Yesterday Was Tau Day

29 Jun 2020 • 5 minute read

You probably know that 3/14 is Pi Day, since pi (π) starts 3.14 and so matches the date, at least the way it is written in the US but not in the rest of the world. Well, yesterday was Tau Day, 6/28. Tau (τ) is 2π and so is 6.28. Some mathematicians believe that we should be using τ everywhere.

Remember when you were in primary school (or was it middle school) and came across the formula for the circumference of a circle. It was 2πr so the very first time you run into π it appears as 2π. Of course, you can say that the diameter of the circle is more "fundamental" than the radius, which is at least a plausible position, but then you have to say that the area of a circle is πd²/4 which nobody ever writes. In advanced math, nobody ever uses the diameter.

As it says in a Scientific American piece Let's Use Tau--It's Easier Than Pi:

In fact, almost every mathematical equation about circles is written in terms of r for radius. Tau is precisely the number that connects a circumference to that quantity. But usage of pi extends far beyond the geometry of circles. Critical mathematical applications such as Fourier transforms, Riemann zeta functions, Gaussian distributions, roots of unity, integrating over polar coordinates and pretty much anything involving trigonometry employs pi.

But what they don't say explicitly is that everywhere it occurs, it is 2π.

It actually goes way beyond things involving trigonometry (and so circles). In fact, some of the topics in the preceding paragraph such as Gaussian distributions (bell curves aka normal distribution) or the Riemann zeta function (distribution of prime numbers) have little to do with trigonometry. One nice thing about looking at equations from advanced math for this topic is that you (I) may not understand the equation but I can see it contains 2π everywhere.

History

pi is wrong It seems that the first discussion of Tau was Bob Palais's 2001 paper π Is Wrong. This appears in The Mathematical Intelligencer Springer-Verlag New York Volume 23, Number 3, 2001, nearly 20 years ago (although the link I gave you is to a PDF since the actual paper seems to be behind a paywall). "I never would have imagined the scale of the discussion" he said about it. One paragraph:

The sum of the interior angles of a triangle are π granted. But the sum of the exterior angles of any polygon from which the sum of the interior angles can easily be derived, and which generalizes to the integral of the curvature of a simple closed curve is 2π.

Then there is Mike Hartl's The Tau Manifesto. It was originally published on Tau Day 2010 but seems to get updated on Tau Day each year, most recently in 2019 (although maybe it was updated yesterday). It is also available in some other languages:

The one place that springs to mind where π occurs without being doubled is in the famous Euler identity that combines the five most significant constants in all of mathematics:

e^iπ+1 = 0

But it is also true that:

e^iτ-1 = 0

Which seems just as good.

Here's a video from the Khan Academy where Vi Hart explains that Pi Is (Still) Wrong:

Approximations to Pi (and Tau)

Let's start with the Bible:

And [Hiram] made a molten sea, 10 cubits from the one brim to the other: it was round all about, and…a line of thirty cubits did compass it round about.

That's 1 Kings 7:23, concluding that π is 3.

In 1897, the Indiana legislature tried to rule that π was 3.2 but luckily a professor from Purdue was in the assembly the day it was put forward and so it never became law.

When I was in school, before the days of electronic calculators, we would use rational approximations. The most famous is probably 22/7, which is actually more accurate than 3.14.

There is actually a Wikipedia page Approximations of π which, as so often, tells a lot about approximations over history.

The most practical approximations (if you don't just hit the π key on your calculator) are 22/7, that I used in school, and 355/113 which has a relative error of 8x10^-8.

How accurate do you need to be? As this Wired article points out, if you consider a:

sphere with the diameter of the observable universe at 93 billion light-years. If we don't know the exact value of Pi, but only 152 digits then we don't know the exact circumference. However, the uncertainty in the circumference is less than the Planck length—the smallest unit of distance measurement that has any meaning. You need even fewer digits of Pi to get an uncertainty in the circumference smaller than the size of an atom.

Actually, it seems you only need 40 digits to calculate the diameter of the universe accurate to a single atom (so 1Â). NASA uses 15 digits for their calculations for their spacecraft.

The record number of digits, so in some sense the most accurate "approximation" to π calculated was by Google Japan employee Emma Haruka Iwao who used Google's Cloud to calculate π to 31 trillion digits, using 25 machines for 121 days. I'm guessing she didn't get billed. If you are interested in exactly how she did it, Google published a post about it Pi in the sky: Calculating a record-breaking 31.4 trillion digits of Archimedes’ constant on Google Cloud.

But it seems that record has already been broken in January of this year when Timothy Mullican announced the computation of 50 trillion digits over 303 days.

Get the Shirt

What better way to show your position on the debate other than wearing the "official" T-shirt...I mean Tau-shirt.

Resolving the Debate

XKCD has a compromise solution:

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