All week, I’ve been looking forward to today, but I was mindful that it might be contentious and provoke rancor among my friends. But, ultimately, truth is more important than partisanship, so, come what may, I want to wish you all a very happy Tau Day!
As you may remember from math class, pi is approximately 3.14159 and, every year, on March 14th, math geeks tend to celebrate. Pi is what you get when you divide a circle’s circumference (the length around the outside) by the diameter (the length straight across. And it’s deucedly inconvenient.
We don’t normally define a circle in terms of its diameter, we use its radius. When you look at the ratio of a circumference to the radius you get tau = 6.283185…. = 2*pi. And that actually turns out to clean up a lot of the formulas and conventions you learned in trig. Look back to the video above to see how radians go from being that weird conversion you have to keep using your TI-89 to double check to a totally intuitive way to talk about angles. Pi takes you halfway around a circle (in radians, pi is equivalent to 180 degrees). Tau is equivalent to 360 degrees so if you want to talk about half a circle, you’re talking about tau/2. Three quarters round a circle is 3*tau/4. There’s none of this 3*pi/2 = 270 degrees nonsense (which I still had to look up to confirm, because it’s not intuitive).
It makes no more sense to have a separate symbol for τ/2 than it does to have a separate symbol for 1/2. Indeed, imagine we lived in a world where we used the letter h to represent “one half” and had no separate notation for 2h. We would then observe that h is ubiquitous in mathematics. In fact, 2h is the multiplicative identity, so how can one doubt the importance of h? But this is crazy: 2h is the fundamental number, not h. Let us therefore introduce a separate symbol for 2h; call it 1. We then see that h=1/2, and there is no longer any reason to use h at all.
This hypothetical scenario becomes reality in the case of circles: what is really going on here is that π is half of something. We have a standard symbol (π) for half a “circular unit”, but we have no standard symbol for the unit itself. Whether we use τ or some other symbol, the circular unit needs a name. If you ever hear yourself saying things like, “Sometimes π is the best choice, and sometimes it’s 2π”, stop and remember the words of Vi Hart in her wonderful video about tau: “No! You’re making excuses for π.” It’s time to stop making excuses.
In other words: