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> Which is pretty insane because these two numbers are not supposed to be related

Not really, there is Euler's identity: https://en.m.wikipedia.org/wiki/Euler%27s_identity

That one's "just" a special case of how complex numbers happen to work. I think the really cool relationship between e and pi is the fact that the Gaussian integral acts as a fixpoint/attractor when sampling and summing data from any distribution (this is the central limit theorem):

∫(−∞ to ∞) e^(-x²) dx = √π

I think the attractor property makes it a little more fundamental in some sense, whereas Euler's identity is "just" one special case of e^ix. The Gaussian is kind of the "lowest energy" or "highest entropy" state of randomness, which I think is really cool.

This is only true for distributions with finite variance (and the edge case of distribution with slowly growing infinite variance).

And for a given variance, gaussian distributions are exactly the maximal entropy distribution.

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