[Physics] Determining the value of Pi empirically


A shower thought hit me so hard I had to come here and ask:

Outside pure mathematics, where should I look, if I wanted to physically measure the best possible approximation for $\pi$? How many significant digits would I be able to get, assuming I had the best contemporary measuring equipment (and if necessary, fabrication facilities) available?

I was originally thinking planetary orbits, but actually measuring those to several significant digits seemed a bit difficult. Any other inverse-square force should also make pretty ellipses, and rotating things make nice circles and sine waves, but is there some especially handy thing or phenomenon that would give an exceptionally good approximation for $\pi$?

If at all possible, it would be nice to have an answer that takes all or most actual measurement problems (like possible disturbances, measurement errors, systematic errors, and the actual surroundings of the experiment) into account.

(Full disclosure: since it was a literal shower thought, and I've been thoroughly influenced by M. Hartl's propaganda, the original question in my mind was "does $\tau$ really exist")

Following the advice of user stafusa, I removed my request for help on improving this question so that we could get this question reopened. (I had tried to keep it constructive but admittedly it did show some unnecessary indignation.)

However, any such help would still be very much appreciated. Thanks in advance!

(I am not familiar with the meta question process, so if you think that would be the proper course of action, please feel free to open such a question.)

Best Answer

Lazzarini's experiment with Buffon's needle is as far I know the only case where somebody has actually tried to "measure" $\pi$ empirically using a physical process (and there is a bit of a cheat/hoax involved). But in principle this could be used to measure the constant. As noted here, using longer needles can make convergence faster.

One can do this kind of thing with other random methods (even a version of the Monty Hall problem). Whether this counts as a physical measurement is somewhat arguable: certainly a physical process of experimentation is occurring, producing an empirical result, but the setup has been such that the result should approach $\pi$. Measuring 1/2 by flipping a coin is the same: the tricky part is getting a fair coin (experimental setup) - most likely the setup will have to be done by checking candidate coins for fairness, that is, ensuring that one gets a known result.

Still, in principle one could run through any physics formulary and get ways of estimating $\pi$. For example, an oscillating spring has period $T=2\pi\sqrt{m/k}$, so $\pi = T/2\sqrt{m/k}$. Planck's law for energy distribution of blackbodies gives $\pi = w(\lambda)\lambda^5 (e^{hc/\lambda kT}-1)/8hc$. So one could perform blackbody experiments and get approximations, although there has to be some care with the values of the ingoing constants - they have to be measured in a way that does not implicitly involve $\pi$.