Tau Day

Happy Tau (τ) day! It’s June 28th, so here at Preciseish we celebrate 6/28 3:19 (6.283185) This is what we should use instead of pi. Everyone loves pi, but it really should not exist. Tau does everything pi does, but better.

Pi is what you get when you divide the circumference of a circle by it's diameter, but tau is the circumference divided by the radius. After you've defined pi, you'll almost never use the diameter again because it is the radius that makes a circle. A circle is defined as the set of points on a plane a constant distance (radius) from a point. If you draw a circle, you maybe do it by using a compass, or a bit of string, but always by enforcing a radius.

So what if we replaced pi with tau? Working in radians would become much more intuitive. We know that as a unit of angle measurement, 360° = 2 π. But why is one time around two times pi? It's confusing. 360° = 1 τ. Much better. Now you can immediately see how big the angle is. If you have ¼ τ, that's just one quarter of the way around, or 90°. Even better, sine and cosine become intuitive. Instead of memorizing some points like sin(π/2) = 1 without understanding what it means, now you can just learn that the sine is just the height on a unit circle at the angle. Now knowing sin(τ/4) is just knowing how high you are on a circle when you're a quarter the way around: you're on top, so 1.

One thing that seems, at first, like an advantage for pi is that the expression for the area of a circle comes out neater: π r² seems simpler than ½ τ r². But if you work with math or physics enough you'll realize that the ½ (constant variable)² form is very natural and makes sense. It comes from calculus. Consider falling. The fundamental constant for falling is gravitational acceleration, g, and our variable is time. To start with, all we know is g. Integrate that and get g t, that’s the speed you’re falling. integrate again and you get ½ g t² which is how far you've fallen. This is what is going on with the circle too. We start with the fundamental constant, tau. integrate and get τ r, that’s the circumference of the circle, integrate again and you get ½ τ r², the area of the circle. It's natural.

How about the beautiful and strange Euler's Identity that features pi so prominently? Euler's Identity is the following: e^{i π} + 1 = 0. It seems amazingly improbable that Euler's number, the imaginary square root of negative one, and pi should so succinctly combine to a non-imaginary, non-irrational zero. But would using tau do any damage? It turns out to give e^i τ = 1. Not only is this more elegant, in my opinion, but it results more directly from the underlying relationship involved: e^{i x} = cos(x) + i sin(x). If x is taken as pi, the result is an awkward -1. It takes some rearranging to get the more pleasing form above. If x is taken as tau, the result is a perfect 1 right off.

If you still aren't convinced, you must watch Vi Hart's quick and entertaining video on the subject, and enjoy this thorough but relative light and quick paced Tau Manifesto.