Stranger Than We Can Imagine (29 page)

T
he universe, we used to think, was predictable.

We thought that it worked like a clockwork machine. After God had put it together, wound it up and switched it on, his job was done. He could relax somewhere, or behave in mysterious ways, because the universe would continue under its own steam. The events that occurred inside it would do so under strict natural laws. They would be preordained, in that they would transpire according to the inevitable process of cause and effect. If God were to switch the universe off and reset it back to its original starting state, then switch it back on again, it would repeat itself exactly. Anyone who knew how the universe worked and understood exactly what state it was in at any point would be able to work out what was coming next, and what would come after that, and so on.

That was not an idea that survived the twentieth century.

When Armstrong, Collins and Aldrin climbed into a tin can perched on top of a 111-metre-tall firecracker, it was Newton’s laws which they were trusting to get them to the moon. In all credit to Newton, the laws he discovered over 250 years earlier did the job admirably. Relativity and quantum mechanics may have shown that his laws didn’t work at the scale of the extremely small and the extremely large, but it still worked well for objects in between.

The mathematicians who performed the calculations needed to send Apollo 11 to the moon were aware that the figures they used would never be exact. They might proceed on the understanding that the total mass of the rocket was 2.8 million kilograms, or that the first-stage engines would burn for 150 seconds, or that the distance to the moon was 384,400 kilometres. These figures were accurate enough for their purposes, but they were always rough
approximations. Even if those numbers were only out by a few hundred thousandths of a decimal place, they would still be out. But this wasn’t a problem, because it was possible to compensate for any discrepancies between the maths and the actual voyage as the mission progressed. If the weight of the rocket was underestimated then it would travel a little faster than expected, or if the angle it left orbit at was slightly off then it would head increasingly off-course. Mission control or the astronauts themselves would then adjust their course by a quick blast of their steering rockets, and all would be well. This made complete philosophical sense to mathematicians. If the variables in their equations were slightly out it would affect the outcome, but in ways that were understandable and easily correctable.

That assumption lasted until 1960, when the American mathematician and meteorologist Edward Lorenz got hold of an early computer.

After he failed to convince President Eisenhower to unilaterally launch a nuclear attack on Russia, John von Neumann, the Budapest-born genius who inspired the character of Dr Strangelove, turned his attention to computers.

Von Neumann had a specific use in mind for computer technology. He believed that computer power would allow him to predict the weather, and also to control it. The weather, in his hands, would be a new form of ‘ultimate weapon’ which he would use to bury all of Russia under a new Ice Age. All the evidence suggests that von Neumann really didn’t like Russia.

He became a pioneer in the world of computer architecture and programming. He designed an early computer that first ran in 1952 and which he called, in a possible moment of clarity, the MANIAC (an acronym for Mathematical Analyzer, Numerator, Integrator and Computer). He also designed the world’s first computer virus in 1949. He was that type of guy.

His intentions for weather control beyond Russia were more altruistic. He wanted to trigger global warming by painting the polar ice caps purple. This would reduce the amount of sunlight that the
ice reflected back into space, and hence warm the planet up nicely. Iceland could be as warm as Florida, he decided, which was fortunate because much of Florida itself would have been under water. Von Neumann, of course, didn’t realise this. He just thought that a hotter planet would, on balance, be a welcome and positive thing. This idea was also expressed by the British Secretary of State for the Environment, Owen Paterson, in 2013. Von Neumann’s thinking took place in the years before the discoveries of Edward Lorenz, so in his defence he cannot be said to be as crazy as Paterson.

Von Neumann died in 1957, so he did not live long enough to understand why he was wrong. Like many of the scientists involved in the development of America’s nuclear weapon, he had been scornful of the idea that radiation exposure might be harmful. And also like many of those scientists, he died prematurely from an obscure form of cancer.

At the time, the idea of accurate weather prediction, and ultimately weather control, did not appear unreasonable. Plenty of natural systems were entirely predictable, from the height of the tides to the phases of the moon. These could be calculated with impressive accuracy using a few equations. The weather was more complicated than the tides, so it clearly needed more equations and more data to master it. This was where the new computing machines came in. With a machine to help with the extra maths involved, weather prediction looked like it should be eminently achievable. This was the reason why Edward Lorenz, a Connecticut-born mathematician who became a meteorologist while serving in the US Army Air Corps during the Second World War, sat down at an early computer and began modelling weather.

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 allowed 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.