These numbers seem to be singled out in the search for a fundamental theory of matter. The more you probe the fundamental structure of matter, the simpler things seem to become. In developing new theories that can encompass the current ones, scientists look for more simplicity in the form of symmetry. In addition to being elegant, symmetry is useful in constraining the number of competing models. The more symmetry there is, the fewer models that fit that symmetry exist.
One useful such symmetry is called supersymmetry, which connects matter in the form of fermions with force carriers in the form of bosons. This is an elegant symmetry relating seemingly different aspects of our universe. Although this symmetry is still theoretical, the Large Hadron Collider, scheduled to start operation by the end of the decade, will look for it experimentally. Fermions and bosons differ by the property known as spin. In quantum units, fermions carry half-integer spin, whereas bosons have integer spin. Supersymmetry relates the spin of particles differing by one-half. For example, the electron, which has spin , is thought to have a partner called the selectron, which has spin 0; in this sense the electron and the selectron are mirror images. All their properties are related to each other by the symmetry. So, too, the boson and fermion can be related in this symmetry.
But there can be more than one supersymmetry, just as there is more than one way to position a mirror. A single supersymmetry connects a boson to a fermion. If there are other such symmetries, they connect more bosons and fermions and thereby unify more aspects of our universe. For example, with additional supersymmetry, the electron and the selectron would have additional partners of spin 0 and 1. The symmetry would also restrict the form in which these partners can interact with each other.
Ultimately, though, too much symmetry simplifies the theory to the point of being trivial. All the particles are unable to interact with each other or with our measuring devices. This is certainly not a good thing for a theorist to construct, so the goal is to get the greatest amount of symmetry that still allows for interesting physics.
A guide in this pursuit is a theorem devised by physicists Steven Weinberg and Edward Witten, which proves that theories containing particles with spin higher than 2 are trivial. Remember each supersymmetry changes the spin by one half. If we want the spin to be between -2 and 2, we cannot have more than eight supersymmetries. The resulting theory contains a spin-2 boson, which is just what is needed to convey the force of gravitation and thereby unite all physical interactions in a single theory. This theory--called N=8 supergravity--is the maximally symmetric theory possible in four dimensions and it has been a subject of intense research since the 1980s.
Another type of symmetry occurs when an object remains the same despite being rotated in space. Because there is no preferred direction in empty space, rotations in three dimensions are symmetric. Suppose the universe had a few extra dimensions. That would lead to extra symmetries because there would be more ways to rotate an object in this extended space than in our three-dimensional space. Two objects that look different from our vantage point in the three visible dimensions might actually be the same object, rotated to different degrees in the higher-dimensional space. Therefore all properties of these seemingly different objects will be related to each other; once again, simplicity would underlie the complexity of our world.
These two types of symmetry look very different but modern theories treat them as two sides of the same coin. Rotations in a higher-dimensional space can turn one supersymmetry into another. So the limit on the number of supersymmetries puts a limit on the number of extra dimensions. The limit turns out to be 6 or 7 dimensions in addition to the four dimensions of length, width, height and time, both possibilities giving rise to exactly eight supersymmetries (M-theory is a proposal to further unify both cases). Any more dimensions would result in too much supersymmetry and a theoretical structure too simple to explain the complexity of the natural world.