Other Worlds, Other Ways of Being

Ever since Einstein, we’ve known that our universe contains at least one extra unseen dimension. Physicists explore the realm of higher dimensions in order to make sense of the world we live in.

Mathematicians, however, don’t worry whether the spaces they study are habitable--only that they are interesting. As it turns out, the third and fourth dimensions are the most mathematically interesting--and mysterious--of all.

The unique complexity of the third and fourth dimensions was not recognized by mathematicians until the 1960s, when most of the pressing problems in dimensions five and higher were worked out. “No one expected that you could solve problems in five dimensions that you couldn’t solve in three or four,” said Columbia University mathematician Joan Birman. “That was a great surprise.”

Mathematicians gathered this summer at the Mathematical Sciences Research Institute in Berkeley to discuss the mysteries of three- and four-dimensional spaces. Although progress has been made of late, much about dimension four is still a mystery.

“I could list 10 theorems that are known to be true in every dimension other than four,” Columbia’s John Morgan said.

For example, structures exist in four dimensions that can’t exist in any other, according to UC San Diego mathematician Michael Freedman, who attended the Berkeley meeting. “There are equations that can be written only in dimension four,” he said, making it possible to do things “that don’t make sense in any other dimension.”

The importance of the order in which things happen also has a special role in the fourth dimension. For example, if you tie a knot, does it matter whether you first put the left end over the right? If you multiply a string of numbers together, does it matter which you multiply first?

Holes, Handles and Twists

“Once you get to higher dimensions, it doesn’t matter,” said William Thurston of UC Davis. “But in dimensions three and four, it does matter. In three and four, there’s an additional knottedness,” he said.

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At times, mathematicians studying these forms seem like so many botanists, categorizing various species and grouping them into families with similar traits. They distinguish the “families” by characteristics: For example, do the forms have holes, or handles? Do they twist, or change their orientation like a Mobius strip? Can they be shrunk down to a point?

Curiously, everyday geometry is not an essential feature for these mathematicians. In topology--the study of these strange surfaces--surfaces are rubbery and can transform into each other as long as nothing has to tear or break. For example, a coffee mug with one handle and a doughnut are topologically equivalent, because they both have one hole.

(Mathematicians like to joke that a topologist is someone who doesn’t know the difference between a doughnut and a coffee cup.)

A sphere, on the other hand, is a different class of object, because it doesn’t have a hole. But if you can’t see the four-dimensional object, how could you tell whether or not it’s a sphere or a doughnut?

The most famous outstanding problem in the world of four-dimensional surfaces concerns just such a question: How can you tell whether any given surface in four dimensions is a sphere? (See graphic.)

One way might be to tie a loop around the outside of the sphere, then shrink it. If the loop can shrink to a point, the shape is a sphere. If it can’t, then you might have a doughnut on your hands.

The hole in the doughnut would prevent the loop from collapsing completely.

But does the ability to shrink a loop to a point guarantee that a shape is a sphere in higher dimensions? It does in dimension five and higher. Freedman proved in the 1980s that the test works in dimension four. But no one knows whether it works in dimension three.

Known as the Poincare conjecture after French mathematician Henri Poincari, who first proposed it, this puzzle is right up there on the top of mathematics’ priority list. “It’s the most famous problem in topology,” Morgan said, “and one of the most famous problems in math.”

UC Berkeley mathematician Rob Kirby, a leader in the field, allowed that most mathematicians think Poincare’s conjecture is true. “But we can’t prove it,” he said. “It’s simple to state. It’s great. A lot of people have been embarrassed by it.”

Analogies and Surgery

How can the mathematicians draw conclusions about forms they can’t even properly see?

Like heart surgeons and brain surgeons, mathematicians who specialize in dimensions three and four tend to use different approaches. “It’s a different set of people,” said Scharleman, who recently switched his specialty from dimension four to three. “They use different tools. What appealed to me about [three-dimensional surfaces] is you can see what you’re doing.”

In dimension four, the mathematicians have to rely on analogies. “We use models,” Morgan said. “We know they aren’t complete. But it’s like Plato’s cave. They can tell you the essential features.”

Not surprisingly, these topologists spend a lot of time turning over shapes in their heads. “We’re pretty good at imagining things,” Kirby said.

They also make use of newly developed mathematical techniques known as “surgery,” taking slices through higher-dimensional spaces the way a CAT scan takes two-dimensional “slices” of the human body. They “drill out” pieces of surfaces--removing, say, a doughnut-shaped hole. They “fill in” the spaces with other forms, or sew them up. “Basically, you slice them up and screw them back together,” Thurston said.

Of course, none of this is performed on real shapes, but rather, through equations and imagination. Thurston and others also translate his visions into computer graphics. At the Berkeley meeting, he invited his colleagues to “look inside” a higher-dimensional knot while he “drilled” and “filled” with abandon.

Using airplane-like shapes to follow the contours of some strange internal spaces, Thurston pointed out the various bends and folds from the perspective inside the higher-dimensional knot. To an observer, it seemed like taking a trip inside the human body, following the floor of blood vessels and nerves.

As the computer “airplanes” encountered twists in space, or bends in contours, they flipped over or disappeared into the distance.

In the Real World

Recently, Thurston has been spending more time in the real world, studying the structure of foam-like materials. He thinks the way they pack together in space can teach him things about spaces in general.

“I’m sure a topologist can learn from nature,” he said. “There’s a good chance that topology and materials science have a lot to say to each other.”

Certainly, the mathematicians have taken their cues from physicists many times before. Over the past 10 years, some of the most important discoveries in four-dimensional space have come from physicists--or at least the solutions were inspired by physicists.

In the last few years alone, several major insights into the nature of four-dimensional space have come from physicists like Witten and Seiberg, who study “string theory.” According to string theory, all matter and forces in the universe boil down not to particles, but to tiny loops of some fundamental stuff that vibrates in 10- or 12-dimensional space, creating everything there is. (Some people have therefore called it the Theory of Everything.)

As a loop of string moves through the dimension of time, it traces out a cylinder. As these cylinders collide and intersect, they produce not only everything in the universe, but also exactly the kinds of shapes these three- and four-dimensional twilight zone mathematicians study.

“It keeps going back to physics because physicists can’t build machines big enough [to find out what they want to know],” said Kirby, “so mathematics is their playground. They give mathematicians all sorts of things to think about.”

In the end, the twilight zone of higher dimensions may be the door that leads to an understanding of our own existence.

The image represents a surface called a “Klein bottle” that can exist only in four-dimensional space. In effect, it consists of two two-dimensional Mobius strips glued together. The “inside” surface smoothly connects with the “outside” surface without intersecting it.

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Image by Thomas Banchoff and Davide Servone

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A Sphere or Not a Sphere?

The most famous outstanding problem in the mathematics of higher dimensions is: How can you tell whether any given surface is a sphere? For a familiar sphere like the surface of the Earth, the solution is simple.

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Loops that can be shrunk

A loop that can be shrunk

A lop that can’t be shrunk

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Try tying a loop around the sphere, then shrink it. If all loops can be shrunk to a point, the shape is a sphere. If they can’t, then it might be a doughnut. The hole in the doughnut would prevent the loop from collapsing. However, just as the two-dimensional surface of the Earth sits in three-dimensional space, a three-dimensional surface sits in four-dimensional space. Does the loop trick work for three-dimensional surfaces? Mathematicians don’t know.

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Source: “The Mathematical Tourist” by Ivars Peterson