What are some interesting cases of π appearing in situations that do not seem geometric?
Ever since I saw the identity ∞∑n=11n2=π26
and the generalization of ζ(2k), my perception of π has changed. I used to think of it as rather obscure and purely geometric (applying to circles and such), but it seems that is not the case since it pops up in things like this which have no known geometric connection as far as I know. What are some other cases of π popping up in unexpected places, and is there an underlying geometric explanation for its appearance?
In other words, what are some examples of π popping up in places we wouldn’t expect?
Suppose we iterate the function f(z)=z2+c starting at z0=0. For example, if c=1/4, the first few terms are
It can be shown that the sequence converges slowly up to 1/2. On the other hand, if c=1/4+δ, where δ>0 (no matter how small), then the sequence diverges to ∞. This corresponds to the fact that c=1/4 is on the boundary of the Mandelbrot set.
We now ask the following: given δ>0, how many iterates N does it take until zN>2? Here are the answers for several choices of δ:
δnumber of iterates until escape0.01310.00013130.00000131410.0000000131415
In fact, if N(δ) represents the number of iterates until the iterate value exceeds two, then it can be proved that