to confess my confusion to a friend as loyal as you.) I will write to you at length when I can see things more clearly.
Mittag-Leffler decided he needed to inform the other judges:
Poincaré’s memoir is of such a rare depth and power of invention, it will certainly open up a new scientific era from the point of view of analysis and its consequences for astronomy. But greatly extended explanations will be necessary and at the moment I am asking the distinguished author to enlighten me on several important points.
As Poincaré struggled away he soon saw that he was simply mistaken. Even a small change in the initial conditions could result in wildly different orbits. He couldn’t make the approximation that he’d proposed. His assumption was wrong.
Poincaré telegraphed Mittag-Leffler to break the bad news and tried to stop the paper from being printed. Embarrassed, he wrote:
It may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible.
Mittag-Leffler was ‘extremely perplexed’ to hear the news.
It is not that I doubt that your memoir will be in any case regarded as a work of genius by the majority of geometers and that it will be the departure point for all future efforts in celestial mechanics. Don’t therefore think that I regret the prize … But here is the worst of it. Your letter arrived too late and the memoir has already been distributed.
Mittag-Leffler’s reputation was on the line for not having picked up the error before they’d publicly awarded Poincaré the prize. This was not the way to celebrate his monarch’s birthday! ‘Please don’t say a word of this lamentable story to anyone. I’ll give you all the details tomorrow.’
The next few weeks were spent trying to retrieve the printed copies without raising suspicion. Mittag-Leffler suggested that Poincaré should pay for the printing of the original version. Poincaré, who was mortified, agreed, even though the bill came to over 3500 crowns, 1000 crowns more than the prize he’d originally won.
In an attempt to rectify the situation, Poincaré set about trying to sort out his mistake, to understand where and why he had gone wrong. In 1890, Poincaré wrote a second, extended paper explaining his belief that very small changes could cause an apparently stable system suddenly to fly apart.
What Poincaré discovered, thanks to his error, led to one of the most important mathematical concepts of the last century: chaos. It was a discovery that places huge limits on what we humans can know. I may have written down all the equations for my dice, but what if my dice behaves like the planets in the solar system? According to Poincaré’s discovery, if I make just one small error in recording the starting location of the dice, that error could expand into a large difference in the outcome of the dice by the time it comes to rest on the table. So is the future of my Vegas dice shrouded behind the mathematics of chaos?
The chaotic path mapped out by a single planet orbiting two suns.
If nature were not beautiful it would not be worth knowing, and if nature were not worth knowing, life would not be worth living.
Henri Poincaré
I wasted a lot of time at university playing billiards in our student common room. I could have pretended that it was all part of my research into angles and stuff, but the truth is that I was procrastinating. It was a good way of putting off having to cope with not being able to answer that week’s set of problems. But in fact the billiard table hides a lot of interesting mathematics in its contours. Mathematics that is highly relevant to my desire to understand my dice.
If I shoot a ball round a billiard table and mark its path, then follow that by shooting another ball off in very nearly the same direction, the second ball will trace out a very similar path to the first ball. Poincaré had believed that the same principle applied to the solar system. Fire a planet off in a slightly different direction then the solar system will evolve in a very similar pattern. This is most people’s intuition: if I make a small change in the initial conditions of the planet’s trajectory it won’t alter the course of the planet much. But the solar system seems to be playing a slightly more interesting game of billiards than the one I played as a student.
Rather surprisingly, if I change the shape of the billiard table this intuition turns out to be wrong. For example, fire balls round a billiard table shaped like a stadium with semicircular ends but straight sides and the paths can diverge dramatically even though they started in almost the same direction. This is the signature of chaos: sensitivity to very small changes in the initial conditions.
Two quickly diverging paths taken by a billiard ball round a stadium-shaped billiard table.
So the challenge for me is to determine whether the fall of my dice is predictable, like a conventional game of billiards, or whether the dice is playing a game of chaotic billiards.
THE DEVIL IN THE DECIMALS
Despite Poincaré being credited as the father of chaos, it is striking that this sensitivity of many dynamical systems to small changes was not very well known for decades into the twentieth century. Indeed, it really took the rediscovery of the phenomenon by scientist Edward Lorenz, when he, like Poincaré, thought he’d made some mistake, before the ideas of chaos became more widely known.
While working as a meteorologist at MIT in 1963, Lorenz had been running equations for the change of temperature in a dynamic fluid on his computer when he decided he needed to rerun one of his models for longer. So he took some of the data that had been output earlier in the run and re-entered it, expecting to be able to restart the model from that point.
When he returned from coffee, he discovered to his dismay that the computer hadn’t reproduced the previous data but had generated very quickly a wildly divergent prediction for the change in temperature. At first he couldn’t understand what was happening. If you input the same numbers into an equation, you don’t expect to get a different answer at the other end. It took him a while to realize what was going on: he hadn’t input the same numbers. The computer printout of the data he’d used had only printed the numbers to three decimal places, while it had been calculating using the numbers to six decimal places.
Even though the numbers were actually different, they differed only in the fourth decimal place. You wouldn’t expect it to make that big a difference, but Lorenz was struck by the impact such a small difference in the numbers had on the resulting data. Here are two graphs created using the same equation but where the data that is put into the equations differ very slightly. One graph uses the input data 0.506127. The second graph approximates this to 0.506. Although the graphs start out following similar paths, they very quickly behave completely differently.
The model that Lorenz was running was a simplification of models for the weather that analysed how the flow of air behaves when subjected
