A typical math problem for an American middle school student might be to calculate the sales tax on a dozen baseball caps. At an after-school program called the Berkeley Math Circle, a typical problem is more… advanced.

As she writes the hyperbolic cosine formula on the blackboard, circle director Zvezda Stankova asks the four dozen or so teens in the room to use it in a differential equation to describe the arc of a wire, hanging between two poles.

It's no problem for 12-year-old Bobby Veres, who says Math Circle teens can handle this. He points out, "It's very prestigious. Most of the top math kids from around the Bay area come here. I like the chance that we get to learn extra math that isn't available in schools, and we get to make friends."

They also get to hear from some top mathematicians and theorists, like astrophysicist Cliff Stoll. Stankova calls him the epitome of what you would think of an internationally renown scientist: "Absent-minded, yet very animated. Very sharp and very enthusiastic."

And very demonstrative. Stoll is jumping high, spreading his arms wide, then running forward until he's nose to nose with the Math Circle students - all this to show the three dimensions of our universe: height, width and depth.

With Stoll in command, the Math Circle takes a wild ride from our 3-D universe to an imaginary world of two dimensions, that's limited to only width and height.

"Can I tie a knot in a string that's in a two-dimensional universe?" he asks, prompting a chorus of "No!"

When one student says "yes," Stoll challenges him to prove it.

As the students scribble math equations to support their answers, Stoll provides a more dramatic proof of the three-dimensional nature of knots. He grabs the cord of a reporter's microphone and ties a knot that is so bulky, it obviously takes up three dimensions.

"Excellent! Excellent!" he declares as the students laugh.

Then, from two- and three-dimensional space, Stoll takes a giant mental leap up to four dimensions.

"Height, width, depth and…"

A student suggests, "Something else."

"Four space," Stoll says, adding, "In that case, the curious thing is that strings go right through themselves. They don't stop like that. They go, 'Phloop!' Right through."

While we can't really imagine or describe that fourth dimension, Stoll says that calculating its theoretical properties is helping scientists map new possibilities. For instance, take how tying knots might work in the fourth dimension . . . or maybe not.

"Can you tie your shoes if you lived in a four-dimensional universe?" Stoll asks.

No, you can't. In fact, in four space, our blood vessels would pass through each other. It'd be hard to have a pumping heart because blood wouldn't stay within a pipe.

These are mind-bending imaginations. But the Math Circle students catch on fast. One points out, "Then there's no such things as shoes, because the string would go through the shoe!"

Stoll agrees, "People who live in four space don't wear tie-down shoes!"

Then he suggests another dilemma: "Will DNA work in four space?"

The students don't think so, and one observes, "If we don't have DNA, then we're dead."

By the end of Stoll's two-hour lecture, the Berkeley Math Circle knows many reasons why a 3-D universe is probably the only place where we can tie our shoes.

While these may seem like mental frolics, they're similar to the speculations that led mathematicians, thousands of years ago, to suggest the world is round, instead of flat. They're similar to imaginings, a century ago, that led to the discovery of DNA. And, Stoll says, they're questions that cosmologists are asking about our universe today.

"The most interesting things in mathematical topology today is called knot theory or string theory," he says.

As for what future dimensions await these students, Bobby Veres suggests, "I'd like to be a doctor, scientist, mathematician." And Zvezda Stankova notes, "We don't yet have Nobel Prize winners, but we are hoping soon we will have one."