## Tau and Pi

### Interlude

You’re at a fancy restaurant and have just finished a sumptuous feast. You don’t think you could eat another bite when the waitress brings out the desert cart, and lo and behold there’s a steaming hot fresh apple pie. The waitress looks at you and asks, “how much pie would you like?”

Let’s say you want an eighth of a pie. You – who we’ll call the pie-eater – of course meant an eighth of the area of the pie. But the waitress, who we’ll call the pie-cutter, can’t cut the pie like a rectangular cake, in straight parallel and perpendicular lines. She has to cut it like a circle, using what we’ll call the pie-cutter’s algorithm:

1. Pick a starting point on the edge of the circle.
2. Go some fraction of the total distance around the circle, around the circle.
3. Cut from that new point to the center.
4. Cut from the center to the starting point.

The question of course is how far around the edge of the pie (circumference of the circle) does the pie-cutter go? There are two competing standards. The one you’re familiar with uses pi or π, which is the circumference of the circle divided by its diameter.

Source: TauDay.com

The other, brand new standard is to use τ (tau), which is equal to circumference of the circle divided by the radius, or equivalently, τ=2π. Why in the world would we abandon the pi we know and love? For the long answer, read the Tau Manifesto; for the short answer, compare the above diagram with this one:

Source: Tauday.com

Using tau, the angles around the circle are fractions of how far around the circle the angle is. If the pie-eater asks for an eighth of the pie, the pie-cutter goes τ/8 around the pie. Simple.

Except we currently ask her to go pi/4 to cut pie/8.

Which is to say that there’s a mismatch between the syntactic pi and the semantic pie. On the other hand, the syntactic tau and the semantic turn match perfectly.

### Rebuttal

And that would be all fine and dandy until someone came along and wrote The Pi Manifesto, which includes this graphic:

Source: ThePiManifesto.com

That’s compelling, until we consider a circle with a radius other than one. In that case, the area is no longer pi, so if I want one eighth of this new pie, I don’t want π/8. But I do still want τ/8. Why? Because while π/8 refers to area, which changes with the size of the circle, τ/8 refers to angular distance, which does not. Only on a circle with radius 1 does the pie-cutter actually travel τ/8 linear distance along an arc, but she always travels τ/8 radians, regardless of the circle size. The pie-cutter’s algorithm always works because it uses angular, not linear, distance. If the pie-cutter is asked for τ/8, she always goes an eighth of the way around the pie and the pie-eater always gets an eighth of the pie, which is not always an eighth pi.

What’s most ironic is that the Pi Manifesto provides little commentary on this image, and immediately accuses the Tau Manifesto of selection bias, when it is the Pi Manifesto that biasedly selects the unit circle, rather than a general circle. The Pi Manifesto is, in my opinion, of inferior quality to the Tau Manifesto. I knew, but had never really considered, that the area of the unit circle and the sum of the angles of a triangle were both equal to pi. (In the case of the triangle, I always thought of it as 180˚, or a straight line). That is, these facts never really came up, which makes them feel forced, where the Tau Manifesto feels natural.

Despite being an awesome pun, I can’t accuse the Pi Manifesto of circular reasoning. It does, however, includes a straw man argument: “there is also a nice formula for the multiple 3π, but that doesn’t mean we should start worshipping 3π”, when no one proposed doing that. In fact, there exist “nice” formulas for every “nice” fraction times either pi or tau, so we should choose the base constant to be something that makes geometric sense. Most ironically of all, the constant people do ”worship” is not 3π but τ/2!

A more worthy point of contention is that when the Pi Manifesto asserts that “the importance of π shines through as it shows up all over mathematics and not just in elementary geometry,” it makes the philosophical claim that complex, higher mathematics is more important than the circles where tau and pi originate. I disagree. It is exactly at the fundamental usage the meaning of something should be defined. Every equation that includes pi or tau must, somehow, be related to circles.

To use the Pi Manifesto’s own table, let’s examine the trigonometric functions:

$\begin{array}{c|c|c} \mbox{Function} & \mbox{Domain} & \mbox{Period}\\ \hline \sin\theta & \mathbb{R} & 2\pi\\ \cos\theta & \mathbb{R} & 2\pi\\ \tan\theta & \theta\neq (n+\frac{1}{2})\pi,~~n\in\mathbb{Z} & \pi\\ \csc\theta & \theta\neq n\pi,~~n\in\mathbb{Z} & 2\pi\\ \sec\theta & \theta\neq (n+\frac{1}{2})\pi,~~n\in\mathbb{Z} & 2\pi\\ \cot\theta & \theta\neq n\pi,~~n\in\mathbb{Z} & \pi\\ \end{array}$

Source: ThePiManifesto.com, with encouragement to “notice that π shows up, along with 2π and nπ. By converting the table to τ we would get even more nasty fractions than are already there.” This is true, except for sine and cosine, which are simpler with τ. I grant that may be convenient to work with π for the four more advanced functions. But just as functions like secant are defined in terms of sine and cosine, the π they use is defined in terms of τ. (The origin of the factor of 1/2 is an intriguing exercise left to the reader. Need help?)

The other failing og the Pi Manifesto’s is its argumentative tone. It uses the word “win” or a variant 11 times to the Tau Manifesto’s zero. It asserts that “π will prevail in the intriguing π versus τ battle.” The combative pomp falls flat is because this isn’t an argument. We do not have to stop using pi, because pi and tau/2 are isomorphic: it’s the same information, thought of in different ways. No one is going to force you to stop using pi because tau does not invalidate pi mathematically, only aesthetically. In areas where tau is more appropriate and natural, and there are many, let us use tau.