I thought it might be fun to ask this question as a way of celebrating Pi Day. One way in which people popularize pi is that they say that even though it's defined in terms of properties of a circle, it shows up in contexts such as the normal distribution which have (apparently) nothing to do with circles, and this is supposed to say something about the mysterious and amazing character of mathematics. So I'm curious what MO have to say about this phenomenon. If I had to venture a guess, it would be something about Fourier analysis, but I know there are many people here who could make such a statement much more precise.
Answers should ideally be given in general terms, but feel free to illustrate your generalities with interesting examples.
Best Answer
Let me play devil's advocate here: I'm not sure that I agree that the ubiquity of π is so mysterious. After all, how do you ever prove that π appears? You have to relate your situation to some known situation where π already appears, so the mystery is solved almost before it occurs. Of course, if you're just shown the appearance of π without the proof then you may be surprised, but that simply means you haven't yet seen the proof. To take an example, the proof that $\int_{-\infty}^\infty e^{-x^2}dx$ involves π uses the rotational invariance of the normal distribution. But rotations are closely connected with circles and hence with π, so it isn't too surprising that π shows up. To take another example, it is amazing that $\sum n^{-2}=\pi^2/6$, but one nice proof of that uses Parseval's identity, calculating the $\ell_2$ and $L_2$ norms of the Fourier coefficients of a certain function and the function itself, respectively. And Fourier coefficients involve trigonometric functions, so the appearance of π is, once again, not a surprise.
Maybe the right thing to say is that the multiple appearances of π are initially striking and mysterious, but the mystery disappears on closer inspection -- like many mysteries. The statement I would dispute is that there is a general mystery of the kind "Why does π appear so much?" I'd give an answer like "Because circles and rotations appear a lot."