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Pi is Great. But There's Something Better.

In 2010, physicist and educator Michael Hartl published something he called The Tau Manifesto, a piece of writing that makes a surprisingly controversial assertion: pi is wrong.
That’s not to say that our measurements of it are off; few things are easier to compute to high precision 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, and the argument for it is actually kind of mind-blowing.

Fun fact: If you know 50 digits of pi, you already know 50 digits of tau, if you're really good at mental math. 
At first, this whole thing might seem unnecessary or trivial. After all, it’s not much harder to write out “2π” than “T”, and changing over would mean rewriting a lot of geometry textbooks. But there's a certain elegance to Hartl's line of thinking. To begin with: angles can be measured in degrees or radians—in degrees, a full circle is 360°, but in radians, it's 2 pi. Wouldn't it be nicer to have the circle constant correspond to a full revolution? Pi shows up everywhere in physics and mathematics, but practically everywhere that pi 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. The Coulomb constant, which relates the magnitude of an electric charge to the strength of the electric field it creates, includes a factor of 4π—two for each of the intersecting planes you need to indicate points a three-dimensional space.

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 quick 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. Integral calculus can be scary, but mathematically it's pretty simple—you add one to the exponent of the variable, and divide the whole quantity by the number you get when you do that. So the integral of g*t is just:
Since t is the same thing as t1. Adding 1 gets us t2, and then we divide the whole thing by the new exponent, 2.

Similarly, we’re asked to consider Newton’s second law, which tells us that the force on an object is proportional to its acceleration, the two being related by 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:
From there, Hartl asks the reader to consider an interesting way of finding the area of a circle: by breaking it up into an infinite number of thin rings. This might seem like a weird way to do it, but the logic behind it is that if a ring is really, really thin, we can find its area by treating it like a rectangle, because the difference between its inner and outer circumference is negligible. If you imagine unrolling the ring below, finding its area is as simple as multiplying its circumference C by its width dr.
Image from The Tau Manifesto, by Michael Hartl

To find the area of the whole circle, we add up the areas of all those rings. This is where integrals come in again—since we're adding up an infinite number of infinitely tiny terms, doing that the old fashioned way would take literally forever. But the same trick from earlier, of moving exponents and dividing, gives us the same answer in a snap! 

The equation for the integral looks like this:
Remember, C is the circumference of our rings, and dr is their width. The long "s" shape indicates that we're integrating, and the 0 and r tell us that we're looking at the sum of the areas of rings with circumferences between 0 and r—the total radius of the circle.

Here, we'll need to make a substitution so that we can write C in terms of r. Fortunately, that's what pi is for—relating the circumference of a circle to the radius:

which means we can substitute to get:
The point of Hartl's demonstration here is that, 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.

Comments

  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.

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