Wednesday, August 26, 2015

On Pi and Tau

In 2010, physicist and educator Michael Hartl published something he called The Tau Manifesto, a piece of writing which put forth arguments for a surprisingly controversial assertion: π is wrong.

That’s not to say that our measurements of it are off, of course; few things are more readily calculable than the ratio of a circle’s circumference to its diameter. Instead, Hartl makes a compelling case for the idea that 2π, the ratio of a circle’s circumference to its radius, is a far more fundamentally significant and useful construct. He calls his new "circle constant" tau.

Fun fact: If you know 50 digits of pi, you already know 50 digits of tau, if you're exceedingly good at mental math. 

At first, this might seem unnecessary or trivial. After all, it’s not much harder to write out “2π” than “T”, and changing over means that a lot of geometry textbooks would need rewriting. However, the line of thinking is not without its appeal; it’s intuitively pleasing that the circle constant should describe one full cycle of a sine wave, rather than half (as is the case with π).  π shows up everywhere in physics and mathematics, but practically everywhere that π shows up, Hartl claims, tau would be simpler.

And indeed, when performing an angular integral, we integrate from zero to 2π. When reducing Planck’s constant h, we use a factor of 2π. Coulomb’s constant carries 4π, two for azimuth and two for altitude. Perhaps the only place where pi appears alone is in the formula for the area of a circle, a point which Hartl brings up and then proceeds to knock down with an argument so simple but striking that it bears repeating here, partly because it should be very familiar to anyone with a physics background.

First, we’re asked to take a small mental detour and consider the case of an object falling in a uniform gravitational field. Its velocity is dependent on how long it’s been falling, related by the proportionality constant g.
In order to find out how far the object has fallen, we can take the time integral of that equation, yielding:

Similarly, we’re asked to consider newton’s second law, where the force on an object is proportional to its acceleration, and where the proportionality constant is the object’s mass m:
To find the object’s kinetic energy, the total work done in accelerating it to a given velocity, we integrate the force to find:

We’re then reminded that the area of a circle can be found by breaking it into an infinite number of rings, of circumference C and width ∂r.
Image from The Tau Manifesto, by Michael Hartl

 from which point we can see that the area should be equal to:
but since
we can substitute to get
Here, in the famous circular area formula, π only appears by itself because the factor of two has been cancelled out by a 1/2 which results from integration. Not only does this insight make it apparent that tau is more mathematically fundamental than π, it encourages a geometric understanding of where the area formula comes from, and unmasks its connection to the other equations discussed above, as the integral of a linear function.

And this resonated with me, because the first time I noticed the similarity between the equations for the potential energy in a spring (which follows a similar force law) and the kinetic energy in a baseball, I felt like I was getting a brief glimpse of something more fundamental about the math that governs our reality. That feeling is part of what made me fall in love with physics, so as someone who hopes to inspire a love of physics in other people, The Tau Manifesto struck me as worth sharing. It reminded me that there are a million of those connections out there, some in plain sight, just waiting to be realized—and that's an exciting feeling, because it means a lifetime experiencing the wonder of discovery.

Thanks go out again to Michael Hartl for writing The Tau Manifesto, which provided the above mathematical argument and which you can read in its entirety here.

1 comment:

  1. Don't forget that there are factors of 1/2 in other areas too. Polygonal area is 1/2 Pa or 1/2 n sin(τ/n), and both of these converge to the area of a circle when n=infinity (and the last one even uses tau). The formula for sector area is 1/2 θr^2, and area of a circle is a special case of it when θ=τ.