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I would like to collect arguments whose conclusion indicate that the dimension of space cannot be anything.

For example, the existence of non-trivial knots implies that the dimension of space is $3$ (since in dimension $1$, there are no knots, in dimension $2$, they are all circles by Jordan's theorem, and in dimension $4$, they are all isotopic to circles since there is enough room to unentangle any curve).

As another example, a radial, divergence-free vector field on $\mathbb{R}^n \setminus \{0\}$ is proportional to $x \mapsto \frac{x}{\vert x \vert^{n}}$; so if the gravitational field created by a point mass is physically supposed to be divergence-free and to have some form, then the dimension of space has to be $3$.

As a kind of a joke, a teacher once told me that in dimension $2$, living beings like us could not exist, since their digestive tubes would disconnect their bodies.

I know that some developments of string theory lead to results like this, but I would expect more elementary examples.

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    $\begingroup$ The "joke" you quoted excludes a specific digestive strategy. Are you interested in other anthropic arguments? $\endgroup$
    – J.G.
    Oct 22, 2021 at 11:56
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    $\begingroup$ As for the detail you can't remember, $\nabla\cdot\frac{x}{|x|^m}=\frac{n-m}{|x|^m}$ (the proof is an easy exercise), which is $0$ iff $n=m$. $\endgroup$
    – J.G.
    Oct 22, 2021 at 12:10
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    $\begingroup$ The Planiverse describes a method where "zipper muscles" attach and detach to pass fluids and nutrients around inside the bodies of organisms, though the digestive systems have a single opening. $\endgroup$
    – notovny
    Oct 22, 2021 at 13:31
  • $\begingroup$ Thank you very much for the links and for the reminder of the exponent for the divergence-free field. I'll wait for other reactions! $\endgroup$
    – Plop
    Oct 22, 2021 at 13:45
  • $\begingroup$ Possible duplicates: physics.stackexchange.com/q/10651/2451 , physics.stackexchange.com/q/110876/2451 and links therein. $\endgroup$
    – Qmechanic
    Oct 22, 2021 at 14:12

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