There is a fifth dimension beyond that which is known to man. In the 1960s Rod Serling’s deep voice intoned that familiar mantra to introduce his popular TV series, The Twilight Zone. Serling’s spooky pronouncement was clearly an invitation to enter the world of the weird. But to mathematicians, the journey to a higher dimension is about as mundane as a trip across town in a taxi. They travel routinely not only to the fifth dimension but also to the seventh, the tenth, and the twenty- sixth. It’s nothing special, says Albert Marden, director of the Geometry Center in Minneapolis. To a mathematician, it’s an everyday event.
Why would mathematicians want to leave the comfort of our familiar three-dimensional world? Because, curiously, by poking their heads up into higher dimensions, they can get a clearer view of complex problems- -they can see relationships that look hopelessly tangled in the squashed and compacted universe of lower dimensions. Similarly, astrophysicists enter higher dimensions to see patterns in star clusters; particle physicists to look for unified theories; engineers to analyze mechanical linkages; and communications specialists to find ways to pack information into tight spaces.
There’s nothing like hopping into a higher dimension to make a complex problem easier. If that sounds counterintuitive, just think about what going to a higher dimension really means. Say you’re living on a one- dimensional line. You can move forward or backward, like a train on its track. But you can’t move sideways. It’s not only out of bounds, it’s out of your universe. Now imagine that your universe suddenly spreads out into two dimensions. You can roam freely over the entire surface: east, west, north, south, or any direction in between. Or, better yet, imagine that you’re a movie character, living your life on a two-dimensional screen. Add a third dimension and suddenly you can step off into the audience. You can simply walk away from that gunman about to shoot you. Thanks to that extra dimension, you have new freedom to move about.
To a mathematician, a dimension is just that: a degree of freedom. For example, take a knotted piece of string. As long as you stay in three dimensions, you’re stuck, says Sylvain Cappell, associate director of New York University’s Courant Institute of Mathematical Sciences. You can’t unknot it. But if you could slip a bit of string through another dimension, you could go around the obstacle and solve it. No matter how knotted it looks, you could go to a higher dimension and solve it. Cappell should know. Among other things, he studies the properties of eight-dimensional knots in ten-dimensional space.
The simplest way to think about a dimension is as a variable-- that is, as a quantity that can have any of a number of different values. It can represent latitude or longitude, time or speed, apples or oranges, particles or stars. You can describe a weather pattern by plugging in values for temperature, humidity, wind velocity, precipitation, and so on. If you need 12 variables to describe a situation, you have a 12-dimensional problem.
