by John Higgs
The machine was a Royal McBee, a mass of wires and vacuum tubes that was built by the Royal Typewriter Company of New York. It was a machine from the days before microprocessors and it would barely be recognisable as a computer to modern eyes, but it was sufficiently advanced for Lorenz to use it to model a simple weather system. His model did not include elements like rain, mist or mountains, but it was sophisticated enough to track the way the atmosphere moved around a perfectly spherical virtual planet.
Like the real weather, his virtual weather never repeated itself exactly. This was crucial, because if his weather conditions returned to the exact same state they had been in at an earlier point, then they would have started to repeat on a loop. His virtual weather would instantly become predictable in those circumstances, and real weather did not work like that. Yet as the constantly clattering output from his printer showed, his virtual weather did not loop. It was something of a surprise that such an unpredictable system could be recreated through a simple string of equations.
One day Lorenz decided to repeat a particularly interesting part of his weather model. He stopped the simulation and carefully reset all the variables to the state they had been in before the period he wanted to rerun. Then he set it going again and went to get a cup of coffee.
When he returned he found his weather system was doing something completely different to what it had done before. At first he thought he must have typed in one of the numbers wrong, but double-checking revealed that this was not the case. The model had started off mimicking its original run, but then the output had diverged. The difference was only slight to begin with, but it gradually increased until it was behaving in a way entirely unrelated to the original.
He eventually tracked the problem down to a rounding error. His machine held the numbers in its memory to an accuracy of six decimal places, but the numbers on the printout he had used when he reset the model were rounded down to three decimal places. It was the difference between a number such as 5.684219 and the number 5.684. It should not, in theory, have made much of a difference. If those numbers had been used to fire Apollo 11 at the moon, such a small difference would still have been accurate enough to send the spaceship in the correct general direction. Lorenz’s weather was behaving as though the spacecraft had gone nowhere near the moon, and was performing an elaborate orbit around the sun instead.
This insight, that complex systems show sensitive dependence on initial conditions, was described in his 1963 paper Deterministic Nonperiodic Flow. This paper gave birth to a new field of study, commonly known as ‘chaos mathematics’. In complicated systems such as the weather, minute variations in one variable could change the outcome in utterly unpredictable ways. Von Neumann’s desire to master the weather would be, once this was understood, quite out of the question.
Lorenz popularised the idea through use of the phrase ‘the butterfly effect’. If a single butterfly in Brazil decided to flap its wings, he explained, then that could ultimately decide whether a tornado formed in Texas. The butterfly effect does not mean that every flap of an insect’s wings leads to tornados or other natural disasters; the circumstances which generate the potential for a tornado have to be in place. The point of the butterfly effect is that the question of whether that potential manifests or not can be traced back to a minute and seemingly irrelevant change in the system at an earlier point.
The idea of the butterfly effect appears in a 1952 short story by the American science fiction author Ray Bradbury called ‘A Sound of Thunder’. In this story hunters from the future go back in time to hunt dinosaurs, but they must be careful to stay on levitating platforms and only kill animals who were about to die, in order not to affect history. They return to the future to find it changed, and realise that the reason was a crushed butterfly on the sole of one of their boots.
Lorenz was surprised to see such an unpredictable outcome from what was, with all the best will in the world, an unrealistically simple model. He wondered how simple a system could be and still never repeat itself exactly. To his surprise, he discovered unpredictability could be found in a simple waterwheel. This was just a wheel with leaking buckets around its rim, which would turn under the force of gravity when water was poured into the top. It was so simple that it appeared it must be predictable. A mathematician, engineer or physicist would, then, have scoffed at the idea that they wouldn’t be able to predict its future behaviour.
Unlike his computer weather, which used twelve variables, the waterwheel could be modelled with just three: the speed the water runs into the wheel, the speed that the water leaks out of the buckets, and the friction involved in turning the wheel.
Some aspects of the model’s behaviour were indeed simple. If the amount of water that fell into the buckets was not sufficient to overcome friction, or if the water leaked out of the buckets quicker than it poured into them, then the wheel would not turn. If enough water fell into the top buckets to turn the wheel, and most of that water leaked away by the time they reached the bottom, then the wheel would turn regularly and reliably. This is the state you’ll find in a well-built waterwheel on the side of a mill. These two different scenarios – turning nicely and remaining still – are two of the different patterns this simple system could fall into. But a third option, that of unpredictable chaotic movement, was also possible.
If the amount of water pouring into the buckets was increased, then the buckets still had water in them when they reached the bottom. This meant that they were heavier when they went up the other side. The weight of the buckets going back up, in these circumstances, competed with the weight of the buckets coming down. This could mean that the wheel would stop and change direction. If the water continued to flow then the wheel could change directions repeatedly, displaying genuinely chaotic and unpredictable behaviour, and never settle down into a predictable pattern.
The wheel would always be in one of three different states. It had to be either still, turning clockwise or turning anticlockwise. But the way in which it switched from one state to another was unpredictable and chaotic, and the conditions which caused it to change – or not – could be so similar as to appear almost identical. Systems like this are called ‘strange attractors’. The system is attracted to being in a certain state, but the reasons that cause it to flip from one state to another are strange indeed.
Strange attractors exist in systems far more complicated than Lorenz’s three-variable model. The atmosphere of planets is one example. The constantly moving patterns of the earth’s atmosphere are one potential state, but there are others. Early, simple computer models of the earth’s weather would often flip into what was called the ‘snowball earth’ scenario, in which the entire planet was covered in snow. This would cause it to reflect so much sunlight back into space that it could not warm up again. A ‘dead’ planet such as Mars was another possible scenario, as was a hellish boiling inferno like Venus.
Whenever early climate models flipped into these alternative states, the experiment was stopped and the software reset. It was a clear sign that the climate models needed to be improved. The real earth avoids these states, just as a waterwheel on a mill turns steadily and reliably. Our atmosphere is fuelled by the continuous arrival of just enough energy from the sun, just as a functioning waterwheel is fuelled by the right amount of water pouring onto it from a river. It would take a major shift in the underlying variables of the system to cause the atmosphere to flip from one of these states into another. Of course, a major shift in one of the underlying variables of our atmosphere has been happening since the start of the industrial revolution, which is why climate scientists are so worried.
History and politics provide us with many examples of complicated systems suddenly switching from one state into another, for reasons that no one saw coming and which keep academics arguing for centuries – the French Revolution, the fall of the Soviet Union in 1991 and the sudden collapse of the global imperial system around the First World War. Strange attractors all
owed mathematicians, for the first time, to see this process unfurl. It was, they were surprised to realise, not some rare exception to the norm but an integral part of how complex systems behaved. This knowledge was not the blessing it might appear. Seeing how systems flipped from one state to another brought home just how fragile and uncontrollable complex systems were.
Thanks to the butterfly effect, climate modelling has proved to be exponentially more difficult than von Neumann expected. But the need for weather forecasting and longer-range predictions has not gone away, so climate modellers have worked hard. In the half-century since Lorenz first coded a virtual atmosphere, climate models have become massively more detailed and computationally intensive. They have to be run many times in order to obtain the statistical likelihood of their results. As the models improved, they became far less likely to flip into ‘snowball earth’ or other unlikely states. And in doing so they confirmed a finding that had shocked the pioneers of chaos mathematics. Whenever they studied complexity and looked deeply at the frothing, unpredictable turbulence, they found the strangest thing. They found the emergence of order.
The fact that our ecosystem was so complicated was what was keeping it so stable.
Benoît Mandelbrot was a magpie-minded Polish mathematician with a round, kindly face and the type of personality that found everything intensely interesting. He was a Warsaw-born Jew who fled the approaching Nazis as a child, first to France and then to America. In 1958 he joined the IBM Thomas J. Watson Research Center in New York to undertake pure research. This allowed him to follow his curiosity wherever it might take him.
In 1979 he began feeding a short equation into a computer. Like Lorenz’s waterwheel model, Mandelbrot’s equation was incredibly simple. It was little more than a multiplication and an addition, the sort of mathematics that could have been attempted at any point in history. The reason why it hadn’t was because the equation was iterative, and needed to be calculated millions of times. The answer that came out at the end was fed back into the start, at which point the equation was performed again, and again, and again. This was why Mandelbrot needed a computer. Even the shabbiest early computer was happy to run a simple bit of maths as many times as you liked.
Mandelbrot wanted to create a visual representation of his equation, so he performed the same iterative mathematics for every pixel on his computer display. The outcome would be one of two things. Either the number would become increasingly small, and ultimately head towards zero, at which point Mandelbrot would mark that pixel on the screen black. Or the number would become massive and race off towards infinity, in which case that pixel would be coloured. The choice of colour varied with the speed at which that number increased. The set of numbers that created this image became known as the Mandelbrot Set.
The result, after the equation had been applied to the whole screen, was an appealing black splodge in the centre of the screen with coloured, crinkly edges. It wasn’t a circle, exactly, but it was satisfyingly plump. The shape resembled a cross between a ladybird and a peach, or a snowman on its side. It wasn’t a shape that anyone had ever seen before, yet it felt strangely familiar.
It was when he looked closely at the edges of the Mandelbrot Set that things got interesting.
The edges of the shape weren’t smooth. They were wrinkled and unpredictable, and sometimes bulged out to form another near-circle. Zooming in to them should have clarified their shape, but it only revealed more and more detail. The closer you looked, the more you found. There were swirls that looked like elephant trunks, and branching shapes that looked like ferns or leaves. It didn’t matter how closely you dived in, the patterns kept coming. There were even miniature versions of the initial shape hidden deep within itself. But at no point did the patterns repeat themselves exactly. They were always entirely new.
Mandelbrot had discovered infinite complexity hidden in one short equation.
It might have been expected that such complexity should have been entirely random and discordant, but that was not the case. It was aesthetically appealing. Mathematicians are notoriously quick to describe whatever they are working on as ‘beautiful’, but for once they had a point. There was something very natural and harmonious about the imagery. They were nothing like the sort of images then associated with computer graphics. Instead, they resembled the natural world of leaves, rivers or snowflakes.
Mandelbrot coined the word fractal to describe what he had discovered. A fractal, he said, was a shape that revealed details at all scales. An example of this would be the coastline of an island. This will always remain crinkly, regardless of whether you are looking at headlands, or rocks, or individual pebbles on the shore. The smaller the scale, the more detail that emerges.
For this reason, measuring the length of a coastline is an arbitrary exercise entirely dependent on the amount of detail factored into the measurements. The length of the British coastline is 17,820 kilometres, according to the Ordnance Survey, yet the CIA Fact-book reckons that it is 12,429 kilometres, or nearly a third shorter. Those measurements are entirely dependent on the scale from which they were taken. The figure is essentially meaningless without that context. The observed can, once again, only be understood if we include its relation to the observer.
Having found fractals on his computer, Mandelbrot looked again at the real world and realised they were everywhere. He saw them in the shapes of clouds, and in the whirls of rising cigarette smoke. They were in the branching of trees and the shape of leaves. They were in snowflakes, and ice crystals, and the shape of the human lungs. They described the distribution of blood vessels, and the path of a flowing river. At one point Mandelbrot was invited to give a talk at the Littauer Center at Harvard University, and was surprised to turn up and find what looked like one of his diagrams already on the blackboard. The diagram Mandelbrot intended to talk about depicted income variation, which was data in which he discerned fractal patterns. The Harvard diagram already on the board had nothing to do with income variation. It represented eight years of cotton prices. Yet that data had also produced remarkably similar fractal patterns.
Every time Mandelbrot stepped outside his front door and into the fractal landscape of nature, he was confronted by a world that now appeared entirely different to the one imagined by the mathematics of Euclid and Newton. A mountain may roughly be the shape of a pyramid, but only roughly. The classic Euclidian geometric shapes of spheres, cubes, cones and cylinders didn’t actually exist in the natural world. There was no such thing as a straight line until mathematicians invented one. Reality was far messier than it had been given credit for. Like it or not, reality was fractal and chaotic.
Through the insights of people like Lorenz and Mandelbrot, and the arrival of brute computational power, a major shift occurred in our understanding of both mathematics and nature. As the research accumulated, two surprising facts became apparent. When you looked closely at what appeared to be order, you would find outbreaks of chaos around the edges, threatening to break free. And yet when you looked deeply into chaos, you would find the rhythms and patterns of order.
The discovery of order in chaos was of intense interest to biologists. The existence of complex life never really appeared to be in the spirit of the second law of thermodynamics, which stated that, in an isolated system, entropy would increase. Ordered things must fall apart, in other words, so it was odd that evolution kept generating increasingly intricate order. With the arrival of chaos mathematics, biologists now had a key that helped them study the way order spontaneously arose in nature. It helped them understand the natural rhythms of life, both in the internal biology of individual animals and in the larger ecosystems around them.
It was not long before someone applied this insight to the largest ecosystem we have: planet Earth itself, and all the life that exists on it.
In the 1960s the English polymath James Lovelock was working at NASA when they were preparing to launch unmanned probes to Mars. Lovelock’s work was focused on studying t
he Martian atmosphere. It led to him inventing chlorofluorocarbon (CFC) detectors, which later proved invaluable when CFCs were discovered to be the cause of the growing hole in the ozone layer.
The atmosphere of a dead planet should be very different to that of a living planet like Earth, and so analysing the Martian atmosphere would prove to be a useful clue in determining whether or not there was life on Mars. As it turned out, the atmosphere of Mars was very close to a natural chemical equilibrium, being predominantly carbon dioxide with very little of the more interesting gases such as oxygen or methane, and this strongly suggested that Mars was a dead planet.
As Lovelock pondered the atmospheric differences between a living planet and a dead one, he became increasingly intrigued by the processes in which living organisms altered their atmosphere. This could occur in many different ways. For example, an increase in temperature would stimulate the growth of phytoplankton, which are tiny plants that live on the surface of the ocean and which excrete a compound called dimethyl sulphide. Those extra plants produced extra dimethyl sulphide, which entered the atmosphere and made it easier for clouds to form. Those extra clouds then reflected more of the sun’s energy back into space, which had a cooling effect on the climate and helped reduce the amount of phytoplankton back towards their initial quantity. The whole system was a feedback loop, which constantly acted upon itself.
Wherever Lovelock looked, at all manner of chemical, biological, mineral and human processes, he found countless similar examples of feedback loops. Order was spontaneously being generated by chaos. The ecosystems of earth were unwittingly stabilising the very conditions they needed to survive.
