# Pi Day vs. Half Tau Day

By now my views on Pi Day are well documented (see earlier posts from 2011 and 2009 if you’re curious).  Recently, though, I’ve decided to try to be a little less curmudgeonly when it comes to math holidays.  Consequently, while it would be easy to provide snarky commentary on articles with particularly egregious mathematical errors, this year I will try to restrain myself.

As I’ve said before, one of my biggest problems with Pi day is that the activities are, for the most part, a little ridiculous, and don’t actually do anything to better the general populace’s understanding of mathematics.  Last year, I explained why contests involving the recitation of digits of $\pi$ are silly, so this year I’d like to offer an alternative.  Why not use the day as an opportunity to debate with students the relative merits of $\pi$ and $\tau$?

Of course, I’m talking about more than Greek letters here; I’m talking about what these letters represent, at least in certain circles (no pun intended).  $\pi$, the golden child of mathematical constants, known by all (though adored by few), represents the ratio of a circle’s circumference to its diameter.  $\tau$, on the other hand, has recently taken on a new identity as the ratio of a circle’s circumference to its radius - in other words, $\tau = 2\pi$.

In the last few years, a debate has emerged over which constant is more fundamental.  Some argue that $\tau$ should be viewed as the true circle constant, rather than $\pi$.  I’ve written about this subject here and here, so you can view the links if you want more details, but good primary sources for the argument in favor of $\tau$ include the Dr. Michael Hartl’s Tau Manifesto and Pi is Wrong! (the latter, in fact, helped to inspire the former).

On the other hand, last year a rebuttal by the name of the Pi Manifesto emerged, written by an author who identifies him or herself only as MSC.  This article attempts to reclaim $\pi$‘s status as top dog by arguing, essentially, that the Tau Manifesto cherry picks formulas in which $2\pi$ naturally appears, but ignores other situations in which $\pi$ itself seems to be the star.  Both manifestos are worth reading, and for students in Geometry or Trigonometry, a debate about the relative merits of these constants could prove quite instructive.

I’d like to point out, though, that there are essentially two questions are being asked, and only one of them is really interesting.  In my mind, the two questions are:

(1) Which constant is more canonical?

(2) Which constant is more pedagogically valuable?

The first question is typically the one mathematicians answer, but the question of $\tau$ vs. $\pi$ isn’t much of an interesting question, mathematically speaking.  After all, any argument involving $\pi$ can be made into an argument involving $\tau$ by a simple substitution $\pi = \tau/2$.  Similarly, any argument involving $\tau$ can be made into an argument involving $\pi$.  So, from a strictly mathematical standpoint, the question of which notation to use doesn’t really matter.

This point is frequently overlooked when this debate is mentioned in the media.  Gimmicky titles like “Pi is Wrong!” don’t really help.  After all, there’s nothing wrong with the mathematical definition of $\pi$ – the issue is whether the ratio of a circle’s circumference to its diameter is a more or less natural thing to consider compared to the ratio of a circle’s circumference to its radius, but either ratio gives rise to a perfectly well-defined mathematical constant.

Regardless of your response the first question, I think the second question is the more interesting one, and based on my own teaching experience, this is where I’d give $\tau$ a slight advantage.  The most compelling reason for using $\tau$ as a teaching tool comes from trigonometry – specifically, determining the sines and cosines of angles is much easier to understand for first timers if things are written in terms of $\tau$ rather than $\pi$.  These trig values are notoriously difficult for students to remember, but thinking in terms of $\tau$ makes things more intuitive.  These pictures, taken from the Tau Manifesto, provide a clearer picture:

Angles as fractions of a circle.

Angles as fractions of tau.

Many students have trouble with trigonometry; if using $\tau$ instead of $\pi$ makes things easier to understand, then it’s certainly worth discussing.

So this $\pi$ day, let’s set aside the silly contests and ridiculous news articles, and instead think about some actual mathematics.  Please note, though, that I still consider it acceptable to commemorate the holiday with a slice of pie (or two!).