Three dimensions

A passage from an influential thesis:

In many ways, we are fortunate to be living in a universe with exactly three spatial
dimensions. It keeps us from falling apart, allows us, if we are willing, to see things
as they really are, makes it possible, perhaps with some practice, to communicate in
a clear and coherent manner, and it provides some of the more advanced members of
civilization with the ability to tie their shoes [1, 2].

Perhaps these frivolous statements deserve some explanation, or at least a translation from their seemingly nonsensical form into something physically meaningful. We begin by pointing out that Newton’s 1/r^2 force-law [3], which arises for Gaussian
central potentials associated with gravitational and electric point charges, is particular to three spatial dimensions. As shown in [4, 5], a Gaussian central potential in D
dimensional space generates a 1/r^(D−1) force law, and this only permits stable orbits
when D = 3. Indeed, this implies that without exactly three spatial dimensions, we
would lose the stable orbits that keep our structure intact from astrophysical scales
down to atomic scales. (Similar results arise from such considerations in the framework of general relativity [6].) Another point of clarification is that transmission of
information signals via light or sound waves is only reverberation-free and distortionless for radiation in D =1, 3 spatial dimensions [7]. Finally, another seemingly
innocuous, but rather important, fact is that three is the exact number of dimensions
that permits nontrivial knots to exist. Any fewer dimensions, and it is impossible to
form a knot in a strand, since there is no “under” or “over,” just “next to.” Any more
dimensions and there is too much spatial freedom, which will make knots unravel,
since one can always move one strand past another by pushing it into one of the extra
dimensions, where it may pass unhindered.

Knowing that three is an interesting dimensionality for space that grants some
rather nice properties, one might be inclined to ask whether three might also be
an interesting dimensionality for spacetime. Indeed, this turns out to be the case,
primarily because of the property regarding whether nontrivial knots are allowed to
exist and the effect this has on particle statistics. In fact, it is exactly this property
that requires particles in three (or more) spatial dimensions to exhibit only the well-
known bosonic [8, 9] and fermionic [10, 11] statistics that play such a crucial role in
the structure and interaction of matter in the universe.