# Interesting and unexpected applications of $\pi$

$\text{What are some interesting cases of$\pi$appearing in situations that do not seem geometric?}$

Ever since I saw the identity $$\displaystyle \sum_{n = 1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$$

and the generalization of $\zeta (2k)$, my perception of $\pi$ has changed. I used to think of it as rather obscure and purely geometric (applying to circles and such), but it seems that is not the case since it pops up in things like this which have no known geometric connection as far as I know. What are some other cases of $\pi$ popping up in unexpected places, and is there an underlying geometric explanation for its appearance?

In other words, what are some examples of $\pi$ popping up in places we wouldn't expect?

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Is there any known explanation for $\pi$ being there? I wouldn't have expected $\pi$ to pop up when dealing with factorials of all things. – Soke Feb 25 '14 at 2:39
@user92774 The explanation actually would make a nice addition to my answer below. It's a special case of the central limit theorem. Binomial distributions tend toward the normal distribution for large $n$, which relates the $n!$ in the normalization factor of the binomial distribution to the $\sqrt{2\pi}$ in the normal distribution's normalization factor. – David H Feb 25 '14 at 3:07
@user92774 The $\sqrt{\pi}$ in the Stirling's formula comes from the Laplace's approximation of a uni-modal function by the Gaussian function which integrates to $\sqrt{\pi}$. So the real question is why it integrates to that. – Vadim Feb 25 '14 at 3:08
For the fun of it: $\pi=\left[\frac{2\sqrt{2}}{9801} \sum_{k=0}^\infty \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}\right]^{-1}$ – Julien Godawatta Feb 25 '14 at 3:35
@JulienGodawatta Good ol' Ramanujan. – Soke Feb 25 '14 at 3:47

What are some interesting cases of $\pi$ appearing in situations that are not geometric ?