Escape From 3-D

When you need to get a clearer view of a complex problem, there's nothing like stretching into 4, 5, 10, or even 26 dimensions. Just ask a mathematician.

By K.C. Cole|Thursday, July 01, 1993
There is a fifth dimension beyond that which is known to man. In the 1960s Rod Serling’s deep voice intoned that familiar mantra to introduce his popular TV series, The Twilight Zone. Serling’s spooky pronouncement was clearly an invitation to enter the world of the weird. But to mathematicians, the journey to a higher dimension is about as mundane as a trip across town in a taxi. They travel routinely not only to the fifth dimension but also to the seventh, the tenth, and the twenty- sixth. It’s nothing special, says Albert Marden, director of the Geometry Center in Minneapolis. To a mathematician, it’s an everyday event.

Why would mathematicians want to leave the comfort of our familiar three-dimensional world? Because, curiously, by poking their heads up into higher dimensions, they can get a clearer view of complex problems- -they can see relationships that look hopelessly tangled in the squashed and compacted universe of lower dimensions. Similarly, astrophysicists enter higher dimensions to see patterns in star clusters; particle physicists to look for unified theories; engineers to analyze mechanical linkages; and communications specialists to find ways to pack information into tight spaces.

There’s nothing like hopping into a higher dimension to make a complex problem easier. If that sounds counterintuitive, just think about what going to a higher dimension really means. Say you’re living on a one- dimensional line. You can move forward or backward, like a train on its track. But you can’t move sideways. It’s not only out of bounds, it’s out of your universe. Now imagine that your universe suddenly spreads out into two dimensions. You can roam freely over the entire surface: east, west, north, south, or any direction in between. Or, better yet, imagine that you’re a movie character, living your life on a two-dimensional screen. Add a third dimension and suddenly you can step off into the audience. You can simply walk away from that gunman about to shoot you. Thanks to that extra dimension, you have new freedom to move about.

To a mathematician, a dimension is just that: a degree of freedom. For example, take a knotted piece of string. As long as you stay in three dimensions, you’re stuck, says Sylvain Cappell, associate director of New York University’s Courant Institute of Mathematical Sciences. You can’t unknot it. But if you could slip a bit of string through another dimension, you could go around the obstacle and solve it. No matter how knotted it looks, you could go to a higher dimension and solve it. Cappell should know. Among other things, he studies the properties of eight-dimensional knots in ten-dimensional space.

The simplest way to think about a dimension is as a variable-- that is, as a quantity that can have any of a number of different values. It can represent latitude or longitude, time or speed, apples or oranges, particles or stars. You can describe a weather pattern by plugging in values for temperature, humidity, wind velocity, precipitation, and so on. If you need 12 variables to describe a situation, you have a 12-dimensional problem.

But variables alone don’t add up to geometry. As William Thurston, director of the Math Sciences Research Institute (MSRI) in Berkeley, California, points out, Having three variables is not the same as having three-dimensional space. And for understanding complex relationships, the shape of the space is often just as important as the number of dimensions it occupies. Take a standard two-dimensional relationship--say, a graph relating interest to consumer spending. Neither has anything to do with geometry, but you can get a better grasp of the situation by looking at the shape of the line. You can easily see where it peaks or bottoms out. You can see the slope of the curve.

The same holds true in five- or even ten-dimensional models. Logically, it may seem like the geometry is lost, that it’s just numbers, says Cappell. But the geometry can tell you things that the numbers alone can’t: how a curve reaches a maximum, how you get from there to here. You can see hills and valleys, sharp turns and smooth transitions; holes in a doughnut-shaped nine-dimensional model might indicate realms where no solutions lie.

Forming pictures of these intricate objects is easiest if you think about adding one dimension at a time, each spreading into space in a different direction. Start with a point. Stretch the point out along one dimension and you get a line, bounded by two points. Pull out the line in a perpendicular direction, and you sweep out a square, an area bounded by four lines. To get a cube, blow up the square into the next dimension, and you get a solid figure bounded by six squares. To get a four-dimensional cube, or hypercube, simply puff up the cube into yet another dimension: you’ll have an object whose borders are eight cubes. Similarly, you can rotate a two-dimensional disk around in a third dimension and get a sphere. You can twirl a sphere around in a fourth dimension and get a hypersphere.

In this sense, adding a new dimension is a kind of unfolding, explains Adam Frank, an astrophysicist at the University of Minnesota who works with six-dimensional spaces and looks at globular clusters by analyzing the geometry of 60,000-dimensional cigars. You stick your head up there and the whole landscape changes, he says. It’s extraordinary how all these complicated motions can reduce to, say, a simple 67-dimensional doughnut. It’s gorgeous!

Naturally, of course, you want to ask: But where are these other dimensions? Or, to start at the beginning, where is our nearest neighbor, the fourth dimension? The answer is simple: it’s perpendicular to all the other dimensions, just as the dimension we call height is perpendicular to what we call length. Perceiving this other dimension, unfortunately, is not so simple. But it’s not impossible, either.

Bill Thurston is known among at least some of his colleagues as the world’s greatest living geometer. At age 36 he won the Field Medal, the mathematical equivalent of a Nobel Prize. These days Thurston is trying to transform MSRI, which is affectionately known as misery in the field, into a more effective communicator of the pleasures of mathematics to the outside world. To that end, he’s begun pronouncing the acronym emissary instead.

To a visitor, Thurston comes across as just a big kid who likes to play around with shapes. Like a kindergarten classroom, his office has a small round table that’s covered with little plastic triangles and pentagons in primary colors. What Thurston does with them, however, is anything but elementary. He snaps four triangles together to form a tetrahedron; he then arranges five tetrahedrons around a common center like a bouquet, pointing out how they almost--but not quite--fit snugly together. That little angle left over, he says, is just enough in the roomy world of four dimensions to pack 600 of these tetrahedrons into a hypersphere.

Perhaps his favorite toy is a soft, three-holed torus, or doughnut shape, fashioned from 24 seven-sided pieces of cloth--a kind of convoluted muff with room for lots of hands to join in the middle. His mother, 75-year-old Margaret Thurston, sewed it last summer when she visited his Geometry and the Imagination course at the Geometry Center in Minneapolis, a course taken by 50 students ranging from high school kids to college professors. The pattern for the complicated figure was created by his two sons. It wasn’t easy. among geometers the form is a famous one, of almost mythological proportions. Few had actually been constructed before. It’s something I’d been wanting to see for a long time, he says.

Thurston believes that our difficulty with perceiving higher dimensions is primarily psychological, that it ultimately has something to do with the division in the mind’s eye between linear, analytical thinking, and geometric visualization of shapes. A lot of mathematics is done using the familiar mode of thinking we call upon for reading and writing and speaking. Algebraic equations, for example, are like sentences. The formula that gives you the area for a cube, x ¥ x ¥ x, can easily be communicated in words. But the shape of the cube is another matter. You have to see it.

When we talk about higher-dimensional spaces, Thurston says, we’re learning to think in and plug into this other spatial processing system. The going back and forth is difficult because it involves two really foreign parts of the brain. We don’t have a good way of communicating this spatial information. The problem isn’t with the substance of the mathematics; it’s with how to think about it.

Sir Arthur Eddington recognized this problem back in the 1920s, when he wrote widely in an effort to explain Einstein’s four-dimensional space-time to popular audiences. He warned readers not to listen to the voice inside them that whispered: At the back of your mind, you know that a fourth dimension is all nonsense. Eddington reminded his readers that the nonsense we routinely take for granted includes solid tables that are really mostly empty space, and transparent air that pushes on us with a force of almost 15 pounds per square inch.

Although some people who work in higher dimensions choose simply to ignore the question of what or where the extra dimensions really are (Eventually you stop trying to visualize it, says Frank. That’s when the roar goes out of your ears), others, like Thurston, work very hard at this business of overcoming common sense. He’s joined in these efforts by his colleagues at the Geometry Center--officially called the National Science and Technology Research Center for Computation and Visualization of Geometric Structures. Here Marden hosts dozens of students and visitors ranging from professors to college apprentices who come to explore the realms of higher-dimensional realities using powerful computers. While it’s certainly possible to describe any higher-dimensional space algebraically, he explains, you learn a lot more by actually seeing it. It’s just like in three dimensions. If you can see something, you can try to see what properties it might have. It raises questions. If you’ve never seen a tree, you won’t be able to come up with questions about it.

One way to view higher-dimensional objects is to slice them. Just as you might cut a three-dimensional block of cheese into two-dimensional slices, so you can cut a four-dimensional block into three-dimensional slices. Daeron Meyer, a senior at the University of Wisconsin at Madison and an apprentice at the center, programmed into his computer an impressive array of tools for exploring four-dimensional spaces, using software developed at the center. The 4-D object exists in there as a virtual object, he explains. You can specify any geometry you want. Then you simply plug in the coordinates in four dimensions. Meyer projects three- dimensional shadows of four-dimensional objects on his computer screen; takes two-dimensional slices of three-dimensional knots tied in four- dimensional space; and rotates a hypercube, the better to inspect its eight faces (which are, of course, cubes). As you explore the internal structure of the hypercube, you feel as if you’re flying through space, zooming right through walls into tenuous chambers that appear and evaporate like ghosts.

The feeling is an accurate perception of reality. After all, a four-dimensional perspective allows you to fly right past, over, or around any solid barrier in three dimensions. A being from the fourth dimension could reach into you and literally tickle your ribs! No wonder philosopher Henry More speculated, as far back as the seventeenth century, that spirits were fourth-dimensional beings.

More down-to-earth thinkers have found higher-dimensional spaces to be surprisingly useful for understanding the nonspiritual world as well. Einstein’s general theory of relativity, for example, describes the force of gravity as a result of warps in four-dimensional space-time produced by the presence of massive objects. The theory is more than just a different way to picture gravity: for calculating the effects of gravity in the vicinity of very massive objects, it gives results dramatically different from those that could be derived from Newton’s conception of gravity alone. Only relativity, for example, produces such exotica as black holes. Following Einstein, physicist Theodor Kaluza proposed in 1921 that the force of electromagnetism could also be understood as an effect of geometry--a wiggle in the fabric of four-dimensional space-time produced by disturbances in an unseen fifth dimension.

In fact, geometers have long stood in very good stead with those studying real properties of the physical world. Back in the nineteenth century, Lord Kelvin, who brought us modern thermodynamics, was impressed by the similarity between the properties of knots and the properties of particles. Knots are a classic problem in topology, which is a kind of fluid and flexible geometry. Where geometry is about rigid lines and angles and areas, topology is about holes, crossing points, tangledness, and so forth. A standing joke among mathematicians defines a topologist as someone who doesn’t know the difference between a doughnut and a coffee cup. Topologically they are equivalent, because a doughnut made of putty could be stretched and twisted into a cup with one handle without tearing or breaking the surface. Knots, similarly, are studied according to whether you can transform one into another, or how many holes they carve in space, or how you can go from one place on a knot to another without intersecting a line or surface.

Lord Kelvin proposed that atoms are knots in the luminiferous ether, an essence that people believed pervaded all space. One reason Kelvin liked this idea was that knots, like atoms, could be categorized into families whose members shared certain characteristics. Furthermore, there were many kinds of knots one could make from the same piece of string, and presumably many kinds of atoms one could form from different kinds of knots in the ether. Finally, viewing nature from this knotty perspective allowed Kelvin to reduce matter and forces to the same substance--the ether. Of course, Kelvin turned out to be wrong, but his ideas had an enormous influence on mathematicians, who began to take the properties of knots very seriously.

In recent years knots and topology have become central to theories that attempt to find common ground between gravity (which is described by the geometry of curved space, or general relativity) and other forces (such as electromagnetism) that are described by quantum mechanics. For example, the idea that space is essentially a weave of imperceptibly small loops linked together like medieval chain mail (see DISCOVER, April 1993) was first formulated when physicists noticed a striking resemblance between the equations that describe knots and certain equations that describe particles in gravitational fields.

Likewise, string theory describes all particles and forces as loops of some fundamental stuff that vibrates in harmonic patterns that give rise to everything from gravity to begonias. String theory works only in 10 or 26 dimensions, which has made many physicists reluctant to pursue it. However, the existence of the extra dimensions does not make string theory inherently more complicated than other theories. In fact, the whole point of the extra dimensions is that they render some relationships simpler: if string theory works, all the particles and forces should be describable in terms of topological properties of the extra, unseen dimensions.

Another way of saying this is that for complex problems, higher dimensions pay off in terms of greater symmetries--of having more ways that something can change and still stay the same. Take a sphere--say, a perfectly uniform beach ball. You can turn it halfway around and it looks exactly the same. You can turn it a quarter turn, or a sixteenth, or a thousandth, or a millionth and it still looks exactly the same. You can rotate it around any axis, by any degree, and it never changes. That’s nearly perfect symmetry. But say a lifeguard comes along and trips and falls on the ball, squashing it into a two-dimensional pancake. If you look at the pancake from above, it still looks round. But from the side it looks like a line. Held at angles in between, it takes the shape of ever-varying ellipses. The symmetry that is so perfectly preserved in the higher dimension is irretrievably broken, as a physicist might say, in the lower.

The same process holds true for any object, in any dimension. For example, a square can sit on any of its four edges. That makes four symmetries. A cube can sit on any of its six square faces, and each square face can be rotated any of four ways. That’s six times four, or 24 symmetries. A four-dimensional cube can sit on any of its eight cubical faces, and each cubical face can be oriented in any of 24 ways. That’s eight times 24, or 192 symmetries. A five-dimensional cube can be set down and rotated so that it looks the same in 1,920 orientations.

If you turn something inside out and it stays the same, that’s another kind of symmetry to add to the rest. Or if you move something in space (say, four inches to the right) or in time (look at it two hours later) and it stays the same, that’s another symmetry. You can have symmetries between forces, or symmetries of scale. If something gets bigger or stronger and all its other properties stay the same, that’s yet another kind of symmetry. Higher dimensions almost always mean more symmetries.

Physicists love symmetry because it means that even things that appear radically different can turn out to be fundamentally the same. And the more apparently different things (say, electricity and magnetism) turn out to be different aspects of the same thing (electromagnetism), the easier it is to explain the physical universe in terms of a few simple laws. If string theory works, it will be partly because the symmetries inherent in some high-dimensional gravity allow physicists to get all the known forces and particles from the same primal stuff.

These symmetries don’t even have to happen in physical space. Physical space is just one kind of space that happens to be particularly interesting and useful, says Sylvain Cappell. Recently he’s published a method of figuring out the number of possible solutions to complex problems based on the geometry of higher-dimensional spaces. But the space he works in is not the kind of geometry you can easily put your finger on.

In some ways, Cappell comes across as the consummate mathematician. He wears a beret. He does most of his work sipping cappuccino at the Cafe Violet in Greenwich Village, just across from the Courant Institute--usually with his longtime collaborator Julius Shaneson from the University of Pennsylvania. He likes to joke that a mathematician is just a machine for turning coffee into theorems.

At the same time, he speaks with disarming sincerity about how math requires you to feel stupid--over and over again. In fact, that’s his explanation for why the most brilliant mathematics is done by the young. When you start on a new problem, you always feel stupid, he explains. You might spend a whole day on a single paper, an hour on a single line. And you still don’t understand it. When you get to a certain position in life, you don’t want to feel stupid anymore. In mathematics, that’s when you’re dead.

Cappell has been practicing the art of useful stupidity for some time. He’s also very interested in having math understood and in finding ways to make it useful. Cappell and Shaneson’s most recent invention is so useful it might be patented.

It works like this: Suppose you have a problem involving a whole lot of variables and a whole lot of possible solutions. For example, say you have up to 15 ships that can carry up to 300 barrels of oil to up to 25 ports over as many as 100 possible routes and during as many as 365 days of the year while costing up to 1,000 dollars each for up to 20 sailors per ship and you want to know which choices will allow you to operate with maximum efficiency. To figure out even the number of possible solutions merely by counting would take close to forever because of the way each variable is linked to the next. (If you take long trips to faraway ports, you can’t do as many trips in a year.) But if you figure out the volume of a seven-dimensional figure based on these seven variables, you immediately know the number of possible solutions--because each point that fits inside the volume indicates a place where all the variables intersect (that is, a solution).

Of course, it isn’t quite as simple as that. The volume doesn’t quite give you the right answer, because the edges and angles mess things up. Imagine a pegboard. Say you draw a multisided figure on it. If you ask how many peg holes the shape encompasses, the answer isn’t clear. Some lines cut through a small fraction of a hole, while others cut off nine- tenths; angles take out chunks of various sizes. Since higher-dimensional figures tend to have a lot of edges and angles, the problem gets accordingly more complicated. But Cappell and Shaneson recently worked out a way to account for the edge effects. They hope, in the near future, to make the method even more useful. They hope to make it possible to find just those solutions that satisfy a certain constraint--for example, that produce a certain profit, where profit is a result of all the other variables combined.

Turning numerical problems into geometric problems is also revealing when examining the complex motions of physical systems--like stars and atoms. Adam Frank, at the University of Minnesota, does most of his work in the Supercomputer Center, just across the street from the Geometry Center. The connection is fitting. Frank and his colleagues work with the geometry of something called phase space. Phase space isn’t a physical space, either. It’s a way of looking at the shape of dynamic systems with large numbers of moving parts.

Take a particle--say, an air molecule floating in a room. It can move in three physical dimensions. But it also has a certain velocity in each dimension. It turns out that if you imagine the particle in a six- dimensional phase space that takes into account both its velocity and position at the same time, its motion looks a lot simpler. Of course, if you have more than one particle, you have more than six dimensions. Two air molecules would have a total of 12 dimensions in phase space. Four air molecules would have 24. Since phase space is normally used for figuring out the dynamics of gases and star systems--which normally contain many thousands of particles--the dimensions add up quickly.

If you looked at the motions in regular space, they would be very complicated--they would keep folding back on themselves, explains Frank. But in phase space they unfold. You no longer have motion in time. The whole motion is built into it. If you had to sit and watch a particle and follow its motions in space, you’d have to wait forever. But in phase space, you just look at the shape. It’s the object that matters. And the entire dynamics is built into the topology of the object.

Calculating the motions of 10,000 stars in phase space, for example, might fill out an object like a simple torus. The shape of the torus tells you a great deal: Is the star system stable? How much energy does it have? How will it evolve over time?

Just how shapes can shed light on the dynamics of a system is somewhat easier to understand if you think in terms of simple mechanical linkages. Thurston starts with the example of a bicycle pedal. The lever that pushes the wheels around can swing in a full circle. That’s called its configuration space. The little pedal attached to the lever--the one you put your foot on--also swings out a full circle. A circle of circles is a torus. So the configuration space of the system is a torus.

Configuration spaces of other kinds of motions can add up to spheres or ellipses or other simple shapes that can be easily analyzed for dynamic properties. The more ways the linkages can move, the more degrees of freedom they have. So the torus or sphere you end up with might not be a simple three-dimensional shape but a six-dimensional torus, or a 50- dimensional one. You want to know what happens when things move, Thurston explains. Mathematicians know a lot about the geometry of these things. For example, it turns out that dynamics on a sphere or a torus can be predictable for a very long time. But a two-holed torus has to be chaotic. It has a high entropy, or tendency toward disorder. So that’s a crucial distinction.

And as Cappell points out, you can even learn a lot by looking at the number of holes. Imagine a set of relationships that fills out a sphere (just as the graph of height and weight might fill out a two-dimensional form). On a sphere, you have only one highest point and one lowest point. But say you have a torus, and you stand it up on end: now you still have the two critical points at the top and the bottom, but you also have two saddle points: one at the bottom of the hole and another at the top, for a total of four critical points. Knowing these critical points can be very important, explains Cappell, because a lot of systems tend to flow toward those points, stabilize there. Say you have a space that describes all possible positions of something and all possible amounts of energy. This tells you how it will evolve.

Just what all these maxima and minima reveal is beside the point. They could be numbers of apples, or energy levels, or strengths of forces. The method works equally well no matter what. Or as the late Richard Feynman put it: The glory of mathematics is that we do not have to say what we are talking about (emphasis his).

Curiously, one thing that does matter very much is what dimension you’re working in. Because dimensions, it turns out, have personalities that are independent of whether they’re dimensions of a problem or of physical space. For example, it turns out that dimension seven has properties very conducive to doing certain kinds of calculus. Dimension eight turns out to be excellent for packing spheres--like packing oranges in a box. While few grocers work in eight dimensions, many people who work in modern communications do. Eight dimensions are just what you need for efficient packing of encoded information transmitted by computer modems--a boon to companies that shuffle information a lot, like banks and airlines.

Looking for just the right dimension has been especially important to string theorists because the theory has to work compatibly with general relativity (the world of the big, ruled by gravity) and quantum theory (the world of subatomic particles). Above all, that means the dimension, to be right, has to demonstrate the same symmetries in both worlds. I don’t think there’s a good physical intuition of why it should work out to be ten, says Dan Freed, a mathematician from the University of Texas at Austin who is currently at the Geometry Center. When I was a graduate student, it was 11; then it went up to 26. God knows where it’s going next. If there’s a physical principle involved, it’s the need to have the right symmetries.

According to Clifford Taubes of Harvard, the problem of finding the right dimension for string theory is also related to finding a dimension for space-time that would produce the right number of forces and particles. One way to think of the fit between dimensions and particles is in terms of a space-time that’s shaped like a pretzel with holes. The number of holes tells you the low-energy states. And that tells you the number of particles. You can’t have too many holes or you’d have more particles than we can see. The way the holes link together is analogous to the way forces link particles together. (Of course, we wouldn’t perceive this pretzellike shape any more than we normally perceive the curved surface of Earth.)

Physicists and mathematicians are fascinated with these problems for different reasons. Mathematics is the investigation of all possible universes, says Taubes. Physics is the investigation of the universe. That makes a lot of people who work in physics mathematicians.

Even those of us who’ve learned to accept the physical reality of space-time as a four-dimensional rubber mat may have trouble following the mathematicians into their special twilight zone, where the knottiness of higher-dimensional spaces seems never to quite, completely, untangle. For example, they might lead us to consider: If our space-time is a four- dimensional rubber mat, then what kind of space does it sit in? For mathematicians, there can be a big difference between the dimensionality of an object and the dimensionality of the euclidean space it occupies. A comic strip character, for example, is a two-dimensional creature embedded in a two-dimensional space, but a hollow ball is a two-dimensional surface that has to sit in a three-dimensional space. There are even two- dimensional surfaces that can exist only in four-dimensional space.

In fact, there are so many kinds of higher-dimensional spaces, says Cappell, that some people collect them like butterflies. And Cappell likes them all. Ask him if he has a favorite dimension and he looks almost horrified: That’s like asking me if I have a favorite child! He muses: You know, every now and then someone tries to hand you a box and says, ‘This is all of geometry.’ And it looks so nice and neat that you’re tempted to believe it. But then you see that there’s something oozing out from the sides. Well, we’re always focusing on what’s oozing out from the sides.
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