Mathematician Studies Fourth Dimension

Kusner directs the GANG Lab (Center for Geometry, Analysis, Numerics & Graphics), while teaching and conducting research in geometric analysis. He and his colleagues at the lab, a bright office space in the Lederle Graduate Research Tower, gaze at computer screens on which neon, multi-colored spheres and doughnut-like shapes roll and twist in ways that are difficult to decipher. That's because the shapes are moving in the fourth dimension, according to James Lawrence, the undergraduate whose computer work created this particular 4-D environment. While the computer images are intriguing and offer a certain level of fun, their real usefulness is as a teaching tool, "especially for very bright students, who often are turned off to traditional methods of teaching and learning math," says Kusner. "That's the main point of the GANG lab."

On the screen before Kusner and Lawrence, the shapes continue to mesmerize and confuse the untrained eye. "There's no inside or outside. There's a forward and backward, but also a hyper-forward and a hyper-backward," says Lawrence, who insists that navigating in 4-D is a learnable skill, like playing a musical instrument.

"It's counter-intuitive at first, because it's not the world we encounter in our everyday lives," Kusner notes. "So it's hard to know where you are." But this doesn't mean that the fourth dimension doesn't exist in the universe, he adds. In fact, theoretical physicists and mathematicians insist that the universe is multi-dimensional. But let's ask the obvious question: why on Earth would anyone want to do this? There are lots of reasons, Kusner says.

One down-to-earth reason is that it may provide a way of tracking enormous amounts of data when looking for correlations, as researchers do in epidemiological studies - studies into why people contract certain illnesses, Kusner says.

"Visualizing many dimensions can help in finding correlations that exist in a huge set of data," Kusner says. "Correlations are where things line up; correlations are where you find answers."

The fourth dimension also provides a graspable, visual explanation for some difficult concepts in higher mathematics. Lawrence pulls a shape up onto the computer screen: it looks like two basket handles, connected at each handle's top point. One handle is sliced by a gray plane, while the other handle, still connected, floats below it. The figure is a complex parabola representing the square root of a negative number, a number that used to be tagged as "imaginary," a number so difficult to comprehend that teachers once told students to disregard it.

Their 4-D visualization also has uses in high-energy physics. Kusner points to last summer's discovery by Japanese and American researchers that the sub-atomic particles called neutrinos have mass - a discovery which, while it may have raised a mainstream eyebrow or two, utterly rocked the physics world. 4-D provides a way for scientists to track neutrinos through time and space, and to determine how the particles interact. In other words, it helps them to understand the very laws that govern the universe.

Kusner explains the fourth dimension this way: "Most people tend to think of the world as three dimensional: left-right, backward-forward and up-down. Perhaps they think of time as the fourth dimension: before-after. Mathematicians assign numbers to these quantities," he explains. "Considering a box, they might ask: How wide is the box? How deep is the box? How high is the box? For how long does the box exist? They might even assign other numbers to describe other features of the box: How heavy is it? What color is it? Thus they would be assigning not three or four numbers, but five or six or even more numbers to the box. In this sense, to the mathematician, the box may have more than the usual three or four dimensions."

Kusner's enthusiasm extends to 3-D work as well. His research in solving a 40-year-old problem of how a sphere could be turned inside-out, without tearing or creasing, won accolades in the prestigious journal Science last summer. More recently, this work was the focus of a lengthy cover story of Science News.

Is this art or science? "What mathematicians do is both correct and beautiful," he says, launching into a discussion of why math is a bit of both. "It's pure creativity, governed by logical rules." But there's also a great deal of craft required, he adds.

"You have to be skilled; you have to know the craft in order to make creative leaps without making mistakes. And you also need an aesthetic sense, because when you make those leaps, there's something ineffable, something intuitive that happens, almost like magic. Only later do you go back and fill in the logical details and calculations."

He turns back to the computer screen, where the basket handles float placidly against the gray plane. "This," he intones, "is not your father's parabola."