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.