Happy Tau Day, the true circle constant

You've probably heard of Pi Day, perhaps the most popular geek party. But I'm here to tell you that Pi Day is wrong, or rather, the whole idea of ​​pi as a mathematical concept is wrong.

It's easy to see why people like Pi Day: it all starts with a mathematical play on words (the date is written as 3/14 in American notation, Pi starts with digits 3.14). It is an easy and fun ritual to see how many digits you can uselessly memorize from the famous and endless, a number that never repeats itself (although 39 digits are more than enough for almost any calculation you need). More pi sounds like cake, and who does not like the cake?

But here's the thing: π as a number is bad, and therefore, that's all the wrong day dedicated to its celebration. It's a lot to assimilate, and I was once like you: they taught me the virtues of Pi for years, going back to the Pi Day parties in high school. But instead of pi, we should hold tau, an alternative circle constant referred to by the Greek letter τ which equals 2π, or approximately 6.28.

I'm not inventing this out of nowhere: the terribleness of pi as The constant was first proposed by the mathematician Bob Palais in his article "π It's wrong!" and then exposed in The Tau Manifesto by Michael Hartl, which serves as the basis for modern tauism. (The famous Internet mathemusician Vi Hart is also a great defender of tau on pi, if you prefer your mathematical arguments in a more entertaining video form)

But the arguments of Palais and Hartl are reduced to a basic mathematics. Go back in time until the first time you learned geometry and remember the simple origins: no matter what circle you are using, if you divide the circumference of the circle by the diameter, you will get the same answer: an infinite number, starting with the digits 3.14159265 .. . (aka pi).

And there's the fundamental flaw. The point is that we do not use the diameter to describe circles. We use the radius, or half the diameter. The equation of the circle uses the radius, the area of ​​a circle uses the radius and the fundamental definition of a circle – "the set of all points in a plane that are at a given distance from a given point, the center" – based on the radius By connecting that to our constant equation of circle, we get a new circle constant equivalent to 2π, or 6.28318530717 …, colloquially referred to by the Greek letter τ (tau). Changing to τ is not making an arbitrary change for the sake of it. It brings one of the most important constants in mathematics according to the way we actually do mathematics.

Now, you may be thinking that this will cause fundamental seismic changes in mathematics. "How the hell could you replace something as important as pi!", You might ask. But if we're honest, π is not really something that we use in daily mathematics to begin with. Unless you are someone who does a lot of geometric calculations in your daily life, chances are you only find pi when it comes time to recite some digits for Pi Day. Sure, it's a good introduction to the idea of ​​irrational numbers, but tau would work just as well for that. And if works a lot with π, replacing it with τ is beneficial for many reasons, mathematically speaking. Again, I'll address it to The Tau Manifesto for the full storyline, but I'll just point to some here.

A great thing that tau fixes are the radian angles. You can remember that as "those annoying pieces of a circle represented by odd fractions of high school math pi", but with tau, it's simple: everything fits where you should do it fractionally. Then half the circle (180 degrees) – τ / 2. 1/12? τ / 12. It is a small change, but it makes the angular notation, a frustratingly obtuse part of the geometry that by the use of pi requires an elitist notion of memorization of angles and conversions, a more welcoming and intuitive perspective for new students .

It also makes the functions of the circle such as sine and cosine easier, since it makes a complete cycle of the function coincide with a full circle turn ( tau), instead of the seemingly arbitrary 2π that you get to use π as the function of your circle. As with radian angles, it makes the sine and cosine values ​​a simple process by simply drawing the function, instead of requiring students to remember that 3π / 2 is for some reason the three-quarter point in the wavelength.

Similarly, it makes a set of other mathematical integrals higher in polar coordinates, the Fourier transform and the formula Cauchy's integral are simpler, since they also work in 2π terms anyway. Using tau simply cuts the middleman. Looking back on the years of math and physics notes with the illuminated lens of tau, I cry for my former self and the cumulative hours of unnecessary conversions and complications presented by pi.

However, they are not just practical purposes. Replacing π with τ makes mathematics more elegant in general. And deep down, is not that what we aspire to do with mathematics? The universe is vast and almost impossible to understand for us, but by distilling it into a system of logical numbers and symbols, we can make a bit of order out of chaos. So, why not embrace a constant circle that makes our equations and formulas more beautiful?

Unfortunately, pi is probably too well installed in traditional mathematics for us to free ourselves from his tyrannical control. Mathematics textbooks still defend the virtues of pi, and instilling such a systemic change in the way we teach mathematics is an uphill battle. (On the other hand, Common Core somehow seems to have succeeded, despite its – for my eyes – incredibly obtuse nature, but imagine). And that's a shame, given how much more sense does tau make as a constant circle even for the most basic functions we use pi for. But the first step is to stop glorifying pi, so I will not celebrate Pi Day this year, and neither should you.

But all is not lost for those looking for a fun day to celebrate mathematics: Tau Day (June 28/6) is here, today!

Update June 28, 9:00 a.m .: Updated publication for Tau Day 2018!

Correction : Accidental transposition of two digits of pi.