I'm still wondering why space is 3-dimensional.
This may seem pedantic, but a lot of assumptions brake down when you look at tiny or high energy things. So, it's really important to mark down all your assumptions.
The mathematics that model space fit conveniently in 3 dimensions without becoming unmanageably complex.
However, by employing additional dimensions, particularly dimensions in which the basis vector multiplied by itself is a product other than itself, you can sometimes simplify the math that describes portions of the universe to a shocking extent. For instance, using geometric algebra with one dimension that squares to positive one and three that square to negative one, Maxwell's Equations reduce to "nabla field_bivector = free_space_permeability * speed_of_light * current_vector".
Sometimes, introducing special-purpose dimensions with interesting geometries and constraints upon the multidimensional representation makes certain types of math easier. For instance, conformal representation employs two dimensions that square to zero to represent the origin and infinity, and regular space is represented as a curved subset of the hyperdimensional space. Otherwise complicated operations become simpler, as a translation and rotation through dimensions that do not exist that almost coincidentally lands the result right back on the constrained hypersurface that represents 3-dimensional space.
When we invent these extra dimensions for the purposes of doing the math, we have no good way of knowing whether they are entirely imaginary or just undetectable and inaccessible to us at our current level of technology. If the universe had a round dimension with a diameter of less than the Planck distance, we would not be able to detect it, because we can't measure a distance that small. But maybe certain properties in the particle zoo can be more easily explained using an angular phase value on a tiny round dimension perpendicular to everything else.
Some non-string theory attempts at formulating quantum gravity have started without an assumption of space-time and attempted to make it an emergent property, and given 3-dimensional space's quite unique properties that feels like it might ultimately be a better course of reasoning.
http://www.reddit.com/r/askscience/comments/42u57e/how_can_a...
A three-dimensional space can be close to flat but have positive curvature as well (or negative curvature, for that matter). Some proposals give our universe positive curvature, rendering its space finite, though still stupidly-big. (I'm not aware of there being a final word on the subject, though). And if different dimensions can have different curvatures, some of them could be much smaller than others.
Of course, if we a assume a multiverse with universes having regularly distributed, positive curvatures, then, the size of the universe would grow asymptotically as the curvature approached flat, so, statistically, life is more more likely to arise on the flatter universes.
That would probably be more easily measured though. I don't know.
A small non-zero curvature would correspond to a non-zero cosmological constant, which is precisely what seems to be driving the accelerated expansion of the universe (there are other explanations but this is the simplest one).
/s ;-)