This isn’t to say that it’s impossible to fake a Pollock—in fact, Taylor has created a piece of equipment he calls the Pollockizer, which paints genuine fractal paintings. Pots containing paint are attached by strings to an electromagnetic coil, which can be programmed to produce chaotic motion, resulting in convincing Pollocks. So although math can help to detect fakes, it can also be used to create images that could well convince the experts.
Fractals are certainly weird shapes if they have dimensions that are not whole numbers—like 1.26 or 1.72—but at least we can draw pictures of them. But things are about to get stranger, because our next step is into hyperspace to explore shapes that exist beyond our three-dimensional world.
HOW TO SEE IN FOUR DIMENSIONS
I can still remember the excitement I felt the day I first “saw” in four dimensions by learning the language that allowed me to conjure up these shapes in my mind’s eye. Seeing in four dimensions is possible by using a dictionary, invented by René Descartes, which changes shapes into numbers. He realized that the visual world was often very hard to pin down and wanted a neat mathematical way that would help.
This puzzle shows you that you can’t always trust your eyes. As Descartes used to say, “Sense perceptions are sense deceptions”:
Figure 2.31 Rearrange the pieces and the area appears to decrease by one unit.
Although the second picture is simply the shapes in the first picture rearranged, the total area seems to have been reduced by one block. How can this be? It’s because although the hypotenuses of the two small triangles look as if they line up, in fact, they are at slightly different angles—just enough that when you rearrange them, you appear to lose a unit of area.
To deal with this problem of perception, Descartes created a powerful dictionary that translates geometry into numbers, and we are now very familiar with it. When we look up the location of a town in an atlas, we find that it’s identified by a two-number grid location. These numbers pinpoint our north–south, east–west location from a point on the equator that lies directly south of Greenwich in London.
For example, Descartes was born in a town in France called . . . Descartes (though when he was born there, it was called La Haye en Touraine), which is at latitude 47° north, longitude 0.7° east. In Descartes’s dictionary, his hometown can be described by two coordinates: (0.7, 47).
We can use a similar process to describe mathematical shapes. For example, if I want to describe a square in terms of Descartes’s dictionary of coordinates, I can say that it is a shape with four vertices located at the points (0, 0), (1, 0), (0, 1), and (1, 1). Each edge corresponds to choosing two vertices where the coordinates differ in one position. For example, one of the edges corresponds to the coordinates (0, 1) and (1, 1).
The flat two-dimensional world needs just two coordinates to locate each position, but if we also want to include our height above sea level, then we could add a third coordinate. We will need this third coordinate, too, if we want to describe a three-dimensional cube in terms of coordinates. A cube’s eight vertices can be described by the coordinates (0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 0), (1, 0, 1), (0, 1, 1), and finally the point farthest from the first corner, located at (1, 1, 1).
Again, an edge consists of two points whose coordinates differ in exactly one position. If you look at a cube, you can easily count how many edges there are. But if you didn’t have this picture, you could just count how many pairs of points there are that differ in one coordinate. Keep this in mind as we move to a shape for which we don’t have a picture.
Descartes’s dictionary has shapes and geometry on one side and numbers and coordinates on the other. The problem is that the visual side runs out if we try to go beyond three-dimensional shapes, since there isn’t a fourth physical dimension in which we can see higher-dimensional shapes. The beauty of Descartes’s dictionary is that the other side of the dictionary just keeps going. To describe a four-dimensional object, we just add a fourth coordinate that will keep track of how far we are moving in this new direction. So although I can never physically build a four-dimensional cube, by using numbers, I can still describe it precisely. It has 16 vertices, starting at (0, 0, 0, 0), extending to points at (1, 0, 0, 0) and (0, 1, 0, 0), and stretching out to the farthest point at (1, 1, 1, 1). The numbers are a code to describe the shape, and I can use this code to explore the shape without ever having to physically see it.
For example, how many edges does this four-dimensional cube have? An edge corresponds to two points in which one of the coordinates is different. Meeting at each vertex there are four edges, corresponding to changing each of the four coordinates one at a time. So that gives us 16 × 4 edges—or does it? No, because we’ve counted each edge twice: once as an edge emerging from the vertex at one of its ends, and again as an edge emerging from the point at its other end. So the total number of edges in the four-dimensional cube is 16 × = 32. And it doesn’t stop there. You can move into five, six, or even more dimensions and build hypercubes in all these worlds. For example, a hypercube in N dimensions will have 2N vertices. From each of these vertices, there will be N edges emerging, each of which I am counting twice, so the N -dimensional cube has N × 2N – 1 edges.
The math gives you a sixth sense, allowing you to play with these shapes that live beyond the bounds of our three-dimensional universe.
WHERE IN PARIS CAN YOU SEE A FOUR-DIMENSIONAL CUBE?
To celebrate the two hundredth anniversary of the French Revolution, the then president of France, François Mitterrand, commissioned the Danish architect Johann Otto von Spreckelsen to build something special in La Défense, the financial district of Paris. The building would line up with several other significant Paris buildings—the Louvre, the Arc de Triomphe, and Cleopatra’s Needle—in what has become known as the Mitterrand perspective.
The architect certainly didn’t disappoint. He built a huge arch, called La Grande Arche, which is so large that the towers of Notre Dame could pass through the middle, and weighs a staggering three hundred thousand tonnes. Unfortunately, von Spreckelsen died two years before the arch was completed. It has become an iconic building in Paris, but perhaps less well-known to the Parisians who see it every day is that what von Spreckelsen actually built is a four-dimensional cube in the heart of their capital.
Well, it isn’t quite a four-dimensional cube, because we live in a three-dimensional universe. But just as the Renaissance artists were faced with the challenge of painting three-dimensional shapes on a flat two-dimensional canvas, so the architect at La Défense has captured a shadow of the four-dimensional cube in our three-dimensional universe. To create the illusion of seeing a three-dimensional cube while looking at a two-dimensional canvas, an artist might draw a square inside a larger square and then join the corners of the squares to complete the picture of the cube. Of course, it’s not really a cube, but it presents the viewer with enough information: we can see all the edges and visualize a cube. Von Spreckelsen used the same idea to build a projection of a four-dimensional cube in three-dimensional Paris, consisting of a small cube sitting inside a larger cube with edges joining the vertices of the smaller and larger cubes. If you visit La Grande Arche and count carefully, you can see the 32 edges that we identified in the previous section using Descartes’s coordinates.
Whenever I visit La Grande Arche at La Défense, it is uncanny how there is always a howling wind that seems to suck you through the center of the arch. So serious has this wind become that the designers have had to erect a canopy at the heart of the arch to disrupt the flow of air. It’s almost as if constructing a shadow of a hypercube in Paris has opened up a portal to another dimension.
There are other ways to get a feel for the four-dimensional cube in our three-dimensional world. Think of how you would make a three-dimensional cube from a piece of two-dimensional card stock. First, you would draw six squares connected in a cross-shape—one square for each face of the cube. Then, you would wrap the cross-shap
e up to form a cube. The two-dimensional card stock is called the “net” of the three-dimensional shape. In a similar fashion, it is possible in our three-dimensional world to build a three-dimensional net, which, if you had a fourth dimension, could be wrapped up to make a four-dimensional cube.
You could set about making a four-dimensional cube by cutting out and assembling eight cubes. These will be the “faces” of your four-dimensional cube. To make the net of the four-dimensional cube, you need to join these eight cubes together. Start by gluing together the first four cubes into a column, one stacked upon the other. Next, take the remaining four cubes and stick them to the faces of one of the four cubes in the column. Your unwrapped hypercube should now look like two intersecting crosses, as shown in Figure 2.32.
Figure 2.32 How to make a four-dimensional cube from eight three-dimensional cubes.
To fold this thing up, you would need to start by joining the bottom and top cubes in the column. The next step would be to join the outward-facing squares of two of the cubes stuck on opposite sides of the column to the bottom cube in the column. Then finally, you’d need to glue the faces of the other two side-cubes to the remaining two faces of the bottom cube. The problem is, of course, that as soon as you start to fold this thing together, you get into a tangle, as there just isn’t enough room in our three-dimensional world. You need a fourth dimension in which to wrap it up as I have described.
Just as the architect in Paris was inspired by the shadow of the four-dimensional cube, so the artist Salvador Dalí was intrigued by the idea of this unwrapped hypercube. In his painting Crucifixion (Corpus Hypercubus), Dalí depicts Christ crucified on the three-dimensional net of a four-dimensional cube. For Dalí, the idea of the fourth dimension as something beyond our material world resonated with the spiritual world beyond our physical universe. His unwrapped hypercube consists of two intersecting crosses, and the picture suggests that Christ’s ascension to heaven is connected with trying to wrap this three-dimensional structure into a fourth dimension, transcending physical reality.
However we try to depict these four-dimensional shapes in our three-dimensional universe, they can never give a complete picture, just as a shadow or silhouette in the two-dimensional world can give only partial information. As we move and turn the object, the shadow changes, but we never see everything. This theme was picked up by novelist Alex Garland in his book The Tesseract, which is another name for a four-dimensional cube. The narrative describes different characters’ views of the central story set in the gangster underworld of Manila. No single narrative provides a complete picture, but by piecing together all the strands, like looking at the many different shadows cast by a shape, you start to understand what the story might be. But the fourth dimension is not just important for constructing buildings, paintings, and narratives. It might also be the key to the shape of the universe itself.
IN THE COMPUTER GAME ASTEROIDS, WHAT SHAPE IS THE UNIVERSE?
In 1979, the computer arcade company Atari released its most popular game, Asteroids. The object of the game was to shoot and destroy asteroids and flying saucers while trying to avoid colliding with passing asteroids or being shot by the saucers’ counterfire. The arcade version was so successful in the United States that many arcades had to fit bigger cash boxes to hold all the quarters that were being fed into the machines.
But it is the geometry of the game that is interesting from a mathematical point of view: as soon as the spaceship flies off the top of the screen, it magically reappears at the bottom. Similarly, if you exit the screen on the left, the spacecraft reappears, entering on the right of the screen. What is happening is that our spaceman is stuck in a two-dimensional world in which the entire universe can be seen on the screen. Although this is a finite universe, it has no boundaries. Because the spaceman never hits an edge, he isn’t living in a rectangle, but is flying around in a more interesting universe. Can we work out what shape his universe is?
If the spaceman exits the screen at the top and reenters at the bottom, then these bits of his universe must be connected. Imagine that the computer screen is made out of flexible rubber, so that we can bend it around and join the top to the bottom. As the spaceman flies vertically, we can now see that he is actually just travelling around and around a cylinder.
What about the other direction? When he exits the screen at the left, he enters again at the right, so the two ends of the cylinder must also be connected. If we mark the points where they are connected, we find that we must bend the cylinder around and join its top to its bottom. So actually, our spaceman is living on a bagel, or what we mathematicians call a torus.
What I’ve illustrated with this piece of rubber is actually a new way in which mathematicians started to look at shapes about a hundred years ago. For the ancient Greeks, geometry (a word that comes from the Greek and means literally “measuring the earth”) was about calculating distances between points and angles. But analyzing the shape of the spaceman’s universe in the game of Asteroids is not so much about the actual distances in our spaceman’s universe but about how it is all connected. This new way of looking at shapes, in which I’m allowed to push and pull them around as if they were made from rubber or Plasticine, is called topology.
Many people use topological maps every day. Although geometric maps are geographically accurate, they’re not very good for finding your way around. Londoners, for instance, use a topological map for finding their way around the Underground. This topological map was first designed in 1933 by Harry Beck; he pushed and pulled the geometric map of London to get something that was much more user-friendly and is now familiar around the world.
Understanding whether a knot can be untangled is also a question of topology, because we are allowed to pull the ropes around but not cut them. This is of fundamental importance to biologists and chemists because human DNA tends to fold up into strange knots. Some diseases, such as Alzheimer’s, might be related to the way DNA knots itself, and math has the potential to unlock these mysteries.
At the beginning of the twentieth century, the French mathematician Henri Poincaré began to wonder how many topologically different surfaces there are. This is like looking at all the possible shapes that our two-dimensional Atari spaceman might be able to inhabit. Poincaré was interested in these universes from a topological perspective, so two universes should be regarded as the same if one universe can be morphed into another continuously and without making any cuts. For example, the two-dimensional surface of a sphere is topologically the same as the two-dimensional surface of a rugby ball because one can be molded into the other. But this spherical universe is a different topological shape to the torus in which Atari’s spaceman is flying around, because you can’t morph a sphere into a doughnut without cutting or gluing the shape. But what other shapes are out there?
Figure 2.33 The first four shapes in Henri Poincaré’s topological classification of how to wrap up two-dimensional surfaces.
Poincaré was able to prove that, however complicated a shape might be, it is always possible to continuously morph it into one of the following shapes: a sphere or a torus with one hole, two holes, three holes, or any finite number of holes. From a topological point of view, this is a complete list of possible universes for our Atari spaceman. It is the number of holes—what mathematicians refer to as the genus—that characterizes the shape. For example, a teacup is topologically identical to a bagel because both have one hole. A teapot has two holes—one in the spout and one in the handle—and can be molded to look like a pretzel with two holes in it. It is perhaps more challenging to understand why the shape in the following figure, which also has two holes, can be morphed into the two-holed pretzel. With the bagels interlocked, it looks like you’d have to cut the shape to morph it successfully, but you don’t. At the end of the chapter, I explain how to undo the rings without any cutting.
Figure 2.34 How can you undo the two interlocked rings by continuously morphing them and without cutting them?
HOW CAN WE TELL THAT WE’RE NOT LIVING ON A BAGEL-SHAPED PLANET?
In ancient times, it was assumed that the earth was flat. But as soon as people started to travel great distances, the question of the large-scale shape of the earth became more important. If the world were flat, then, it was agreed, if you travelled far enough you would fall off the edge—unless you never reached the edge because the world went on forever.
Many cultures began to realize that the earth was most probably curved and finite. The most obvious proposal for the shape is, of course, a ball, and several ancient mathematicians gave incredibly accurate calculations for the size of this ball based only on analyzing how shadows changed throughout the day. But how could scientists be so sure that the surface of the earth wasn’t wrapped up in some other interesting shape? How could they tell that we weren’t living on, for example, the surface of some huge bagel, rather like the Asteroids spaceman stuck in his two-dimensional, bagel-shaped universe?
One way is to go on an imaginary journey in these alternative worlds. So let’s set down an explorer on the surface of a planet and tell him that it is either a perfect sphere or a perfect bagel shape. How can he distinguish between the two? We get him to head off in a straight line across the surface of the planet with a brush and a bucket of white paint, which he uses to mark out his path. Eventually, the explorer will return to where he started, having traced his path as a huge white circle round the planet.
We now give him a bucket of black paint and tell him to head off in another direction. On a spherical earth, for whatever direction he chooses, the black path will always cross the white path before he gets back to his starting point. Remember that the explorer is always travelling in a straight line on the surface. The point where the two paths cross will be at the “pole” opposite the point where the explorer started.
The Number Mysteries: A Mathematical Odyssey through Everyday Life Page 9