In anti-honor of “Pi Day,” I’d like to direct your attention to Michael Hartl’s The Tau Manifesto. The Manifesto is inspired by Bob Palais’ article is wrong! and presents a list of simple, but compelling, reasons that , not , is the more fundamental constant.
These ideas have been discussed on the blathosphere before, e.g. on Bill Gasarch and Lance Fortnow’s blog Computational Complexity. There Terence Tao makes the following remark:
It may be that is an even more fundamental constant than or . It is, after all, the generator of . The fact that so many formulae involving depend on the parity of is another clue in this regard.
The basic argument for this point of view can be summarized as follows: is a special function because it is the unique eigenvector of eigenvalue of the derivative operator acting on, say, complex-analytic functions on , and this function has period . So we see that this constant pops directly out of a definition of and a definition of the derivative of a complex-analytic function: no arbitrary choices were necessary. (The closest thing to an arbitrary choice here is the decision to identify the tangent space of a point in a vector space with the vector space itself, but this is completely invariant.)
The here is precisely the circumference of a unit circle, which is distinguished among all circles because in it is the only circle of positive radius closed under multiplication. This is a fundamental number because of the relationship between the unit circle and Pontrjagin duality (which has the Fourier transform and Fourier series as special cases), and is responsible for all appearances of in mathematics that I know of.
For example, the reason there is a factor of in the definition of the Gaussian distribution (which is where the factor of comes from in Stirling’s formula) is that the Gaussian distribution is its own Fourier transform. This factor is commonly cited as an application of that has nothing to do with circles, but of course the Fourier transform has everything to do with circles.
Edit, 3/15/11: Vi Hart also explains the wrongness of in video form. I have to admit I think I read the title of her post and then promptly forgot I had done so when writing this post.