Wednesday, July 1, 2015

Belated Tau Day

I realize I missed it by two days, but happy belated Tau Day everyone! What's Tau Day, you ask? It's like Pi Day (3/14), but for Tau (6/28). "But what's tau," you're probably asking.

Simply put, tau is twice pi, or \(\tau=2\pi\). Like pi, tau is irrational and transcendental. But what's so special about \(2\pi\)? Why not \(3\pi\), or \(4\pi\), or any other integer multiple?

The reason has to do with circles, geometry, and the ancient Greeks. Everyone who's taken geometry is familiar with the formula for the circumference of a circle, \(C=2\pi r\), where C is the circumference and r is the radius. This, however, is the modern formulation. The ancient Greek geometers who first came up with the concept were thinking in slightly different terms: namely, that two times the radius of a circle equals its diameter. Thus, to them, the formula would be expressed as \(C=\pi d\) where d is the diameter (\(d=2r\)).

That formulation looks nice and compact, just four symbols total (although that itself is a modern artifact, as symbolic algebra as we know it wasn't invented until the Middle Ages). However, it's not quite as fundamental as it could be, because the diameter of a circle is a less fundamental concept than its radius. A circle can be—indeed is—defined as the set of all points whose distance from a single central point is exactly equal to the radius. There's no such simple way to define a circle using a diameter other than to cut it in two and use the radius anyway. Hence, the modern formulation uses radius rather than diameter.

Unfortunately, using the radius introduces that factor of two into the picture. You see, the modern formulation goes only halfway. The ancient Greeks liked their version because it was clean and simple: take any line segment to be the diameter of a circle, multiply it by this one single, simple constant (\(\pi\)), and boom, you have the circumference. And you can see the simplicity there that all mathematicians strive for.

So let's say we want to remake the modern formulation to get rid of that ugly extra 2 in there. We can simply multiply pi by it and get tau, to make our new formula \(C=\tau r\). It's simple, elegant, and—importantly—more fundamental than using pi. Take any line segment as the radius of a circle, multiply by tau, and bam—you've got your circumference.

Pi was a good first attempt at getting a very important number in math and physics, the circle constant that shows up everywhere cyclical processes are involved. It's just the wrong constant. It's the ratio of the circumference to the diameter, when the more fundamental constant is the ratio of the circumference to the radius—tau.

It's getting late and I should wrap this post up, so I will end by providing a link to The Tau Manifesto for anyone who's still interested. It's where I first encountered the idea of using tau instead of pi, and shows in detail how tau simply makes more sense to use in many areas of math and physics. I was reflexively against the idea when I first read it, but over time I've come to see how much sense it makes—to the point I'm even writing about it here on my blog. Funny how these things happen. A hui hou!

2 comments:

  1. That is so incredibly simple! I'm amazed anyone bothered with pi. And now I have another math day to celebrate. Cool.

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    Replies
    1. It really is a lot simpler, isn't it? Sometimes it takes the longest time to notice the simplest things.

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