The title of this post is misleading, because you might think there's a substantial connection between its two halves. The connection is only that I happened to come across both of these things this morning and it seemed silly to make separate posts about them.
Todd Trimble gives two proofs of the continued fraction expansion for e, namely e = [2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, ...]. This is in the usual notation for continued fractions, so it actually means
e = 2 + 1/(1 + 1/(2 + 1/(1 + 1/(1 + 1/(4 + ...)))))
but all the extra 1s obscure the pattern. This is one of those things that it's much easier to state than it is to prove. And I must admit it's always seemed a bit strange to me that e has such a nice continued fraction expansion, while π doesn't -- e has a special place in continued-fraction land.
From the arXiv, via the physics arXiv blog: A Monte Carlo Approach to Joe DiMaggio and Streaks in Baseball, by Samuel Arbesman and Steven Strogatz, which is what it sounds like. This expands upon this piece in the New York Times on the same subject, which I wrote about back in March. And no, I'm not sure why it's in a physics category at the arXiv.