Figure 2.35 Two paths on a sphere cross at two places.
On the surface of a bagel-shaped planet, things are rather different. The white journey could take him around the inside of the bagel, through the hole, and out the other side. But if on his black journey he set off at 90 degrees to the white path, he’d walk around the hole without going inside it. So it is possible to make two journeys that will meet only at the place you started from.
Figure 2.36 There are paths on a torus that cross only once.
The problem is that the surface of a planet is not generally perfectly spherical or bagel-shaped—it is distorted. Dents in the surface where meteorites have struck will have bent it out of shape, so as the explorer travels in a straight line, hitting one of these dents or bumps would send him off in a new direction. In fact, it’s quite possible that if the explorer heads off in a straight line, he will never return to his starting point. Since the dented shapes are still just distorted versions of the sphere or bagel, perhaps there are other ways to distinguish them. This is where the subject of topology is so powerful, because it’s concerned not so much with the shortest path between two points, but with whether one path can be molded into another.
Let’s now send our explorer off with a white elastic rope, which he lays down behind him. He keeps going until he comes back to the beginning again and then joins the ends of the rope so that he has a noose around the planet. He then heads off in a new direction with a black elastic rope and again keeps going until he returns to his starting point. If the planet is essentially a ball or sphere with a few dips and peaks in it, then without cutting either rope, he can always move the black rope so that it lies completely over the white rope. But on a bagel-shaped planet, this isn’t always possible. If the black rope is wrapped through the inside of the hole of the bagel and the white rope is laid down on a circle going around the outside ring of the bagel, then there is no way to pull the black rope to match up with the white rope without cutting it. So the explorer can tell whether there’s a hole in the planet by making journeys around it and without ever leaving the surface to find out what shape it is.
Here are two other curious ways to tell whether you are on a ball-shaped planet or a bagel-shaped planet. Imagine that both planets are covered in fur. The explorer on the furry bagel will find that he can comb the hair so that it all lies down smoothly—for example, by combing the hair into the hole and back out the other side. But the explorer on the furry ball is going to be in trouble. However hard he tries to comb the hair on the ball-shaped planet, there will always be a crown where the hair sticks up.
Bizarrely, this has a strange implication for the weather on these two planets, as the hairs can be thought of as the direction in which the wind is blowing on these different worlds. On the globe, there is always somewhere where there is no wind blowing—at the crown—but on the bagel, it is possible for the wind to be blowing everywhere on the surface.
Another difference between these two planets is in the maps that can be drawn on them. Divide each planet into different countries, and then try to color the maps so that no two countries with a common border have the same color. On the surface of the spherical earth, you can always get away with just four colors. On a map of Europe, look at the way Luxembourg is boxed in by Germany, France, and Belgium—you can see that you need at least four colors. But the extraordinary thing is that you don’t need any more colors—there is no way to redraw the boundaries of Europe that will force you to invest in a fifth color. However, to prove this was no easy matter. The proof that there really wasn’t some crazy map that would need a fifth color was one of the first in mathematics that had to resort to employing a computer to check several thousand maps—coloring all those maps by hand would have taken too long.
Figure 2.37 Four colors are needed to color a map of Europe.
What about the cartographers living on the bagel-shaped planet—how many buckets of paint are they going to need? There are maps on the bagel-shaped planet that actually need as many as seven colors. Remember from the Asteroids game that we can wrap up the rectangular screen to make a bagel in which the top and bottom are joined to make a cylinder, and then the left- and right-hand sides of the screen that make up the ends of the cylinder are joined to make the bagel. Here’s a map on the surface of the bagel before it is joined up; it needs seven colors once it has been sewn together.
Figure 2.38 Wrap up this map into the shape of a bagel by joining the top and the bottom and then joining the two ends. You’ll find that you need seven colors to color it.
And now, having journeyed through the mathematics of bubbles and bagels, and fractals and foam, we are ready to tackle the ultimate question of the mathematics of shape.
WHAT SHAPE IS OUR UNIVERSE?
This is one question that has obsessed humankind for millennia. The ancient Greeks believed that the universe was bound by a celestial sphere on the inside of which the stars were painted. This sphere would rotate every 24 hours, which explained the movement of the stars. However, there is something rather unsatisfying about this model: if we travelled out into space, would we eventually hit a wall? And if so, what is on the other side of that wall?
Isaac Newton was one of the first to propose that perhaps the universe has no boundary—that it is infinite. But there is a third possibility that the universe is finite yet has no boundary. How is this possible?
This is a similar problem to the one faced by our explorers on a world that has a finite surface area but no edges or boundaries. But instead of being stuck on a two-dimensional surface, we are inside a three-dimensional universe. Is there an elegant way to find the shape of this universe and resolve the apparent paradox of it having no boundary yet still being finite?
It took the invention of four-dimensional geometry in the middle of the nineteenth century for shapes to appear that provided a possible answer. Mathematicians realized that the fourth dimension gave them the room to wrap up our three-dimensional universe to create shapes that are finite in volume yet have no boundaries, just as the two-dimensional surface of the earth or the surface of a bagel is finite in area but has no edges.
We have already seen how a finite two-dimensional universe like the Asteroids universe is actually the surface of a three-dimensional bagel, but we are three-dimensional travellers who can travel in a third dimension. Perhaps the universe we live in behaves the same way as the universe in Asteroids. To start with, imagine freeze-framing the universe just after the big bang, when it has expanded to the size of your bedroom. This bedroom-sized universe is finite in volume, but it doesn’t have any boundaries—because the bedroom is connected together in a rather curious way.
Imagine that you’re standing in the middle of the bedroom facing a wall. (I’m assuming that your bedroom is cube-shaped.) As you walk forward, instead of hitting the wall in front of you, you actually emerge through the wall that was behind you. Similarly, passing through the wall behind you sees you emerge through the wall in front. If you change direction by 90 degrees and head toward the wall on your left, then as you pass through it you emerge through the wall on the right, and vice versa. So the way we have connected up your bedroom is just like in the game of Asteroids.
But we are three-dimensional space travelers, and there’s a third direction we can head in. When we fly up into the ceiling, instead of bouncing off of it we pass through it and find ourselves emerging through the floor. Travelling in the opposite direction will take us out through the floor and back in through the ceiling.
The shape of this universe is actually the surface of a four-dimensional bagel, or hyperbagel, but just like the spaceman who is trapped in the game of Asteroids and can’t get out of his two-dimensional world to see how his universe is wrapped up, we can never see this hyperbagel. But by using the language of mathematics, we can still experience its shape and explore its geometry.
Our universe has now expanded way beyond the size of a bedroom, but it might still be stuck together
like a hyperbagel. Think about the light that travels out in a straight line from the sun. Maybe, rather than disappearing into infinity, this light can loop around and come back and hit the earth. If so, then one of the distant stars out there could be our sun seen from the opposite side, since the light would have travelled all the way around the hyperbagel and finally back to earth. We could therefore be staring at our own sun when it was much younger.
This seems incredible, but just think of sitting in your mini bagel–bedroom universe and striking a match. Looking toward the wall in front, you see the light of the match right in front of you. Now turn around. Looking at the back wall of your bedroom, you will see the match again, only a little farther off because the light from the match heads toward the wall that was in front of you and then reemerges through the back of the bedroom and hits your eye.
Instead of a hyperbagel, we could be living on the surface of a four-dimensional soccer ball. Some astronomers believe that we might be living on a shape that looks like a 12-faced dodecahedron where, as in your mini bedroom universe, when you hit one of the faces of the dodecahedron, you reenter the universe on the opposite face. Perhaps we have come full circle and returned to the model Plato proposed two thousand years ago, according to which our universe was enclosed in some kind of glass dodecahedron with stars stuck on its surface. Maybe modern mathematics can make sense of this model, where the faces of this shape are joined up to make a universe with no glass walls.
But are there other shapes that the universe could be? Remember that Poincaré classified all the possible shapes that a two-dimensional surface, like the surface of our planet, could have. The surface can be wrapped up like a soccer ball; a bagel; or a pretzel with two holes, three holes, or more holes. Poincaré proved that any other shape that you might try to make can be morphed into a ball or a pretzel with holes in.
So what about our three-dimensional universe—what shape could it be? Called the Poincaré conjecture, this is the million-dollar problem for this chapter. It is rather special because in 2002, news emerged that the Russian mathematician Grigori Perelman had solved it. His proof has been checked by many mathematicians, and it is now acknowledged that he has indeed classified all the possible shapes the universe might be. This was the first of the million-dollar problems to be solved, but when Perelman was offered the million dollars in June 2010, he amazingly decided to turn it down. For him, the prize wasn’t money but solving one of the biggest problems in the history of mathematics. He had already turned down a Fields medal, the mathematicians’ equivalent of a Nobel prize. In this age of celebrity and materialism, there is something rather noble about a man who gets his kicks out of solving theorems and not winning prizes.
With the acceptance of Perelman’s proof, mathematicians have sorted out the possible shapes there could be. Now it is up to astronomers to look into the night sky and pin down which one best describes the elusive shape of the universe.
SOLUTIONS
Imagining Shapes
The slice cuts all six faces, and each face contributes an edge to the new face. The shape has to be symmetrical, so you get a hexagon.
Unlinking the Rings
This is how to undo the two interlocked rings by continuously morphing them into a double-holed torus:
Figure 2.39
Three
THE SECRET OF THE WINNING STREAK
Playing games is an essential part of the human experience. Games are a safe way of exploring real-life situations. Monopoly is a microcosm of the economy, chess is an 8 × 8 battlefield, and poker is an exercise in assessing risk. Games allow us to develop ways of predicting how, given certain rules, events will unfold and to plan accordingly. They teach us about chance and unpredictability, which play such essential parts in nature’s game of life.
From ancient civilizations all around the world, we have a fascinating assortment of games: stones thrown in the sand, sticks tossed in the air, tokens placed in hollows carved into wooden blocks, hands used to compete, and pictures drawn on cards. From ancient mancala to Monopoly, from the Japanese game of go to the poker tables of Vegas, games are invariably won by whoever is best at taking a mathematical, analytical approach. In this chapter, I will show you how math is the secret to the winning streak.
HOW TO BECOME THE ROCK-PAPERSCISSORS WORLD CHAMPION
Jan-ken-pon in Japan, ro-sham-bo in California, kai-bai-bo in Korea, and ching-chong-cha in South Africa—the game of rock-paper-scissors is played all around the world.
The rules are very simple: on a count of three, each player makes his or her hand into one of three shapes: a fist for a rock, a flat hand for paper, or two fingers in a V for scissors. Rock beats scissors, scissors beats paper, and paper beats rock. Two of the same is a draw.
Now, the rationale for the first two wins is clear: rock blunts the scissors, scissors cut the paper. But why does paper beat the rock? A sheet of paper isn’t much protection against someone throwing a rock at you. But it may be that this convention goes back to ancient China, in the days when a petition to the emperor was symbolized by a rock. The emperor would indicate whether he’d accepted the petition by placing a piece of paper above or below the rock. If the rock was covered by the paper, the petition was refused and the petitioner defeated.
The origins of this game are hard to trace. There is evidence that it was played in the Far East and by Celtic tribes, and even possibly as far back as the ancient Egyptians, who used to play finger games. However, all these cultures seem to have been beaten to the discovery by a group of lizards that have been playing the game in the fight for survival long before humankind was making fists.
The west coast of the United States is home to a species of lizard called Uta stansburiana, more commonly known as the common side-blotched lizard. The male comes in three different colors—orange, blue, and yellow—and each color has a different mating strategy. Orange lizards are the strongest and will attack and beat blue lizards. The blue lizards are bigger than the yellow lizards and are happy to engage in battle with them and beat them. But though the yellow lizards are smaller than the blue and orange males, they look like female lizards, and that confuses the orange lizards. So the orange lizards, who are looking for a fight, don’t notice the yellow lizards slipping under their gaze and mating with the females. The yellow lizards are sometimes referred to as “sneakers” for the devious way in which they outflank the orange ones. So orange beats blue, blue beats yellow, and yellow beats orange—an evolutionary version of rock-paper-scissors.
Figure 3.1
These lizards have been playing the game in the course of perpetuating their genes for a long time, and it would be interesting to know whether they have developed a strategy for winning. Their population tends to follow a six-year cycle in which first orange dominates, then yellow, then blue, then orange once again. The pattern that emerges is precisely the one that people will use in trying to win the game in one-to-one combat. See too many rocks being thrown and you start to offer paper, but once your opponent sees the run of paper beating the stone, he or she wises up and switches to scissors to cut off your paper run. You soon pick up your opponent’s change of behavior and shift to rock.
At its heart, winning this game is all about spotting patterns, and that’s a very mathematical trait. If you can predict what your opponent is going to do next because of a pattern of behavior he or she has established, then you’re in. The problem is that you don’t want there to be any immediately obvious rhythm in the way you respond, or your opponent will gain the upper hand. A huge amount of psychology is going on as each contestant tries to spot patterns in his or her opponent’s play, each second-guessing what the other might do next.
Rock-paper-scissors has recently grown from a playground game to an international contest. Each year, a prize of $10,000 awaits the winner of the coveted title of rock-paper-scissors world champion. The roll of honor has been dominated by contestants from North America, but in 2006, Bob “The Rock” Cooper fr
om North London held his nerve to take the title. His training for the tournament? “Several hours of hard practising in front of the mirror each day.” I guess this helps to build up the psychology of dealing with your opponent reading your mind. And the secret to his success? His nickname makes other players assume that he’ll throw a rock more often than not, which allows him to cut in with scissors to beat the paper that they try to catch him with. But once they’ve seen through this ruse, Bob “The Rock” uses a more mathematical approach.
From a mathematical, rather than a psychological, point of view, your best strategy is to make your choices random. Your opponent then has nothing at all to go on, because in a truly random sequence, what has gone before will not influence what follows. If I toss a coin ten times, the first nine tosses will have no influence on the outcome of the last toss. Even if you’ve tossed nine heads, that doesn’t mean that the tenth has got to be a tail to balance things out. A coin has no memory.
The strategy of randomizing things gives you only an evens chance of winning, as it makes playing a game of rock-paper-scissors no different from tossing a coin to see who wins. But if I were going up against the world champion, I’d take any strategy that gave me an evens chance of winning. I can’t think of many sports in which you can devise a strategy to give you a fifty-fifty chance of beating the world champion. The one hundred–meter sprint? I don’t think so.
But how do you come up with a sequence of choices that you can be certain is random and doesn’t have some hidden pattern? It’s a real problem: we humans are notoriously bad at producing random sequences—we are so addicted to patterns that we tend to let structure seep into any random sequence we try to put together. To help you win the game, you can download a PDF from the Number Mysteries website containing a rock-paper-scissors die you can assemble and use to help you randomize your choices.
The Number Mysteries: A Mathematical Odyssey through Everyday Life (MacSci) Page 10