Happy Tau Day, 2016!

Happy Tau Day everyone! I've talked about tau before, but if you're new (or don't remember), tau is what pi should have been: instead of the ratio of the circumference of a circle to the diameter, tau is the ratio of the circumference to the radius, which is much more natural given circles are defined by their radii. Mathematically, \(\tau=C/r\). Numerically, the first couple of digits are 6.283185… (hence why June 28 is Tau Day!).

For an excellent introduction to tau and why it's a better choice than pi, look no further than The Tau Manifesto by Michael Hartl. He explains much better, and in much more detail than I can here, why tau makes much sense mathematically, and, perhaps even more importantly, how it simplifies learning more advanced math like trigonometry.

I've talked about tau before, so today I'm just going to briefly explain an experiment you can do to estimate tau. This uses a problem known as Buffon's needle, originally posed in the 18th century. The problem can be stated as:

Suppose we have a floor made of parallel strips of wood, each the same width t, and we drop a needle of length l onto the floor. What is the probability that the needle will lie across a line between two strips?

If we assume that the needle is shorter than the distance between two lines (not necessary, but it simplifies things), then the answer (given in full on the Wikipedia page linked above) is simple: the probability P is given by \(P=\frac{2l}{t\pi}\). This formula can be rearranged to find pi: \(\pi=\frac{2l}{t P}\). Remembering that \(\pi=\frac{1}{2}\tau\), we get \(\tau=\frac{4l}{t P}\).

If you drop n needles (or toothpicks, or matchsticks, etc.), and h of them cross a line, then the probability P can be approximated by \(h/n\). This mean you can approximate tau by:
\[\tau\approx \frac{4l\,n}{t\,h}\]The more needles you drop, the better your estimate is likely to be (barring some sort of systematic bias in how you're dropping the needles).

Anyway, that's it for today! Have a great Tau Day, and do let me know in the comments if you actually try this experiment!