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 hypothetic $(1+d)$-dimensional world. I have two answers in mind.
- The Huyghens 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 reasonnable extent, I assume that the Physics of a hypothetic 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, ...)