Yes, pi is wrong. Sorry about that.

Which is the more fundamental constant: π or 2π? As pi day approaches, I think it’s important for everyone to state their thoughts on this contentious matter.

Here are the main options:

  • π is more fundamental. (the establishment position)
  • 2π is more fundamental. (the progressive position)
  • They’re about equally fundamental. (the “both sides” position)
  • Any rational multiple of π would be just as good. (the nihilist position)

While I wish that π were more fundamental, it’s just not. I’m firmly in the camp that says 2π is more fundamental.

My memory’s not great, but I think I figured out on my own as a teenager or young adult that 2π was probably more fundamental than π. You don’t have to be a detective to notice that a whole lot of math and physics formulas have a “2” before the “π”. My opinion may have been bolstered by some discussion I read on Usenet. This was before Bob Palais’s classic π Is Wrong! article gave the issue widespread attention.

When I read the Palais article, and saw the reaction to it, what surprised me was the realization that much of the mathematical community apparently did not already know this — and many disagreed with it. What can I say? Even smart people who should know better can get emotionally attached to something, and fall victim to wishful thinking.

For more information on this topic, visit the Tau Day website. Modern proponents of 2π have taken to using the symbol τ (tau), where τ=2π. I’ll adopt that practice.

Basically, τ is more fundamental than π for the simple reason that it’s the circumference of the unit circle, and consequently it’s the period of the trigonometric functions. (I’m aware that the tangent function, for example, degenerates to a period of π. But that’s a technicality of no importance. As a group, their period is 2π.) Nothing that π can do for you ever really makes up for this.

So how, historically, did we end up with a symbol for 3.14… instead of for 6.28…? This is pure speculation, but I bet it’s ultimately because it’s human nature to think of the diameter of the circle as more fundamental than its radius. So, we measured the circumference of a circle in diameterians, and got 3.14…

At some point, we figured out that math works better when we use radians instead of diameterians, and the formula for circumference became more complicated, going from from \pi d to 2 \pi r. But the formula for area of a circle became simpler, from {1 \over 4}\pi d^2 to \pi r^2. So there was still no strong pressure to change the circle constant.

I could make good excuses for why the area-of-a-circle formula is simpler in pi world than in tau land, but I don’t deny that it is. But don’t be fooled by other formulas that look like they’re simpler with π, when they really aren’t. For example, {\pi^2} \over 6 would not be written as { { ( \tau / 2 )}^2} \over 6 in tau land. It would be written as {\tau^2} \over {24}, which is no more complex than {\pi^2} \over 6.

Of course, there’s no way to prove that one constant is more fundamental than the other. There’s no standard way to measure “fundamentalness”. But I reject the notion that it’s purely subjective. It’s highly improbable that these two options would be about equally good. And I think I know which one reality prefers.

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