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https://en.wikipedia.org/wiki/Spacetime#Privileged_character_of_3.2B1_spacetime

Does Coulomb's Law, with Gauss's Law, imply the existence of only three spatial dimensions?

Why are so many forces explainable using inverse squares when space is three dimensional?

This wikipedia article claims that the existence of inverse square laws (like Newton's Law of Gravitation) has a connection with the fact that our universe has three large scale spatial dimensions.

Can someone point me to some book/paper where this topic is treated in a detailed and formal manner?

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I think what you're looking for is the divergence theorem: https://en.wikipedia.org/wiki/Divergence_theorem

It tells you that the flux of a vector field through a surface is equal to the integral over the divergence inside the Volume.

Since in 3 Dimensions Objects have surfaces scaling with $r^2$ and mass, charge etc. can be seen as positive divergence, or sources, of the corresponding force fields, we get inverse square laws. in Dimension d however surfaces scale with $r^{d-1}$ and we get $\frac{1}{r^{d-1}}$ laws.

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It's pretty simple. In 3 dimensions when you transmit a wave uniformly in all directions the energy at any distance as it propagates is the same. Since the area of a sphere at distance r is $4\pi r^2$ the energy per unit area at any distance is inversely proportional to $r^2$. If the spatial dimensions were n it'd be inversely proportional to $r^{n-1}$

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