You can see footage of Roberto Carlos’s free kick at www.youtube.com/watch?v=Pb2qykj6_ZU&feature=fvst, or use your smartphone to scan this code.
Carlos’s strike, far from defying physics, took advantage of the science of moving soccer balls. The effect of spin on a soccer ball can cause it to do some crazy things. Kick a ball without giving it any spin, and it will travel as if it’s moving across a fixed two-dimensional sheet of paper, tracing out a parabola. But put some spin on the ball, and suddenly the mathematics of its motion becomes three-dimensional. As well as moving up and down, it can also swing left or right.
So what is pushing the ball to the left or right as it flies through the air? It’s a force called the Magnus effect, named after the German mathematician Heinrich Magnus, who in 1852 was the first to explain the effect of spin on a ball. (The Germans have always been good at soccer.) It’s similar to how lift on an airplane’s wing is created. As I explained on page 222, the different speeds of the air flowing over and under the wing cause a decrease in pressure above the wing and a higher pressure below it, producing a force that pushes the wing up.
To get the ball to swing from right to left, Carlos kicked it so that its left side spun toward him (spinning around a vertical axis through the center of the ball). The ball’s spin was then effectively helping to push the air past it more quickly on the left side. That caused the air on the left side to travel faster past the ball, decreasing the pressure—the same as happens on the top of an airplane’s wing. The pressure on the right side of the ball went up because the air speed there was decreasing as the surface of the ball was spun into the path of the air flowing past it. The increased pressure translated into a force pushing the ball from right to left, which eventually took the ball into the back of the net.
The same principle is used to make a golf ball travel farther than is predicted by the equations formulated by Galileo. This time, the axis of spin is horizontal and perpendicular to the motion of the ball. When the ball is driven off the tee, the head of the club spins the ball so that the bottom of the ball is spinning in the direction of flight. That reduces the airflow speed and, by the Bernoulli effect, increases the pressure below the ball, which creates an upward force on the ball that counteracts gravity. In fact, the ball is almost weightless as it flies through the air, as if the spin is giving it a helping hand carrying it down the fairway.
There’s one extra ingredient that we haven’t included, and it explains why Carlos’s free kick swung to the left so late: the drag on the ball. As with the ups and downs of the lemming populations, the secret to Carlos’s magic turns out to be a transition from chaotic to regular behavior. The airflow behind a soccer ball can be either chaotic or regular. The chaotic airflow is called turbulence and happens when the ball is travelling very fast. The regular airflow is called laminar flow and happens at slower speeds. Where the switch from one to the other kicks in depends on the type of ball.
Figure 5.13 Chaotic turbulence causes less drag than regular “laminar” flow.
You can see the different sorts of airflow caused by different wind speeds quite easily. Walk in a straight line holding a flag (or a strip of fabric) so that it trails behind you, and watch it float along. Now do the same thing at a much faster speed, either by holding the flag out of a car window or by running as fast as you can into a strong wind. It will now be flapping around wildly. The reason is that air going around an object such as a flag behaves differently at different speeds. At lower speeds, the airflow is easily predictable, but at higher speeds, it’s much more chaotic.
What effect does this change from turbulence to laminar flow have on taking a free kick? It turns out that chaotic turbulence causes much less drag on the ball. So when the ball is moving quickly, the spin doesn’t have such a great effect on its direction, and the force of the spin is spread out over a larger part of the trajectory. When the ball slows and passes the transition point, the turbulence gives way to laminar flow, which causes much more drag. It’s like someone slamming on the brakes. In that moment of transition, the air resistance increases by 150 percent. Now the effect of the spin can kick in, and the ball suddenly swerves much more dramatically. The extra drag also increases the lift, causing the Magnus effect to increase and pushing the ball even harder to the side.
Roberto Carlos needed a free kick far enough from the goal so that he could hit it hard enough to get the chaotic turbulence and for there to be time for the ball to slow down and bend before going out of play. When the ball is kicked at about 110 kilometers an hour, the airflow around it is chaotic, but about halfway through its flight, as it slows down, the turbulence changes. The brakes are applied, the ball’s spin takes over, and Barthez is beaten.
It’s not just games of soccer that are affected by this math. The way we travel is affected by chaos, too, particularly in the air. Most people associate the word turbulence with a request to fasten their seatbelt and being tossed to and fro by chaotic air currents. Airplanes travel much faster than a soccer ball, and the chaotic airflow over their wings—turbulent flow—increases the air resistance to the plane’s flight, meaning more fuel has to be burned, at a higher cost.
One study concluded that a 10 percent reduction in turbulent drag could increase an airline’s profit margin by 40 percent. Aeronautical engineers are always looking at ways to change the texture of a wing’s surface to make the airflow less chaotic. One idea is to introduce a row of tiny parallel grooves along the wing, spaced as closely as the grooves on a vinyl record. Another is to cover the wing surface with minuscule tooth-like structures called denticles. Interestingly, the skin of a shark is covered with natural denticles, showing that nature discovered how to overcome fluid resistance long before engineers did.
Although it’s been studied intensely, the turbulence behind a soccer ball or an airplane’s wing is still one of the big mysteries of mathematics. There is some good news: we’ve managed to write down the equations that describe the behavior of air or fluid. The bad news is that nobody knows how to solve them! These equations aren’t important just to the likes of Beckham and Carlos. Weather forecasters need to solve them to predict air currents in the atmosphere, medics need to solve them to understand blood flow through the body, and astrophysicists need to solve them to figure out how stars move around in galaxies. All these things are controlled by the same mathematics. At the moment, forecasters, designers, and others can only use approximations, but because there is chaos hiding behind these equations, a small error can have a big effect on the outcome—so their predictions could be completely wrong.
These equations are called the Navier-Stokes equations, after the two nineteenth-century mathematicians who formulated them. They are not simple. A common representation of them is the following:
If you don’t understand some of the symbols in these equations, don’t worry—not many people do! But for those who know the language of math, these equations hold the key to predicting the future. They are so important that there is a million-dollar prize on offer for the first person to solve them.
The great German physicist Werner Heisenberg, who created quantum physics, once said, “When I meet God, I am going to ask him two questions: Why relativity? And why turbulence? I really believe he will have an answer for the first.”
When Roberto Carlos was asked how he’d discovered the secret to bending balls so dramatically, he replied, “I have practised on my free-kick accuracy since I was a child. I used to stay at least one hour after each training session and practise extra on my free-kick accuracy. It’s like with everything: the more pain and sweat, the more you will get.”
I guess that applies to math as well. The more difficult the problem, the more satisfying it is when you crack it. So if the mathematical going gets tough, just remember the words of Roberto Carlos: “the more pain and sweat, the more you will get.” And when you finally crack one of the big mathematical enigmas of all time, everyone will be thinking just what Barthez was thinking as
he stared at the ball in the back of his net: how on Earth did you do that!
PICTURE CREDITS
CHAPTER ONE
1.1 Soccer Players in Prime Numbers © Joe McLaren
1.2 Cicada Seven-Year Cycle © Joe McLaren
1.3 Cicada Nine-Year Cycle © Joe McLaren Cicada and Predator © Joe McLaren
1.4 Messiaen Score of Quartet for the End of Time © Editions Durand, Paris. Reproduced by arrangement with G. Ricordi, & Co. (London) Ltd., a division of Universal Music Publishing Group
1.5 Arecibo message—reproduced with the kind permission of NASA
1.6 Number 200,201 in Cuneiform © Joe McLaren
1.7 Egyptian counting © Joe McLaren
1.8 Number 71 in Ancient Babylonian © Joe McLaren
1.9 Symbol for Number 10 © Joe McLaren
1.10 Counting beans © Raymond Turvey
1.11 Symbol for 3,607 © Joe McLaren
1.12 Counting hands © Raymond Turvey
1.13 Mayan counting © Joe McLaren
1.14 Hebrew counting © Joe McLaren
1.15 Chinese counting © Joe McLaren
1.16 Bamboo sticks © Raymond Turvey
1.17 Bamboo sticks © Raymond Turvey
1.18 Sieve © Raymond Turvey
1.19 Sieve © Raymond Turvey
1.20 Sieve © Raymond Turvey
1.21 Prime numbers hopscotch © Joe McLaren
1.22 Rabbits © Joe McLaren
1.23 Fibonacci spiral © Joe McLaren
1.24 Rice chessboard © Joe McLaren
1.25 Prime Dice © Joe McLaren
CHAPTER TWO
2.1 Watts’s Tower © Joe McLaren
2.2 Volume sliced © Raymond Turvey
2.3 Soccer Balls © Joe McLaren
2.4 Platonic Solids © Joe McLaren
2.5 Tetrahedron © Raymond Turvey
2.6 Great Rhombicosidodecahedron © Raymond Turvey
2.7 Soccer Balls © Raymond Turvey
2.8 Soccer Ball © Raymond Turvey
2.9 Two partial sphere bubbles © Joe McLaren
2.10 Double bubble © Joe McLaren
2.11 Fused bubbles © Joe McLaren
2.12 Wire frame © Joe McLaren
2.13 Triangle © Joe McLaren
2.14 Octohedron © Raymond Turvey
2.15 Two soap films © Raymond Turvey
2.16 Two shapes packed together © Raymond Turvey
2.17 Rhombic Dodecahedron © Raymond Turvey
2.18 Ball and stick model © Raymond Turvey
2.19 Three maps of Britain © Joe McLaren
2.20 Rope to measure coastline © Raymond Turvey
2.21 Rope to measure coastline © Raymond Turvey
2.22 Rope to measure coastline © Raymond Turvey
2.23 Coastline © Thomas Woolley
2.24 Coastline © Thomas Woolley
2.25 Coastline © Thomas Woolley
2.26 Scottish coastline at three magnifications © Steve Boggs
2.27 Four grids © Thomas Woolley
2.28 Six fractals © Thomas Woolley
2.29 Five maps of the UK © Thomas Woolley
2.30 Fractal dimensions painting © Joe McLaren
2.31 Descartes puzzles © Raymond Turvey
2.32 Four dimensional cube © Joe McLaren
2.33 Sphere, Torus © Raymond Turvey
2.34 Interlocking bagel © Raymond Turvey
2.35 Sphere © Raymond Turvey
2.36 Bagel © Raymond Turvey
2.37 Map of Europe
2.38 Seven shades © Joe McLaren
2.39 Unlocking rings © Raymond Turvey
CHAPTER THREE
3.1 Lizards © Joe McLaren
3.2 Lottery Ticket © Raymond Turvey
3.3 Winning Lottery Ticket © Raymond Turvey
3.4 Tetrahedral Dice © Raymond Turvey
3.5 Platonic Dice © Raymond Turvey
3.6 Icosahedron © Raymond Turvey
3.7 Archimedean Solid © Raymond Turvey
3.8 Dice pyramid © Raymond Turvey
3.9 Chocolate Roulette © Joe McLaren
3.10 Arranged chocolate Roulette © Joe McLaren
3.11 Cake stand © Raymond Turvey
3.12 Dürer’s Magic Square © Joe McLaren
3.13 Bridge connections © Raymond Turvey
3.14 Envelope © Raymond Turvey
3.15 Bridges of Königsberg, eighteenth century © Joe McLaren
3.16 Bridges of Kaliningrad, twenty-first century © Joe McLaren
3.17 Traveling salesman problem © Raymond Turvey
3.18 Dinner party problem © Raymond Turvey
3.19 Country borders © Raymond Turvey
3.20 Minefields © Raymond Turvey
3.21 Minefields © Raymond Turvey
3.22 Loading truck problem © Joe McLaren
3.23 Traveling salesman solution © Raymond Turvey
CHAPTER FOUR
4.1 Babington Code © Joe McLaren
4.2 Enigma Machine © Joe McLaren
4.3 Chappe Machine © Joe McLaren
4.4 Chappe Code © Joe McLaren
4.5 Nelson semaphore © Raymond Turvey
4.6 Semaphore code © Joe McLaren
4.7 Beatles cover © Joe McLaren
4.8 Beatles cover corrected © Joe McLaren
4.9 CND Symbol
4.10 Morse Code Alphabet © Raymond Turvey
4.11 Morse Code © Raymond Turvey
4.12 Hexagram © Raymond Turvey
4.13 Hexagram © Raymond Turvey
4.14 Photograph of Leibniz’s binary calculator © Marcus du Sautoy
4.15 Coldplay cover © Raymond Turvey
4.16 Baudot code © Raymond Turvey
4.17 Clocks © Raymond Turvey
4.18 Elliptical curve © Steve Boggs
4.19 Elliptical curve © Steve Boggs
CHAPTER FIVE
5.1 Elliptical plane © Joe McLaren
5.2 Hand-drawn equations © Marcus du Sautoy
5.3 Boomerang © Raymond Turvey
5.4 Boomerang © Raymond Turvey
5.5 Elliptical orbits © Raymond Turvey
5.6 Pendulum © Raymond Turvey
5.7 Magnetic fields © Joe McLaren
5.8 Lemming © Joe McLaren
5.9 Lemming graph © Raymond Turvey
5.10 Lemming graph © Raymond Turvey
5.11 Lemming graph © Raymond Turvey
5.12 Lemming graph © Raymond Turvey
5.13 Turbulence illustration © Joe McLaren
INDEX
Abbasid dynasty, 161
airplane wing, lift of, 222–3, 247, 249
air traffic control, elliptical curve cryptography and, 207
Alexandria, Egypt, 31–2
algebra, 207–8, 220–1
algorithm, 31, 151, 155
Apollo, 15, 215
Archimedes: Archimedean solids, 63–5, 67, 70, 74, 78, 131–2
geometry, dedication to, 63, 65
proposes sphere as containing smallest surface area containing a fixed volume, 57–8
Arecibo radio telescope, 19
Aristagoras, 158
Aristotle, 215
Arup, 70, 76–7
Asteroids (computer game), 97–8, 101, 104–6
Australia, fractal dimension of coastline, 88–9
Babbage, Charles, 165–6
Babington, Anthony, 164
Babylonia, Ancient, 21–5, 29, 128, 218–20
backgammon, 128–9
Barthez, Fabien, 246, 249, 251
Baudot, Émile, 179, 181–3, 188
Beck, Harry, 98
Beckham, David, 2, 6–8, 16, 217–8, 221, 246, 250
Beethoven, Ludwig von, 176, 178
Berg, Alban, 16
Bernoulli equation, 222, 247
Bessy, Bernard Frénicle de, 143
big bang, 105, 161
binary code, 18–9, 138–9, 179–81
Birch and Swinnerton-Dyer conjecture, 207
Bletchley Park, 169–70
&nb
sp; boomerang, why does it come back?, 221–5, 237
Britain, how long is the coastline of?, 80–5, 88, 89
bubbles, 55, 56–9, 70–7
double-bubble conjecture, 72–5
foam and, 73–7
Water Cube (Beijing Olympic swimming center), within design of, 70, 76–7
why are they perfectly spherical?, 55, 56–9
“butterfly effect,” 234–5
Caesar shifts, 160–1
Caesar, Julius, 7, 160
cake stand game, 140–1
calculus, 58
Campaign for Nuclear Disarmament (CND) symbol, 176
Carlos, Roberto, 8, 246–51
casino, mathematics of, 124–7, 237–8
Caspar, Donald, 69
chaos theory, 230–51
“butterfly effect,” 234–5
coin tossing and, 235–8
discovery of, 230
future of the universe and, 235
in the casino, 237–8
lemming populations and, 238–42
pendulums and, 230–3, 235–6
science of moving
soccer balls and, 246–49
turbulence and, 248–50
weather forecasting and, 234–6
chaotic turbulence, 248–9
Chappe, Claude, 171–2
Chappe, Ignace, 171
China, 28–9
gender and number in, 29
negative number, concept of, 29
prime numbers in, 28–9
chocolate roulette, 136–8
cicada, 17-year cycle of an American species of, 10–11, 15
City of God (St. Augustine), 27–8
Clay, Landon, 1–2
clock calculator, 36
The Number Mysteries: A Mathematical Odyssey through Everyday Life Page 23