# 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 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, ...)

• I'm sure you already know this, but the (general) Huygens principle for the wave equation testifies as to the importance of having odd number of spatial dimensions rather than that odd number being exactly $3$. For completeness sake, it also applies to sound. – M.G. Nov 17 '17 at 9:31
• With more spatial dimensions we wouldn't be able to tie our shoe laces. Moreover, $3$- and $4$-manifolds are rather mysterious both in their exceptional behavior and the difficulties they present (as opposed to their higher-dimensional counterparts). I always had a feeling that this should be somehow physically significant as to why $1+3$. – M.G. Nov 17 '17 at 9:38
• Also see the physics.SE discussion physics.stackexchange.com/q/10651/99268 – Nemo Nov 17 '17 at 12:58
• Martin Rees explores this question a little bit in chapter 10 of Just Six Numbers (written for non-professionals). Glancing through it, I think his strongest point concerning why 1+3 is special is the one mentioned in Nemo's answer: orbits are stable under a force obeying an inverse square law, but not under one obeying an inverse cube law. – Will Brian Nov 17 '17 at 14:38
• Related: mathoverflow.net/q/47569/13917 – Qmechanic Nov 17 '17 at 16:17

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.:

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). "Atoms in Higher Dimensions". Nuovo Cimento. 14 (27): 636.

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).

• Well, conceivably we could be once-punctured tori living on a high-genus surface, and have a non-separating digestive trait. (But motion would be a bit weird.) – Marco Golla Nov 17 '17 at 14:54
• It is also conceivable (though unpleasant to think of) that we could eject unwanted matter by the same "opening" by just reversing the direction. – tst Nov 17 '17 at 16:28

One answer in the form of a paper is Tegmark’s “On the dimensionality of spacetime” at https://arxiv.org/abs/gr-qc/9702052