Intricate Dimensions

The blackboards crawl with bizarre creatures etched in chalk--strangely twisted alien species with spikes sticking out at odd angles, winding into corkscrews or sprouting holes, handles, tails. Like convoluted soap films, the outer skins of these forms contort into pretzels, fold into mazes or clump into loopy mounds of fettuccine, festooned with knots and braids.

Men and women holding paper plates of Caesar salad crowd around the forms, waving their pens and pieces of chalk like batons, speaking in strange tongues. They babble about “non-singular leaves” and “Z boundaries” exhibiting “nasty properties” and “bad behavior.” They compare notes about “performing surgery,” “drilling” and “filling.”

No, these are not biologists comparing new specimens from an exotic rain forest, or dentists from “The X-Files.” They are mathematicians exploring the perhaps even more exotic territory of four-dimensional space.

Welcome to the Twilight Zone.

Except Rod Serling had it wrong. The spookiest dimension is not the fifth, as he used to intone in the introduction to the original TV series. Mathematicians find their way around five dimensions fairly easily, as well as 10 or 16 or 25.

In fact, the only dimensions that still leave mathematicians stumped are those that make up the world we live in: the third and the fourth (three of space, plus one dimension of time).

It’s a sobering fact. The very space we live in is the most incomprehensible of all.

“And it’s not just because our understanding is incomplete,” said Columbia University mathematician John Morgan at a meeting in Berkeley recently. “It’s really that complicated.”

Mathematicians are drawn to the fourth dimension in part because it’s a puzzle they want to figure out. “Philosophically, we’re in a strange situation,” said UC Irvine mathematician Ron Stern. “It’s too bad we don’t live in a higher-dimensional world we could understand.”

But this seemingly esoteric subject also has real-world ramifications. Our universe evolves in a still-not-well-understood, four-dimensional framework. Every particle that physicists study is a four-dimensional entity.

Indeed, it was the work of two physicists that led to the recent rapid progress toward understanding the fourth dimension. While studying aspects of 12-dimensional “string theory,” Edward Witten and Nathan Seiberg of the Institute for Advanced Studies discovered equations that vastly simplified the problem.

Their work, Stern said, produced an explosion of research and led to the discovery of many previously unknown four-dimensional surfaces.

Not surprisingly, such a juicy puzzle attracts the best and the brightest. Among the mathematicians who gathered this summer at the Mathematical Sciences Research Institute in Berkeley to discuss the mysteries of three- and four-dimensional spaces were four Fields Medalists--the mathematics equivalent of Nobel laureates.

Alas, even such a concentrated surge of mathematical brainpower didn’t manage to solve the major puzzles of four- (or even three-) dimensional surfaces. “One of the things I’ve learned,” said UC Santa Barbara mathematician Marty Scharleman, one of the organizers, “is that it’s even more complicated than we feared.”

The Stage on Which Everything Else Occurs

But what is a fourth dimension, anyway?

If such a concept seems beyond comprehension, take heart. Some of the best minds in science have felt the same way.

When Albert Einstein first established that our universe is embedded in a four-dimensional framework of space-time, many smart people balked. The great popularizer of Einstein’s theories, the physicist Sir Arthur Eddington, warned his readers not to listen to the voice inside that whispers: “At the back of your mind, you know that a fourth dimension is nonsense.”

In truth, Eddington pointed out, a fourth dimension is no more nonsensical than most other everyday things we take for granted: the fact that seemingly solid objects are mostly empty space, or that invisible air presses down on everything with a force of almost 15 pounds per square inch.

Like it or not, coming to terms with the fourth dimension is essential, if only because the universe we live in cannot be understood without it. Dimensions are the stage on which everything else happens. The number of dimensions determines what’s possible in a universe and what isn’t.

In the simplest sense, a dimension is a coordinate. On a two-dimensional surface, such as the surface of the Earth, any position can be exactly defined with only two numbers: latitude and longitude. For positions above or below sea level, a third coordinate is needed--altitude. To specify a time, add a fourth coordinate.

When you agree to meet a friend at 2 p.m. at the corner of Hollywood and Vine on the 12th floor, you require four dimensions to pin down your location.

More important, the number of dimensions determines the qualities of a space, limiting the number of ways it’s possible to move about. Higher dimensions expand the possible options. That’s one reason that mathematicians call dimensions “degrees of freedom.”

Consider, for example, the life of a two-dimensional comic strip character, Cathy. If Cathy is stuck on a bad date, she cannot move out of her frame. She can move vertically in the strip, or horizontally, but she cannot escape the two-dimensional surface of the paper to scale the walls of her cartoon world. Such a move into the third dimension would be literally out of her universe--as unthinkable to Cathy as a fourth spatial dimension is to us.

However, if Cathy could leap over the comic strip frame, she would be free. Going to a higher dimension presents opportunities to avoid obstacles. There’s a lot more room to move around.

The same would be true if Cathy were a real three-dimensional person trapped in a three-dimensional house. Again, she could use a higher dimension to get out of her fix. By traveling backward in the dimension of time, she could move back in history to a time before the house was built. Presto, she would be free!

And so on, up to higher and higher dimensions. You can leap over a four-dimensional barrier in five-dimensional space, untie an eight-dimensional knot in nine-dimensional space.

Mathematicians do it routinely. In space or time, higher dimensions always allow much more freedom.

Indeed, the main reason the third and fourth dimensions are so rich is that they allow just enough room for things to get trapped and tangled, but not enough room to unravel the knots. A two-dimensional world is so simple that nothing as interesting as a knot--much less a molecule--can even exist.

Another advantage to higher dimensions is that they offer a better view of what’s below--just as the pattern of streets in a city may look simpler from a plane than from the ground.

That’s one reason physicists are so attracted to theories that make use of 10 or 12 dimensions. As it turns out, our complex four-dimensional world actually looks simpler from that higher-dimensional perspective.

12 Dimensions Evolving to 4

Physicists would like nothing better than to explain everything in the universe in terms of one fundamental entity--some primordial “stuff” that produced every bit of force, matter, space and time that makes up our world. Although such a fundamental essence would have a single identity in higher dimensions, it would shatter into many seemingly different forms in the lower dimensions we live in.

This is the same thing that happens to a shadow of a three-dimensional form (say, a person) when it gets projected onto the two-dimensional pavement. Depending on the angle of the sun, the shadow may look distorted. It can look very different at noon than it does at sunrise or sunset. But the person--the higher-dimensional object that cast the shadow--doesn’t change. Only its projection does.

In this sense, our four-dimensional world is a shifting shadow projection of a higher reality in 10 or 12 dimensions that existed only in the first blinks of time. Of those original dimensions, all but four have curled up so tightly that they cannot be perceived.

But why are only four dimensions left? That’s a question physicists would very much like to answer. In particular, they’d like to understand how and why the original 12 dimensions evolved into four.

At least some physicists suspect that the answer may be somewhat self-serving. That is, the 12-dimensional uber-universe probably evolved into every possible variety of universe. But only in this four-dimensional space-time could atoms and planets and people evolve to take note of it.

In other words, all those other universes may well be out there, but they could not have mathematicians to study them. The only universe we can perceive is a Goldilocks universe, where the dimensions are just right to produce us.

Next week: Mathematics of the fourth dimension.