenberg said, "I honestly think He may have an answer to the first question."

Turbulence is one of the most difficult long-standing problems in physics. It occurs when a smooth flow of a gas or liquid becomes clogged, forming whorls and eddies. Turbulence is all around us, and it's a major issue for engineers. We can be fascinated as a cigarette's smoke ring climbs slowly before breaking up into small curls. We're more inclined to panic if the same thing happens beneath the wings of an aeroplane we're flying in.

For a long time, researching fluid dynamics appeared hopeless, so physicists delegated it to engineers who had to deal with its practical ramifications. Chaos theory, on the other hand, was expected to throw new light on fluid dynamics.

Here's the main point: Strange attractors aided scientists in their understanding of turbulence's complex dynamics.

Prior to chaos theory, the most influential theory of turbulence in physics was proposed by Russian scientist Lev D. Landau, who argued that any liquid or gas is made up of a large number of tiny particles whose motion is influenced by the motion of their neighbours. Particles in a smooth flow have limited degrees of freedom. When a flow becomes turbulent, however, particles obtain more degrees of freedom, increasing turbulence.

Harry Swinney and Jerry Gollub, two of Landau's American colleagues, worked up in 1973 to prove that turbulence grows up in a linear method. They built a system of two cylinders, one spinning inside the other, with a liquid flowing in the gap between them, to explore fluid motion in action. The liquid runs smoothly at first. However, as the cylinders' rotation speeds up, the liquid begins to flow in wavy bands. At higher speeds, the motion becomes chaotic, resulting in turbulence. The progression did not appear to be progressive at all. Most notably, even in turbulence, the liquid flow was not evenly disordered – smooth flow zones jostled with turbulence regions.

David Ruelle, a Belgian physicist, came to the rescue. He was working on an alternative to Landau's fluid motion theory when he heard Steve Smale speak on chaos in the early 1970s. He plotted the motions of dynamical systems in phase space to visualize the onset of turbulence. A phase space is an abstract space that tracks all potential states of a system at any given time and aids scientists in visualizing how it evolves. In phase space, certain systems have "attractors," such as a stable state where they find equilibrium or dynamic states where they cycle.

Many nonlinear dynamical systems have what Ruelle calls "strange attractors," as he discovered. These systems revolve around specific places in phase space, but never in the same cycle. After Ruelle's study was published, other scientists began creating their own unusual attractors and discovering them in nature's chaos. The orbit of stars around the core of "globular clusters," for example, corresponds to a weird attractor, according to astronomer Michel Hénon.

Edward Lorenz had arrived first once more. When he was calculating his first nonlinear system, he noticed an eternally looping set of butterfly wings. The Lorenz attractor was the world's first odd attractor.

7. Mitchell Feigenbaum raised chaos theory to new heights of legitimacy by discovering the universal principles of nonlinear systems.

In 1974, workers and police at the Los Alamos National Laboratory in New Mexico were becoming apprehensive. At night, a dishevelled-looking man was pacing frantically about campus, chain-smoking.

Mitchell Feigenbaum, a young mathematical physicist on the cusp of a mind-boggling breakthrough, was revealed to be the oddball.

Feigenbaum was regarded by his colleagues as a savant among savants. He began his study of chaos theory by looking at extremely simple nonlinear equations of phase transitions, comparable to those investigated by ecologist Robert May for animal populations. Feigenbaum, like May, was enthralled by the notion that basic systems can exhibit highly complicated behaviour, and that some systems never achieve equilibrium.

Here's the main point: Mitchell Feigenbaum raised chaos theory to new heights of legitimacy by discovering the universal principles of nonlinear systems.

Feigenbaum was particularly interested in systems that were almost intransitive. These are nonlinear systems that oscillate for a long period around one average state before kicking into a whole new average state at random. For example, climate experts have long presented an image of a White Earth. If enough ice and snow blanketed the world, it would reflect the sun's heat so well that it would settle into a considerably cooler climate. The White Earth scenario is so feasible that scientists are baffled as to why it hasn't occurred in the billion-year history of the planet.

Feigenbaum was fascinated by the point where order and chaos collide and a new average state emerges. He determined the period-doubling bifurcations of various nonlinear equations using a pocket calculator. Feigenbaum discovered an unusual regularity when he wrote the numbers for one of the equations. The numbers were geometrically converging - coming together. Feigenbaum arrived at the number 4.6692016090 when calculating the ratio of convergence for the period-doubling bifurcations of the equation.

But it was when he evaluated the ratio for a different nonlinear equation that he made the truly astonishing discovery. He received the same number. He repeated the computations for all nonlinear equations he could discover, with the same result. Even scientist Robert May recalls coming across this number while researching nonlinear equations for altering animal populations later in his career. Feigenbaum began busily working on his new hypothesis in his Los Alamos lab. He'd finally outlined the general concept of the Feigenbaum constants after two months of working 22 hours a day.

The Feigenbaum constants indicated a significant new characteristic of chaos theory's expanding field: universality. Until that time, scientists assumed that each nonlinear dynamical system had to be treated individually. However, Feigenbaum demonstrated that certain characteristics of nonlinear systems stay constant and can even be predicted.

The mathematical proof of Feigenbaum's idea didn't come until 1979. The discovery of the Feigenbaum constant, on the other hand, was enough to bring the new area of chaos theory together and give it the credibility it lacked in the eyes of established scientists.

8. To popularize chaos theory, a group of young mathematicians at UC Santa Cruz employed computer graphics and ordinary phenomena.

By 1977, most scientists had heard about chaos theory, a bizarre new branch of physics. In the same year, the first major chaos conference was conducted in Como, Italy. However, if you were a young math or physics student interested in chaos, you had no mentors — let alone academics – to help you.

A group of young mathematicians at the University of California's new Santa Cruz campus took matters into their own hands. Robert Stetson Shaw, a modest young graduate student, was the catalyst. He'd heard of the Lorenz attractor and began experimenting with it by graphing its equations on the campus's large analogue computer. Shaw was able to change the variables in the equations by turning knobs on the computer, which helped him visualize the sensitive dependency on the initial conditions discovered by Lorenz.

His work swiftly gained a following, and students and mathematicians Doyne Farmer, Norman Packard, and computer expert James P. Crutchfield soon joined in. The Dynamical Systems Collective was their name, yet their students referred to them as the Chaos Cabal.

The main message is that a group of young mathematicians from UC Santa Cruz popularized chaos theory by using computer graphics and daily happenings.

Other plotters, converters, and filters were added to the Dynamical Systems Collective's computer lab. They could see the random motion of nonlinear systems and uncover patterns in the chaos using all of their computers and equipment. They looked into what the shape of a weird attractor indicated about the system is depicted, for example.

Their connection of chaotic studies and information theory is one of their lasting contributions. The storage and transmission of digital information is the subject of information theory. The concept of entropy, which argues that our world and all physical systems progress toward higher levels of disorder, is central to information theory. Pouring ink into a swimming pool, for example, will cause the ink to spread and evaporate until the water and ink molecules are thoroughly combined.

Strange attractors, according to the Dynamical Systems Collective, are information engines. They increase a system's entropy, resulting in chaotic, novel, and unpredictable behaviour. The group suggested that this chaotic information production could be at the root of our mental processes and biological evolution's trajectory.

They didn't only bring chaos theory to the computer age, though. They also demonstrated how it may be applied to ordinary events. They used to sit in a café together and wonder aloud, "Where is the nearest unusual attractor?" Is a shaking automobile fender considered chaotic? Is it possible for a flag to act nonlinearly when it is caught in the wind? Robert Shaw demonstrated that even a dripping faucet can produce a random, endlessly creative pattern in one experiment.

The Dynamical Systems Collective brought chaos theory to the pinnacle of its popularity using common examples and cutting-edge computer visualizations. Scientists from the fields of economics, ecology, and meteorology took notice, and the field exploded.

9. Nonlinear dynamical systems can be found all across nature, and they're especially vital to our biology.

After the concept of chaos gained traction in the scientific community, researchers began to notice nonlinear dynamical systems all around them.

Albert Libchaber, a French physician, collaborated with an engineer on an experiment to validate Feigenbaum's hypothesis of turbulent fluid motion. Libchaber built a small box that held liquid helium between two metal plates that he could heat independently. The helium began to move once a temperature difference of a few millidegrees was created between the plates. The helium fluid first structured itself into two rolling cylinders as the warm liquid ascended and the cool liquid sank. However, as Libchaber increased the temperature, he noticed a pattern of period-doubling bifurcations.

Nonlinearity, according to Libchaber, is a natural protection against noise, glitches, and errors. When a linear system is nudged, it stays off track indefinitely. When a nonlinear system is given the same nudge, it eventually returns to its original state.

The main point is that nonlinear dynamical systems may be found everywhere in nature, and they are especially crucial to our biology.

Physicians began to confirm Libchaber's theory on chaos and biology in the 1980s.

Physicist Bernardo Huberman presented a nonlinear model of eye movement in individuals with schizophrenia at a large conference on chaos in medicine. People with this diagnosis, and sometimes their families, have difficulty tracking objects with their eyes, such as a swinging pendulum. Their eyes leap around frantically instead of gliding smoothly with the motion, never quite hitting the goal.

Doctors tend to think of the body as a collection of different organs, each with its own microstructure and function. The universal laws of motion apply in the human body just as they do everywhere else, according to Huberman's address. Random motion patterns, nonlinear oscillation patterns, and bifurcating rhythms can all be observed in biological structures.

Take the heart for example. Our heartbeat is regular, and any deviations can be life-threatening. Ventricular fibrillation is one such abnormality. When the synchronized pulse of contraction that underpins our heartbeat breaks down, it throws individual muscle cells and pacemaking nodes out of rhythm. The heart writhes like a bag of worms instead of contracting and relaxing in a regular manner. Every year, hundreds of people die suddenly as a result of heart fibrillation.

Physicians now understand that the heart is a nonlinear dynamical system whose motion can devolve into chaos if given the incorrect nudge. It can be pushed past the bifurcation line and into fibrillation by a tiny impulse at the incorrect time, whether from an internal fault or an external shock. A tiny jolt can also assist a fibrillating heart get back on track, which is fortunate for us. Doctors employ defibrillators to do exactly that: they give the heart a mild electrical jolt to get it back to its regular oscillating rhythm.

More dynamic diseases, including lung difficulties, a type of leukaemia, and possibly even schizophrenia, have been uncovered by doctors today.

Einstein famously stated, "God does not play dice with the universe." Physicist Joseph Ford was prompted by the discovery of chaos theory to oppose him: God does play dice, but they're loaded dice. "To find out by what rules they were loaded," Ford contended, was the goal of modern physics.

The important message in this summary is that scientists have been discovering the chaos hidden in simple physical systems since Edward Lorenz's weather simulations in the 1960s. Simple laws, they discovered, can yield complicated, unpredictable, and chaotic behaviour, much to their surprise. The weather, animal populations, and even human heartbeat are all examples of nonlinear dynamical systems. The chaos, on the other hand, is never aimless. Mathematicians like Benoit Mandelbrot and physicists like Mitchell Feigenbaum have demonstrated that the chaos of our planet has a weird, beautiful order.

Practical Suggestions:

Try out the chaos game.

Michael Barnsley, a British mathematician, created the chaos game to demonstrate how basic laws can lead to complex patterns of chaos. All you need is a penny, some paper, and a pen. Begin writing anywhere on the sheet of paper. Then make a rule for head or tail, such as "move 25% closer to the centre for the head" or "move two inches south for the tail." Begin flipping the coin and making a fresh mark on the paper for each new point. The chaotic game does not yield a random scattering of dots; instead, it begins to reveal a particular shape. Because there is order in the chaos, even with random processes.