Tuesday, June 28, 2016

Tau (τ) > Pi (π): The circle constant you knew is all wrong

Happy Tau Day! What is tau (τ), you ask? Tau is twice the value of the famous mathematical constant pi (π), 6.28... Hence, 6/28 is "Tau Day" in echo of celebration of "Pi Day" on 3/14.

As it turns out, we've been mythologizing exactly the wrong circle constant, and τ is the more natural choice. Of course, the math works out either way, since you just need a factor of two to convert between them. So why would you choose τ over π? Why should you choose τ over π?

(For a more detailed argument rather than the following short summary, see the excellent Tau Manifesto which has over the last few years really popularized this idea.)

τ is the circumference of a unit circle

A "unit shape" is a shape whose critical measures are equal to 1. For a unit circle, that means it has a radius of 1. To obtain the radius of any circle, then, you would just need to express it simply as τr, and not 2πr. If you go back to trigonometry, do you remember how annoying it was to convert between degrees and radians? Using τ makes this more intuitive, as half a circle is half of τ, a quarter of a circle is a quarter of τ, etc. τ as "one full turn" similarly makes understanding the trigonometry functions easier, as the period of the function is equivalent to τ.

Although the area function is simpler with π, τ produces relationships that match other similar physical equations

Consider that the equation of force, F = ma, is integrated to obtain the kinetic energy equation E = (mv^2
)/2. Now consider the circle's circumference with τ, C = τr, which is integrated in the same way to obtain the area equation A = (τr^2)/2!

Outside of the special case of a circle, equations are simpler with τ than with π

This part is more advanced so I won't really get into it, but the upshot is that using τ is more natural and general for describing spheres of arbitrary numbers of dimensions, while π only naturally appears in the special case of 2 dimensions. 2π also routinely shows up in other fundamental equations, including Gaussian distributions, Fourier transforms, Planck's constant, and more, which means they can be more concisely expressed with τ.

What does it mean?

For most intents and purposes, not much. Nothing about the math really changes. But it might just make the equations a bit more intuitive to learn and express. And that's really the main reason we use symbolic constants, isn't it?

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