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Physicists just love the idea of extra dimensions. This article is a brief account of the history and nature of this idea.
Needless to say, they thought this was pretty cool. There's an obvious problem though: our universe certainly looks like it's only got four dimensions. The standard way to deal with this in Kaluza-Klein theories is to say that the extra dimension is rolled up ("compactified") into a little tiny loop that's too small for us to notice. If you remember when I was talking about bundles a while ago, this means that the universe is really a bundle over a four-dimensional spacetime whose fibre is a little loop.
Now, it might have been kinda neat to wrap up electromagnetism and gravity like that, but there are a few problems. Firstly, it only works for classical electromagnetism but we know that electromagnetism is really a quantum theory. Secondly, there are two other forces that really should be included: the strong and weak nuclear forces. Still, it looked like a good start. The obvious idea was to add more dimensions to spacetime and compactify them all but the three of space and one of time that we usually see. Each event in spacetime would then not really be a point but rather a little manifold. Somewhere along the way the idea of supersymmetry (which I won't explain here) got thrown into the mix and the result was a theory called "supergravity" that only really works in eleven dimensions. Then physicists got interested in smearing out point particles into little loops and when they added that idea they ended up with a theory (or rather a collection of theories) called "superstrings". In recent years they've decided that even these aren't radical enough, and have turned the particles into various kinds of vibrating membranes to give something called M-theory (some people say the "M" stands for "Magic"!).
One way to look for large extra dimensions is to look for deviations from Newton's law of gravitation at small distances (this requires lots of cleverness in the experimental design but not much in the way of expensive hardware). The idea here is that the variation of the gravitational force with distance depends on the dimension of space. For instance, if you imagine the gravitational field lines spreading out into a three-dimensional space from a point mass then you'll see that they increase in spacing as the square of the distance (because the same number go through each concentric spherical shell, but the areas of these shells increase as the square of the distance). If you think about how this works in different numbers of dimensions then you'll see that if space is n-dimensional then gravitational forces fall as the (n-1)-th power of the distance. Now, if there are compactified extra dimensions, then if you look at distances that are about as big as the little screwed up dimension then it will look like the universe has more than the usual number of dimensions, so the strength of the gravitational force will fall faster at small distances. At larger distances the extra distances, the extra dimensions are "filled up" with field lines so the usual inverse square law will be restored.
If the extra dimensions are pretty small, then the best way to detect them will be to look for "missing energy" in high-energy particle collisions - we don't see the particles that go zipping off into the other dimensions so it looks like some energy just disappears. I don't know much about how that would work, but I do know it will be a much more expensive set of experiments.