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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, ...)

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    $\begingroup$ 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. $\endgroup$
    – M.G.
    Commented Nov 17, 2017 at 9:31
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    $\begingroup$ 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 $1+3$. $\endgroup$
    – M.G.
    Commented Nov 17, 2017 at 9:38
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    $\begingroup$ Also see the physics.SE discussion physics.stackexchange.com/q/10651/99268 $\endgroup$
    – Nemo
    Commented Nov 17, 2017 at 12:58
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    $\begingroup$ 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. $\endgroup$
    – Will Brian
    Commented Nov 17, 2017 at 14:38
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    $\begingroup$ Related: mathoverflow.net/q/47569/13917 $\endgroup$
    – Qmechanic
    Commented Nov 17, 2017 at 16:17

3 Answers 3

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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

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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).

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    $\begingroup$ 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.) $\endgroup$
    – Marco Golla
    Commented Nov 17, 2017 at 14:54
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    $\begingroup$ It is also conceivable (though unpleasant to think of) that we could eject unwanted matter by the same "opening" by just reversing the direction. $\endgroup$
    – tst
    Commented Nov 17, 2017 at 16:28
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    $\begingroup$ but what is the problem with being disconnected? many of us are, in a sense $\endgroup$
    – Pietro Majer
    Commented Aug 25, 2021 at 8:57
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    $\begingroup$ You might look at Dewdney's Planiverse to see how these restrictions on two-dimensional animals can be circumvented. $\endgroup$
    – rimu
    Commented Aug 25, 2021 at 10:09
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    $\begingroup$ We are not solid beings; our cells do move around, and molecules in them break up and combined. Such a 2D animal could process food like our cells do: let it in through a membrane; close the membrane behind it; absorb it; and expel through the membrane. $\endgroup$
    – Michael
    Commented Aug 25, 2021 at 16:18
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One answer in the form of a paper is Tegmark’s “On the dimensionality of spacetime” at https://arxiv.org/abs/gr-qc/9702052

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