The specificity of dimension $1+3$ for the real world I have been asked sometimes, and I ask myself, to what extent the dimension $1+3$ is important for our real world, say compared to an hypothetical $(1+d)$-dimensional world. I have two answers in mind.

*

*The Huygens principle. If you switch off a point source of light, then a point situated at distance $L$ will be in dark after time $\delta t=L/c$ ($c$ the speed of light). This would be false in dimension $1+2$ for instance, even if the energy would be very low after $\delta t$.

*Chemistry is a consequence of quantum mechanics. Mathematically it involves the linear representations of the rotation group. In $1+2$ dimensions, the group is $SO_2$, which is abelian and isomorphic to a circle ; its representations are one-dimensional, associated with linear characters. In our world, the groups $SO_3$ is not abelian and the situation is way richer. In particular, we have a notion of spin.


What are other manifestations of the dimension $1+3$ in our real world ?

In order to limit this discussion to a reasonable extent, I assume that the Physics of a hypothetical world would be based on equations similar to those we already know. In particular, second-order differential operators would be at stake, because of their nice mathematical properties (maximum principle, ...)
 A: It was shown by Paul Ehrenfest in 1917 that Coulomb interaction is unstable at spatial dimensions higher than three,

*

*P. Ehrenfest, In what way does it become manifest in the fundamental laws of physics that space has three dimensions?, Proc. Netherlands Acad. Arts Sci., 20 (1917), 200 - 209 (pdf).

To quote:

In $R_n$ for $n>3$ the planet falls on the attracting centre or flies away infinitely. In $R_n$ for $n>3$ there do not exist motions comparable with the elliptic motion in $R_3$,- all trajectories have the character of spirals.

This analysis has been extended to quantum mechanics in

*

*Tangherlini, F. R. (1963). Schwarzschild field in $n$ dimensions and the dimensionality of space problem, Nuovo Cimento. 14 (27): 636 https://doi.org/10.1007%2FBF02784569
A: One answer in the form of a paper is Tegmark’s “On the dimensionality of spacetime” at https://arxiv.org/abs/gr-qc/9702052
A: Hawking has an extensive discussion in “A Brief History of Time”. Excerpt:


Two space dimensions do not seem to be enough to allow for the development of complicated beings like us (...) If a two-dimensional creature ate something it could not digest completely (...) because if there were a passage right through its body, it would divide the creature into two separate halves: our two-dimensional being would fall apart (Fig. 11.8).

