TITLE: What are some fundamental "sources" for the appearance of pi in mathematics?
QUESTION [17 upvotes]: 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.

REPLY [2 votes]: Here is another devil's advocacy: We (mathematicians) are not familiar with many transcendental numbers such as $\pi$ or $e$. When in a formula appears some transcendental number that can be expressed as a function of $\pi$, we are happy with that and we have our "solution" for the problem. But if it is not related to $\pi$ (or $e$, or etc...), we just think "OK, we don't have a solution yet." 
Another way of saying pretty much the same thing is the fact that since we know very well $\pi$ and its properties, as soon as it is "in the neighborhood" of our problem, we see it, but if it is not here (and neither $e$ or another famous constant), we do not know where exactly to look to find an acceptable solution.
My 2-cent guess is "the better we know a number (through formulas for instance), the more often it will appear in new formula in the future."