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| Patterns composed of ever-smaller branches are very common in nature; some examples are (clockwise from top left): plants, rivers, pulmonary bronchi and blood vessels. (Credits: Catherine Macbride; Jassen Todorov; Guzel Studio; Inozemtsev Konstantin.) |
IN SOME WAY, the movement of the celestial bodies across the sky resembles that of a colossal clockwork device. Stars and planets follow orbits which are neatly described by mathematical equations, such that by knowing their position at a given point in time, we can foretell their positions at any future time. This is the basis of our ability to predict lunar phases, solar and lunar eclipses, meteor showers and other astronomical phenomena. It is perhaps unsurprising that, until quite recently, science believed the whole universe to operate in such a way, following predictable, ‘clockwork’ processes. Such a view is termed ‘determinism’ since, according to it, the future is completely predetermined by mathematical equations, such that no randomness or unpredictability would be left, were we able to work out those equations. This was perfectly put by the prominent eighteenth-century mathematician Pierre-Simon Laplace, who wrote:
However elegant this might sound, over the second half of the last century it became increasingly clear to scientific minds that the universe is very little like this. Various natural processes, even though they could be perfectly described using mathematics, were found to display a behaviour that was, on all accounts, impossible to forecast. In time, these and other baffling discoveries led us to realise that unpredictability is an intrinsic property of the universe; not only this, but it is this very property that grants inanimate matter the dazzling ability to spontaneously organise itself into the complex shapes and structures of the natural world. The uncanny force responsible for the unpredictability of the world received what is, in fact, a very popular name — chaos.
Although the term ‘chaos’ commonly serves as synonym for disorder or mayhem, its mathematical meaning is more specific. In a system that can be completely described by deterministic mathematical equations, without any unknown or random component, chaos is the property that makes this system capable of behaving unpredictably. In order words, even if we know the state of the system at a given moment, and the equations that describe the system’s evolution, it is impossible for us to predict its future behaviour.
One of the first to describe a system with chaotic behaviour was the meteorologist Edward Lorenz, who in the early 1960s was trying to model the weather using mathematics. The dominant view at the time was that the weather was a deterministic phenomenon, and thus could be forecasted using equations. But once Lorenz had written down a set of equations that captured the dynamics of air masses, he found that these did not yield any useful predictions. Actually, his system appeared to suffer from extreme sensitivity to even the slightest change in its starting conditions; these variations, initially tiny, rapidly amplified across the system as this evolved, causing it to deviate from its predicted course, and thus leading to completely unexpected outcomes.
Lorenz presented his findings in a talk which he titled: ‘Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?’. This concept would rapidly capture the public’s imagination, giving birth to the popular expression ‘the butterfly effect’. What Lorenz’s results imply is that, even if we should, in theory, be able to forecast the weather by measuring a set of variables (such as atmospheric pressure, temperature, humidity, wind speed, etc.) and solving some equations that describe the evolution of atmospheric conditions, these equations are so sensitive to even infinitesimal changes in their initial values, that we cannot possibly measure the variables we need with such accuracy as to be able to reliably predict the weather beyond a couple of days from now. It turns out that the butterfly effect, that is, a high sensitivity to the initial conditions, is actually a hallmark of chaotic systems.
The butterfly effect is easier to understand thanks to a simpler phenomenon than the weather. In the 1970s, biologist Robert May was working on an equation to model the changes in animal populations over generations. This is known today as the logistic equation, and is indeed very simple. Given a value, Current Size, that represents the current size of a population in relation to its maximum possible size (for example, a value of 0.5 means that the population has half the maximum size), using the logistic equation, we can easily find out the size of the population in its next generation, Future Size:
An intellect which, at a certain moment, would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect, nothing would be uncertain, and the future, just like the past, would be present before its eyes.
However elegant this might sound, over the second half of the last century it became increasingly clear to scientific minds that the universe is very little like this. Various natural processes, even though they could be perfectly described using mathematics, were found to display a behaviour that was, on all accounts, impossible to forecast. In time, these and other baffling discoveries led us to realise that unpredictability is an intrinsic property of the universe; not only this, but it is this very property that grants inanimate matter the dazzling ability to spontaneously organise itself into the complex shapes and structures of the natural world. The uncanny force responsible for the unpredictability of the world received what is, in fact, a very popular name — chaos.
Although the term ‘chaos’ commonly serves as synonym for disorder or mayhem, its mathematical meaning is more specific. In a system that can be completely described by deterministic mathematical equations, without any unknown or random component, chaos is the property that makes this system capable of behaving unpredictably. In order words, even if we know the state of the system at a given moment, and the equations that describe the system’s evolution, it is impossible for us to predict its future behaviour.
One of the first to describe a system with chaotic behaviour was the meteorologist Edward Lorenz, who in the early 1960s was trying to model the weather using mathematics. The dominant view at the time was that the weather was a deterministic phenomenon, and thus could be forecasted using equations. But once Lorenz had written down a set of equations that captured the dynamics of air masses, he found that these did not yield any useful predictions. Actually, his system appeared to suffer from extreme sensitivity to even the slightest change in its starting conditions; these variations, initially tiny, rapidly amplified across the system as this evolved, causing it to deviate from its predicted course, and thus leading to completely unexpected outcomes.
Lorenz presented his findings in a talk which he titled: ‘Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?’. This concept would rapidly capture the public’s imagination, giving birth to the popular expression ‘the butterfly effect’. What Lorenz’s results imply is that, even if we should, in theory, be able to forecast the weather by measuring a set of variables (such as atmospheric pressure, temperature, humidity, wind speed, etc.) and solving some equations that describe the evolution of atmospheric conditions, these equations are so sensitive to even infinitesimal changes in their initial values, that we cannot possibly measure the variables we need with such accuracy as to be able to reliably predict the weather beyond a couple of days from now. It turns out that the butterfly effect, that is, a high sensitivity to the initial conditions, is actually a hallmark of chaotic systems.
The butterfly effect is easier to understand thanks to a simpler phenomenon than the weather. In the 1970s, biologist Robert May was working on an equation to model the changes in animal populations over generations. This is known today as the logistic equation, and is indeed very simple. Given a value, Current Size, that represents the current size of a population in relation to its maximum possible size (for example, a value of 0.5 means that the population has half the maximum size), using the logistic equation, we can easily find out the size of the population in its next generation, Future Size:
Future Size = 3.7 × Current Size × (1 – Current Size).
The 3.7 value in the equation above is arbitrary, and as suitable as almost any other value between 3.6 and 4. Given this equation, if the current population size, Current Size, were, for instance, 0.27 (27% of the maximum size), then the size in the next generation would be:
The logistic equation is as simple as it looks, and yet it has a property which is shared between all chaotic systems: feedback. In other words, the equation’s result — in this case, 0.729 — is fed back into the equation, since this value will be the new Current Size when we try to determine the population size in the following generation (the next Future Size). Because the population size in each generation depends on the size in the previous generation, it is easy to see how even the smallest variation in our initial value will grow larger as we solve the equation for more and more generations. If the initial value of the system is a real amount that we need to measure, this implies that we can never measure it accurately enough as to be able to predict its future values indefinitely. But the more precision we achieve in our measurements, the longer we will reliably predict the system’s behaviour.
To show the logistic equation’s chaotic behaviour, let us use again the value 0.27 as our initial value for Current Size. We can use this value to calculate Future Size, which will then become the new Current Size, and repeat this process for many generations, making use of the logistic equation each time. Now, imagine that we did not measure the initial population size with absolute accuracy, and that the actual initial size was not 0.27, but 0.270001 (this represents a change of just 0.00037% in the initial value). In this case, it turns out that with our ‘inaccurate’ initial value of 0.27, we will only be able to predict the future population size for twenty-three generations, and no further. Beyond the twenty-third generation, the system will no longer abide by our predictions, and so we say that it behaves chaotically after this point.
Chaos is not a rare phenomenon at all; it actually crops up everywhere, from climate to living systems to the stock market. The world is inevitably shrouded in unpredictability; on the other hand, the fact that chaos is so embedded in the fabric of the universe is what makes nature capable of spawning the amazing designs we see around us, from the shapes of clouds to the structure of our circulatory system. For, as a visionary mathematician named Benoît Mandelbrot discovered in the seventies, chaos is at the heart of a special kind of geometry that can be used to describe the rough and irregular shapes of nature. Mandelbrot realised that Euclidean geometry, which is concerned with perfect shapes, such as lines, triangles and spheres, is not able to explain the physical world around us; for neither are the mountains triangles, nor are the clouds spheres. Nature seems to have a preference for characteristically rough, ‘imperfect’ structures, and before Mandelbrot, no one knew how to measure and describe that roughness. Mandelbrot’s new geometry, fractal geometry, was one of the greatest mathematical revolutions of the twentieth century. Mandelbrot realised that there is a property common to almost all the shapes of nature, something called self-similarity. This can be described as the property by which an object is composed of parts that look themselves like small versions of the whole object. The closer we look at mountains, trees, clouds and sea waves, the more detail we see, and this new detail always repeats a similar geometric pattern. A tree branch resembles a small tree, just as a rock resembles a small mountain, depending only on how closely we observe them. Amazingly, the pattern of progressively smaller branches adopted by plants is also found in the structure of our blood vessels, our nerves and our lungs — just to mention some.
Mandelbrot discovered that this kind of irregular, self-similar forms, which he christened fractals, are described by simple mathematical equations that have the property of feedback, just like May’s logistic equation and Lorenz’s atmospheric model. This sparked an incredible breakthrough: the realisation that chaos is the force behind nature’s amazing ability to self-organise into the multitude of complicated structures and patterns we see in the world. Chaos is the property that empowers simple mathematical rules to spontaneously give rise to unimaginably complex systems. Our intuition of complexity as something that cannot suddenly arise from something much simpler, but that necessarily implies a process of complex, even conscious, design, needs to be reappraised. For nature is, at the same time, marvellously complicated and marvellously simple.
Future Size = 3.7 × 0.27 × (1 – 0.27) = 3.7 × 0.27 × 0.73 ≈ 0.729.
The logistic equation is as simple as it looks, and yet it has a property which is shared between all chaotic systems: feedback. In other words, the equation’s result — in this case, 0.729 — is fed back into the equation, since this value will be the new Current Size when we try to determine the population size in the following generation (the next Future Size). Because the population size in each generation depends on the size in the previous generation, it is easy to see how even the smallest variation in our initial value will grow larger as we solve the equation for more and more generations. If the initial value of the system is a real amount that we need to measure, this implies that we can never measure it accurately enough as to be able to predict its future values indefinitely. But the more precision we achieve in our measurements, the longer we will reliably predict the system’s behaviour.
To show the logistic equation’s chaotic behaviour, let us use again the value 0.27 as our initial value for Current Size. We can use this value to calculate Future Size, which will then become the new Current Size, and repeat this process for many generations, making use of the logistic equation each time. Now, imagine that we did not measure the initial population size with absolute accuracy, and that the actual initial size was not 0.27, but 0.270001 (this represents a change of just 0.00037% in the initial value). In this case, it turns out that with our ‘inaccurate’ initial value of 0.27, we will only be able to predict the future population size for twenty-three generations, and no further. Beyond the twenty-third generation, the system will no longer abide by our predictions, and so we say that it behaves chaotically after this point.
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| Graph representing the evolution of the size of a population, as described by the logistic equation, for 50 generations and initial values of 0.27 (upper, in blue) and 0.270001 (lower, in red). The system follows the same evolution for the first 23 generations; after this point (discontinuous line), the system displays a different behaviour in each case. Therefore, with an initial value of 0.27 it is impossible to predict the system if the true initial value is not exactly 0.27. This is known as chaotic behaviour. |
Chaos is not a rare phenomenon at all; it actually crops up everywhere, from climate to living systems to the stock market. The world is inevitably shrouded in unpredictability; on the other hand, the fact that chaos is so embedded in the fabric of the universe is what makes nature capable of spawning the amazing designs we see around us, from the shapes of clouds to the structure of our circulatory system. For, as a visionary mathematician named Benoît Mandelbrot discovered in the seventies, chaos is at the heart of a special kind of geometry that can be used to describe the rough and irregular shapes of nature. Mandelbrot realised that Euclidean geometry, which is concerned with perfect shapes, such as lines, triangles and spheres, is not able to explain the physical world around us; for neither are the mountains triangles, nor are the clouds spheres. Nature seems to have a preference for characteristically rough, ‘imperfect’ structures, and before Mandelbrot, no one knew how to measure and describe that roughness. Mandelbrot’s new geometry, fractal geometry, was one of the greatest mathematical revolutions of the twentieth century. Mandelbrot realised that there is a property common to almost all the shapes of nature, something called self-similarity. This can be described as the property by which an object is composed of parts that look themselves like small versions of the whole object. The closer we look at mountains, trees, clouds and sea waves, the more detail we see, and this new detail always repeats a similar geometric pattern. A tree branch resembles a small tree, just as a rock resembles a small mountain, depending only on how closely we observe them. Amazingly, the pattern of progressively smaller branches adopted by plants is also found in the structure of our blood vessels, our nerves and our lungs — just to mention some.
Mandelbrot discovered that this kind of irregular, self-similar forms, which he christened fractals, are described by simple mathematical equations that have the property of feedback, just like May’s logistic equation and Lorenz’s atmospheric model. This sparked an incredible breakthrough: the realisation that chaos is the force behind nature’s amazing ability to self-organise into the multitude of complicated structures and patterns we see in the world. Chaos is the property that empowers simple mathematical rules to spontaneously give rise to unimaginably complex systems. Our intuition of complexity as something that cannot suddenly arise from something much simpler, but that necessarily implies a process of complex, even conscious, design, needs to be reappraised. For nature is, at the same time, marvellously complicated and marvellously simple.
References:
The Secret Life of Chaos. BBC (2010).
Butterflies, Chaos and Fractals. Lecture by Prof. Raymond Flood, Gresham College (2013).
Benoît Mandelbrot. The Fractal Geometry of Nature (Henry Holt & Co., 1982).
The Secret Life of Chaos. BBC (2010).
Butterflies, Chaos and Fractals. Lecture by Prof. Raymond Flood, Gresham College (2013).
Benoît Mandelbrot. The Fractal Geometry of Nature (Henry Holt & Co., 1982).
