## Happy Tau Day?

In the past, I’ve used this blog as a platform to make clear my mixed feelings about Pi Day, a math themed holiday celebrated every year on March 14th (3/14, har har) in honor of the beloved mathematical constant π.  My thoughts on the subject can be found here.

It would seem that I am not alone in my frustration.  Michael Hartl, an educator and entrepreneur (as well as a Ph.D. graduate from Caltech), has just today launched a website in favor of Tau Day as a replacement for Pi Day.  However, his argument (based on a 2001 paper by Bob Palais) goes a step farther - he argues that π day shouldn’t be celebrated because π isn’t the fundamental constant we should be considering!  Rather, he argues that the true fundamental constant is 2π, which is approximately 6.283185… .  Hartl argues that this should be the fundamental constant of interest, and renames it τ (for reasons given on the website).

Why should this be viewed as a more fundamental constant?  Recall how π is defined - it is the ratio of a circle’s circumference to its diameter.  But a circle itself is more naturally defined in terms of the radius, i.e. as the set of points whose distance from the center is equal to the radius.  Because of this, doesn’t it seem more natural to consider the ratio of a circle’s circumference to its radius, rather than the ratio of circumference to diameter?  Put another way, isn’t a more natural constant given by the circumference of a circle with radius 1 rather than the circumference of a circle with radius ½?  He offers plenty of other aesthetic examples for why 2π should be viewed as more fundamental, including references to the Bernoulli numbers and simple quadratic forms.

On the one hand, this may seem like a trivial issue - after all, the difference between π and τ is only a factor of 2, and different normalizations of quantities are quite common in mathematics.  On the other hand, Hartl does make a convincing argument from a pedagogical point of view.  His strongest argument comes from trigonometry.  When students learn to convert between radians and degrees, they learn that 2π corresponds to full revolution.  From this, one sees that half of a revolution corresponds to an angle of π, ¼ of a revolution corresponds to an angle of π/2, and so on.  But if we define the fundamental quantity to be τ, then in radians, half a revolution is τ/2, a quarter of a revolution is τ/4, and the measure of c revolutions is given by cτ for any number c.

Hartl concludes the following: “The unnecessary factors of 2 arising from the use of π are annoying enough by themselves, but far more serious is their tendency to cancel when divided by any even number. The absurd results, such as π/2 for a quarter circle, obscure the underlying relationship between angle measure and the circle constant. To those who maintain that it “doesn’t matter” whether we use π or τ in teaching trigonometry … from the perspective of a beginner, using π instead of τ is a pedagogical disaster.”

It’s an interesting argument, and one I think students would benefit from seeing. π is fairly entrenched, so I’m not sure how much of a following Hartl will gain, but even if π remains the standard, offering students this viewpoint can only help them as they learn trigonometry.  For that reason, I for one will be endorsing Tau Day (6/28, get it?).  It certainly doesn’t sound as delicious as Pi Day, and the fact that students are out of school is a bit of a problem, but today is apparently the inaugural Tau Day, and these are wrinkles that I’m sure can be ironed out.

So happy Tau Day to you, no matter your preference!

(Big ups to James Hawkins for sending me the Tau Day link.)