From 3-D to 4-D
It's difficult for us to accept the idea because when we try to imagine even a single additional spatial dimension -- much less six or seven -- we hit a brick wall. There's no going beyond it, not with our brains apparently.
Imagine, for instance, that you're at the center of a hollow sphere. The distance between you and every point on the sphere's surface is equal. Now, try moving in a direction that allows you to move away from all points on the sphere's surface while maintaining that equidistance. You can't do it. There's nowhere to go -- nowhere that we know anyway.
The square in Flatland would have the same trouble if he were in the middle of a circle. He can't be at the center of a circle and move in a direction that allows him to remain equidistant to every point of the circle's circumference -- unless he moves into the third dimension. Alas, we don't have the four-dimensionsal equivalent of Abbott's three-dimensional sphere to show us the way to 4-D. (In mathematics, moving into ever higher dimensions is a walk in the park. See Multidimensional Math.)
How about 10-D?
In 1919, Polish mathematician Theodor Kaluza proposed that the existence of a fourth spatial dimension might allow the linking of general relativity and electromagnetic theory. The idea, later refined by the Swedish mathematician Oskar Klein, was that space consisted of both extended and curled-up dimensions. The extended dimensions are the three spatial dimensions that we're familiar with, and the curled-up dimension is found deep within the extended dimensions and can be thought of as a circle. Experiments later showed that Kaluza and Klein's curled-up dimension did not unite general relativity and electromagnetic theory as originally hoped, but decades later, string theorists found the idea useful, even necessary.
The mathematics used in superstring theory requires at least 10 dimensions. That is, for the equations that describe superstring theory to begin to work out -- for the equations to connect general relativity to quantum mechanics, to explain the nature of particles, to unify forces, and so on -- they need to make use of additional dimensions. These dimensions, string theorists believe, are wrapped up in the curled-up space first described by Kaluza and Klein.
To extend the curled-up space to include these added dimensions, imagine that spheres replace the Kaluza-Klein circles. Instead of one added dimension we have two if we consider only the spheres' surfaces and three if we take into account the space within the sphere. That's a total of six dimensions so far. So where are the others that superstring theory requires?
It turns out that, before superstring theory existed, two mathematicians, Eugenio Calabi of the University of Pennsylvania and Shing-Tung Yau of Harvard University, described six-dimensional geometrical shapes that superstring theorists say fit the bill for the kind of structures their equations call for. If we replace the spheres in curled-up space with these Calabi-Yau shapes, we end up with 10 dimensions: three spatial, plus the six of the Calabi-Yau shapes, plus one of time.
If superstring theory turns out to be correct, the idea of a world consisting of 10 or more dimensions is one that we'll need to become comfortable with. But will there ever be an explanation or a visual representation of higher dimensions that will truly satisfy the human mind? The answer to this question may forever be no. Not unless some four-dimensional life-form pulls us from our three-dimensional Spaceland and gives us a view of the world from its perspective.
