Yes, it's a cheap and old pun. I don't care.
I've mentioned how extra dimensions are one of the main ways particle theorists
The idea of extra dimensions goes all the way back to 1921, when Theodor Kaluza suggested identifying electromagnetism as the gravitational effect of the extra dimension. Five years later, Oskar Klein expanded on his idea, which is thus known as Kaluza-Klein theory. More generally, the process of taking a five- (six-, etc) dimensional theory and figuring out what it looks like in four dimensions is known as the Kaluza-Klein expansion. The specific model of these two men is no longer in favour, as it implies that electric charge and mass should be correlated (they are not).
Modern theories relevant to experiments like the LHC go back to only 1998, when Nima Arkani-Hamed, Savas Dimopoulos, and Gia Dvali proposed the existence of large extra dimensions. They where inspired by string theory, so large means at most millimetre-sized. This is still much, much bigger than the Planck scale; more to the point, it can lead to phenomena observable at current experiments.
Based on my earlier post, you might wonder how extra dimensions as big as a millimetre are not trivially ruled out. After all, the human eye can resolve things down to a fraction of a millimetre; while with optical microscopes we can look at things at the micrometre scale or even smaller. The answer has to do with the structure of spacetime; in short, things like light and matter are constrained to a four-dimensional brane1 within the higher-dimensional space.
Branes are objects that show up all over the place in string theory, and were first discovered in the mid-90s. You can think of a brane as a bubble wall, which is a two-dimensional surface in a three-dimensional space. In much the same way, it is possible to have a four-dimensional hypersurface in a larger spacetime. Without going into a lengthy diversion about string theory, it is natural to confine anything and everything to branes except gravity.
So the size of the extra dimensions in the ADD scenario is limited not by measurements made with light, which are very precise; but rather, by those with gravity, which are much less so. Hence the relatively large size that is allowed. In practice, even a millimetre is ruled out by indirect constraints, though there is some freedom in what assumptions you make.
The main motivation behind all of this is the hierarchy problem. If gravity can travel through other dimensions but other stuff cannot, then gravity will appear weaker; some of the gravitational attraction has 'leaked out' away from our brane. To see how this affects the gravitational mass scale, remember Newton's Law:
A weaker gravitational force means that G becomes smaller. But G is related to the Planck scale aswhere c is the speed of light and h is Planck's constant. So, gravity appearing weaker than it actually is means that its fundamental mass scale is increased. Turning this around, the existence of extra dimensions would lower the mass scale; if it can lower it to equal the electroweak scale, the hierarchy problem is solved.
Finally, I should comment that this idea is under pressure from LHC measurements.2 The search is based on the existence of gravitational resonances in the extra dimensions, as well as the possibility of low-mass black holes. The resonances provide the tighter constraints, since the signal is cleaner; it rules out the true scale of gravity being below several TeV, or over ten times the electroweak scale. Still, the experimenters will continue to look, since even if extra dimensions don't solve the whole hiearchy, they might be able to solve some of it.
1. So my title makes sense!↩
2. Once again the LHC has failed to find anything interesting :)↩