The Number Mysteries: A Mathematical Odyssey through Everyday Life
Page 22
Alas, even with today’s supercomputers, a forecast for the weather two weeks in advance is still not reliable. We can’t know precisely what today’s weather will be, let alone what it will be further ahead. Even the best weather stations have only a limited accuracy. We can never know the exact speed of every particle in the air, the precise temperature at every point in space, or the exact pressure all across the planet—and just a small variation in any of these could produce a wildly different weather forecast. This has given rise to the phrase “the butterfly effect”: a butterfly flapping its wings makes tiny changes in the atmosphere that just possibly could ultimately cause a tornado or a hurricane to form on the other side of the world, wreaking havoc, taking lives, and causing millions of dollars of damage.
For this reason, meteorologists run several weather predictions simultaneously, each starting with a slight variation in the measurements taken from the worldwide networks of weather stations and satellites. Sometimes, all the predictions come up with broadly similar results, and the meteorologists can then be fairly confident that the weather—though technically chaotic—will be stable for the next week or two. But on some runs, the predictions differ completely, and the forecasters know there’s no way they can accurately predict the weather even days ahead.
With our chaotic pendulum swinging between the three magnets, there were regions in the picture predicting the pendulum’s behavior where a small change in the initial position of the pendulum wouldn’t have caused it to end up at a different magnet. And it’s the same with the weather. Think of the large black region in Figure 5.7 as the weather in a desert: it’s always going to be hot there, however hard a butterfly flaps its wings—and similarly for the arctic, which is like the magnet staying in a white region. But the weather for the United Kingdom is like the pendulum starting at a place where the colors in the picture change rapidly with just a small shift in the pendulum’s position.
If we knew the precise positions and speeds of all the particles in the universe, we could predict the future with certainty. The problem is that if you get one of those starting positions even slightly wrong, the future can turn out to be completely different. The universe may behave like clockwork, but we’ll never know the positions of the cogs accurately enough to take advantage of its deterministic nature.
HEADS OR TAILS?
The 1968 European Football Championship was held before penalties were introduced as a way of deciding a drawn match. So when Italy and the Soviet Union were still goalless after extra time in their semifinal, a coin was tossed to decide which of them would go through to the final. It has been universally acknowledged since Roman times that a coin is a fair way to decide a dispute. After all, it’s impossible to tell as it spins through the air how it will land. Or is it?
Theoretically, if you knew precisely the position of the coin, how much it was being spun, and when it would land, you could calculate how the coin would land. But like the weather, wouldn’t a minute change in any one of these factors potentially cause a completely different outcome? Persi Diaconis, a mathematician at Stanford University, California, decided to test whether coin tossing is as unpredictable as we think. If the conditions are the same whenever you toss the coin, then the math will always produce the same outcome. But is the signature of chaos hidden inside the toss of a coin? What if we change those starting conditions very slightly—do the changes get amplified, so that by the time the coin lands it’s impossible to know whether it will be heads or tails?
With help from his engineering friends, Diaconis built a mechanical coin-tossing machine that could replicate the conditions of the coin toss over and over again. Of course, there would be very minor differences from one toss to the next, but would these differences result in wildly different outcomes, like the pendulum swinging between the three magnets? Diaconis found that every time he repeated the experiment with his mechanical coin-tosser, the coin would always land the same way. He then trained himself to be able to flip the coin in an identical way each time so that he could get ten heads in a row. Make sure not to gamble on the toss of a coin with the likes of Persi Diaconis.
But what about average human tossers, who will change the way they launch a coin from one flip to the next? Diaconis wondered whether there might still be some bias. To begin his mathematical analysis, he needed an expert in spinning objects. He knew he had his man when he met Richard Montgomery, whose claim to fame was proving the falling-cat theorem—a theory that explains why a cat dropped from any angle always manages to land on its feet. Together with statistician Susan Holmes, they showed that a spinning coin launched with a flick of the thumb has a bias toward landing with a particular side up.
To convert the theory into actual numbers, they needed to do some careful analysis of how a spinning coin moves through the air. With the help of a high-speed digital camera that shoots ten thousand frames per second, they captured the motion of a coin and fed the data into their theoretical model. What they found may come as a surprise: there is indeed a bias in a true toss of the coin. It’s small: 51 percent of the time, the coin tended to land with the same face upright as when it was spun into the air. The reason seems to relate to the physics of the boomerang or gyroscope. It appears that the spinning coin also precesses like a gyroscope, and so spends slightly more time in the air with the face that was first showing pointing upward. The difference is inconsequential for one throw, but in the long run, it could be very significant.
One organization that definitely cares about the long run is the casino. Their profit depends on long-term probabilities. For every throw of the dice or spin of the roulette wheel, they rely on your failing to predict how the dice or ball will land. But just as with the tossed coin, if you knew precisely the starting positions of the roulette wheel and the ball, and their starting speeds, you could in theory apply Newtonian physics to determine where the ball would land. Start the roulette wheel in exactly the same position and with exactly the same speed, and have the croupier launch the ball in exactly the same way each time, and the ball will land in exactly the same place. The problem here is the same one that Poincaré discovered: even a very small change in the starting positions and speeds of the roulette wheel and ball can have a dramatic effect on where the ball ends up. And it’s the same with dice.
But that doesn’t mean that mathematics can’t help you narrow down where the ball might end up. You can watch the ball spinning around the wheel a few times before you place your bet, so you have the chance to analyze the trajectory of the ball and predict its final destination. Three Eastern Europeans—a Hungarian woman described as “chic and beautiful” and two “elegant” Serbian men—did just that. They used mathematics to make a killing at the roulette wheel at the London Ritz casino in March 2004.
Using a laser scanner hidden inside a mobile phone linked to a computer, they recorded the spin of the roulette wheel relative to the ball over two rotations. The computer worked out a region of six numbers within which it predicted the ball would fall. During the third rotation of the wheel, the gamblers placed their bets. Having reduced their odds of winning from 37:1 to 6:1, the trio placed bets on all six numbers in the section where the ball was predicted to finish. That first night, they netted £100,000. On the second night, they won a staggering £1.2 million. Despite being arrested and kept on police bail for nine months, they were eventually released and allowed to keep their winnings. Legal teams concluded that they had done nothing to tamper with the wheel.
The gamblers realized that although there is chaos in a roulette wheel, a small change in the starting positions of the ball and wheel don’t always lead to huge changes in the outcome. This is what meteorologists rely on when they are predicting the weather. Sometimes when they run their computer models, they find that changing the conditions of the weather today doesn’t have a dramatic effect on the forecast. The gamblers’ computer was doing the same thing, running through thousands of different scenarios to see where the ball might
end up. It couldn’t identify the position precisely, but a region of six numbers was enough to turn the odds in the gamblers’ favor.
You might think from what you’ve read so far that nature divides itself into problems that are simple and predictable (like a ball falling from the top of the Leaning Tower of Pisa) and problems that are chaotic and hard to predict (like the weather). But it’s not so clear-cut. Sometimes what starts out as easily predictable can become chaotic if one small thing changes by just a fraction.
WHO KILLED ALL THE LEMMINGS?
Some years ago, environmentalists noticed that every four years, the numbers of lemmings seem to plummet dramatically. A popular theory was that every few seasons, these Arctic rodents made their way to a tall cliff and threw themselves over the edge, plunging to their deaths on the rocks below. In 1958, Walt Disney Productions’ natural history unit included footage of this mass suicide in its award-winning film White Wilderness. The sequence looked so convincing that the word lemming came to be used for anyone who unquestioningly follows the masses with potentially disastrous consequences. The animals’ behavior even spawned a video game in which players had to save the lemmings from their mindless march toward the cliff’s edge.
To see footage from White Wilderness, go to www.youtube.com/watch?v=xMZlr5Gf9yY or use your smartphone to scan this code.
In the 1980s, it was revealed that the film crew of White Wilderness had faked the whole sequence. According to a Canadian TV documentary, the lemmings, which had been bought specially for the filming, refused to leap over the cliff on cue—so members of the film crew “encouraged” them over the edge. But if it isn’t mass suicide that’s causing these sudden drops in the number of lemmings every four years, then what is the explanation?
Figure 5.8
Yet again, it turns out that mathematics has the answer. A simple equation tells us how many lemmings there will be from one season to the next. We start by assuming that, because of environmental factors such as food supply and predators, there’s a maximum population that can be sustained. We’ll call that N. We’ll say that L is the number of lemmings that survived from the previous season, and that after the births in the new season, the population rises to K lemmings. A proportion of these K lemmings will not survive. The proportion that dies is namely, the number of lemmings in the previous season divided by the maximum population possible. So K × die, leaving
lemmings at the end of this season. To make our calculations simple, let’s say that the maximum population is N = 100.
This equation, although simple, has some surprising outcomes. Let’s start by looking at what happens if the lemming population doubles each spring, so that K = 2L. Of these, 2 L × will die. Let’s suppose that in the first season, there are 30 lemmings. The equation predicts that at the end of the second season, there will be 60 – (60 × ) = 42 lemmings. They carry on increasing in number until by the fourth season, there are 50 of them.
From then onward, the number of lemmings that survive each season remains constant at 50. The surprising thing is that, whatever the original population at the start of the first season, the number of lemmings left at the end of each subsequent season will always eventually home in on half the maximum number, where it will remain. So once you hit 50 lemmings, the number doubles to 100 during the next season, but by the end of the season, 100 × = 50 will have died, leaving a population of 50 lemmings again.
Figure 5.9
If lemmings double their numbers each spring, the population reaches a stable value however many lemmings there are to start with.
What happens if the lemmings are more productive? If the number of lemmings slightly more than triples from one season to the next, the population doesn’t stabilize but instead flips between two values. In one season, the number of lemmings that survive to the end of the season is quite high; in the following season, it falls.
Figure 5.10 If the lemmings triple their numbers in the spring, the population starts to oscillate.
When the lemmings become even more productive, the population starts to fluctuate in a strange way. If the population increases by a factor of 3.5, then the total number of lemmings oscillates between four values, repeating this pattern every four seasons. (The precise factor at which four values first appear is 1 + √6, which is approximately 3.449.) And this is where we find that in one of these four years there can be a significant drop in the number of lemmings, not because of a mass suicide pact but because of the math.
Figure 5.11 When lemming numbers increase in the spring by a factor of 3.5, the population oscillates between four different values.
The really interesting change in the population dynamics happens when the lemmings increase their numbers by a factor of just over 3.5699. Then, their numbers from one season to the next jump around seemingly without any rhyme or reason. Even though the equation that calculates the population is a simple one, it has started to produce chaotic results. Change the initial number of lemmings, and the population dynamics are completely different. Beyond the threshold where the chaos kicks in, 3.5699, it’s almost impossible to predict how the population will vary. The equation controlling the population numbers can start out very predictable, but with just a small change in lemming fecundity, chaos can suddenly erupt.
Figure 5.12 When lemming numbers increase in the spring by a factor of 3.5699 or more, the population variations become chaotic.
How to Play the Fishy Formula Game
This is a game for two players. Download the PDF from the Number Mysteries website and cut out the ten fish and the fish tank. The game explores how the number of fish varies over ten seasons. Each fish corresponds to one season, and there is a box on the side of the fish in which you can keep track of the number of fish in the tank corresponding to that season. The fish tank can sustain a maximum of 12 fish. The fish survive for one year, and during that year, they have a certain number of offspring and then die.
Roll two dice. The number of fish that start in the tank is then the number on the dice minus one (so it is a number between 1 and 11). Call this number N0. The first player chooses a number, K, between 1 and 50. This will determine how many offspring each fish has. If there are N0 fish in the tank to begin with, then during the first year, they give birth to () × N0 fish. The number of fish is therefore multiplied by , a number between 0.1 and 5.
Not all the new fish survive. If there were N fish in the tank at the end of the previous year, then by the end of the next year, the number of fish will be
You must round up or down to get a whole number of fish (4.5 fish is rounded up to 5).
Let the fish tank “run” for ten years. The first player scores the number of fish in the tank at the end of odd-numbered years, and the second player scores the number of fish in the tank at the end of even-numbered years.
Let Ni be the number of fish in year i. So
player 1 scores N1 + N3 + N5 + N7 + N9,
player 2 scores N2 + N4 + N6 + N8 + N10.
By writing on the fish you cut out, you can keep track of the population numbers from one year to the next. If all the fish die at some point, then player 1, who chose the multiplier K, loses automatically.
Here’s an example. The players throw the two dice and score a 4. So there are three fish in the tank at the beginning: N0 = 3. Player 1 chooses K = 20. The number of fish at the end of year 1 is therefore
In year 2, there are
fish, and in year 3, there are
fish. The number of fish has now stabilized because 6 gets repeated when it is put into the formula. So
player 1 scores 5 + 6 + 6 + 6 + 6 = 29 fish,
player 2 scores 6 + 6 + 6 + 6 + 6 = 30 fish,
and player 2 wins. See what happens when you vary the multiplier K.
Because we are rounding numbers up and down, the game doesn’t have the full subtlety of the chaotic model that killed off the lemmings.
For an online tank simulator to accompany this game, visit http://www.rigb.org/christ
maslectures06/50_20.html or use your smartphone to scan this code.
In this version of the game, the number of fish displayed has been rounded up or down, but the fraction of fish is fed into the formula for the next year. For example, if you set K = 27 and N0 = 3,
N1 = 6.075, rounded to 6 fish;
N2 = 8.09873, rounded to 8 fish;
N3 = 7.10895, rounded to 7 fish;
N4 = 7.8233, rounded to 8 fish;
N5 = 7.352, rounded to 7 fish;
N6 = 7.68872, rounded to 8 fish;
N7 = 7.45835, rounded to 7 fish;
N8 = 7.62147, rounded to 8 fish;
N9 = 7.50844, rounded to 8 fish;
N10 = 7.58804, rounded to 8 fish.
Player 1 scores 6 + 7 + 7 + 7 + 8 = 35 fish,
player 2 scores 8 + 8 + 8 + 8 + 8 = 40 fish.
HOW TO BEND IT LIKE BECKHAM OR CURL IT LIKE CARLOS
David Beckham and Roberto Carlos have hit some extraordinary free kicks in their soccer careers, kicks that seem to defy the laws of physics. Perhaps the most amazing was the one Carlos took for Brazil against France in 1997. The free kick was awarded 30 meters from the goal. Most soccer players would just have kicked the ball to a teammate and gotten play under way again. Not Roberto Carlos. He placed the ball on the ground and stepped back, ready to shoot.
The French goalkeeper, Fabien Barthez, lined up a defensive wall, though he couldn’t really have believed that Carlos was going to aim the ball anywhere near his goal. And sure enough, when Carlos ran up and struck the ball, it appeared to be heading well wide of its mark. Spectators to one side of the goal started ducking, expecting the ball to fly into the crowd. Then suddenly, at the last minute, the ball veered to the left and flew into the back of the French goal. Barthez couldn’t believe what he’d just seen. He hadn’t moved an inch. “How on earth did the ball do that?” you could see him thinking.