# Happy Tau Day, the true circle constant

June 28, 2018

You’ve probably heard of Pi Day, perhaps the most popular of geeky holidays. But I’m here to tell you that Pi Day is wrong — or rather, the entire idea of pi as a mathematical concept is wrong.

It’s easy enough to see why people like Pi Day: the whole thing starts with a mathematical pun of sorts (The date is written as 3/14 in American notation. Pi starts with the digits 3.14. You get it.) It’s an easy, fun ritual to see how many digits you can pointlessly memorize of the famously never-ending, never-repeating number (even though 39 digits is more than sufficient for almost any calculations you’ll ever need). Plus pi sounds like pie, and who doesn’t like pie?

But here’s the thing: π as a number is bad, and therefore, so is the entire misguided day dedicated to its celebration. It’s a lot to take in, and I, too, was once like you: I was taught the virtues of pi for years, going back to Pi Day parties in middle school. But instead of pi, we should celebrate tau, an alternative circle constant referred to by the Greek letter τ that equals 2π, or approximately 6.28.

I’m not just making this up out of nowhere: the terribleness of pi as a constant was first proposed by mathematician Bob Palais in his article “π Is Wrong!” and later expounded upon in *The Tau Manifesto* by Michael Hartl, which serves as the basis for modern tauism. (Internet-famed mathemusician Vi Hart is also a major proponent of tau over pi, if you prefer your mathematical arguments in a more entertaining video form.)

But Palais and Hartl’s arguments both boil down to some basic math. Step back in time to when you first learned geometry and recall the simple origins: no matter what circle you’re using, if you divide the circumference of the circle by the diameter, you’ll get the same answer: an endless number, starting with the digits 3.14159265… (aka pi).

And right there is the fundamental flaw. The thing is, we don’t actually use diameter to describe circles. We use the radius, or one-half the diameter. The circle equation uses the radius, the area of a circle uses the radius, and the fundamental definition of a circle — “the set of all points in a plane that are at a given distance from a given point, the center” — is based on the radius. Plugging that into our circle constant equation gives us a new circle constant equivalent to 2π, or 6.28318530717…, colloquially referred to with the Greek letter τ (tau). Switching to τ isn’t making some arbitrary change for the sake of it. It’s bringing one of the most important constants in math in line with how we actually *do *math.

Now, you may be thinking that this will cause fundamental, seismic changes in math. “How on earth could you possibly replace something as important as pi!?” you could ask. But if we’re being honest, π isn’t really something we use in day-to-day math to start with. Unless you’re someone who does a lot of geometric calculations in your daily life, chances are you only ever really encounter pi when it comes time to rattle off some digits for Pi Day. Sure, it’s a good introduction to the idea of irrational numbers, but tau would work just as well for that. And if you *do* work with π a lot, replacing it with τ is beneficial for a whole host of reasons, mathematically speaking. Again, I’ll direct you to *The Tau Manifesto* for the full argument, but I’ll just point out a few here.

One big thing tau fixes is radian angles. You may remember that as “those annoying chunks of a circle represented by weird fractions of pi from high school math,” but with tau, it’s simple: everything matches up where it should fractionally. So half the circle (180 degrees) — τ/2. 1/12th? τ/12. It’s a small change, but it makes angular notation — a frustratingly obtuse part of geometry that through the use of pi demands an elitist notion of memorization of angles and conversions — a more welcoming and intuitive prospect for new students.

It also makes circle functions like sine and cosine easier, since it makes one full cycle of the function match up with one full turn of the circle (tau), instead of the seemingly arbitrary 2π that you get using π as your circle function. Like with radian angles, it makes deriving sine and cosine values a simple process from simply drawing the function, instead of demanding that students remember that 3π/2 is for some reason the three-fourths point on the wavelength.

Similarly, it makes a bunch of other higher math — like integrals in polar coordinates, the Fourier transform, and Cauchy’s integral formula simpler, since they also already work in terms of 2π anyway. Using tau just cuts out the middleman. Looking back at years of math and physics notes with the enlightened lens of tau, I weep for my former self and the accumulated hours of needless conversions and complications introduced by pi.

It’s not just practical purposes, though. Replacing π with τ makes mathematics more elegant overall. And at the heart of it, isn’t that what we aspire to do with math? The universe is vast and almost impossible for us to ever fully comprehend, but by distilling it down into a system of logical numbers and symbols, we can make some order out of the chaos. So why not embrace a circle constant that makes our equations and formulae more beautiful?

Unfortunately, pi is probably too well-ensconced in traditional mathematics for us to ever break free from its tyrannical hold. Math textbooks still espouse the virtues of pi, and instilling such a systemic change to how we teach math is likely an uphill battle. (On the other hand, Common Core somehow seems to have managed it, despite its — to my eyes — incredibly obtuse nature, but go figure.) And that’s a shame, given how much more sense tau makes as a circle constant for even the more basic functions that we do use pi for. But the first step is to stop glorifying pi, so I won’t be celebrating Pi Day this year — and neither should you.

But all isn’t lost for those looking for a fun day to celebrate math: Tau Day (6/28, or June 28th) is here, today!

**Update June 28th, 9:00am: ***Updated post for Tau Day 2018! *

**Correction***: Accidentally transposed two digits of pi. *