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Intricate Dimensions

Mathematicians have no trouble dreaming up worlds of five dimensions or even 25. They are stumped only when trying to explain the four-dimensional universe we live in.


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.

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