You are probably familiar with Pi Day, perhaps the most popular of geeky vacations. Hoera. But I am here to tell you that Pi-day is wrong – or rather, the whole idea of pi as a mathematical concept is wrong, and why should you instead celebrate Tau Day (which is happening today!)
It's easy enough to see why people like Pi Day: the whole starts with a mathematical play of sorts (the date is written as 3/14 in the American notation. Pi starts with the numbers 3.14.) You understand.) It's an easy, fun ritual to see how many digits you can meaninglessly remember from the famous infinite, never-recurring song (though 39 digits is more than sufficient for almost all calculations you will ever need). Plus pi sounds like pie and who doesn't like pie?
But here is the thing: π because a song is bad, and therefore the whole misguided day is devoted to celebrating it. It is a lot to record, and I too was once like you: I have learned the virtues of pi for years, going back to Pi Day high school parties. But instead of pi, we should celebrate tau, an alternative circle constant referred to by the Greek letter τ equal to 2π, or about 6.28.
I don't just make this out of the blue: the horror of pi as a constant was first represented by a mathematician Bob Palais in his article "π Is Wrong!" and explained later in The Tau manifesto by Michael Hartl, that serves as the basis for modern language. (Internet-famous mathematician Vi Hart is also one important proponent of tau over pi, if you prefer your mathematical arguments in a more entertaining video form.)
But the arguments of Palais and Hartl both boil down to some basic math. Go back in time to when you first got to know the geometry and recall the simple origin: no matter which circle you use, if you divide the circumference of the circle by the diameter, you get the same answer: an endless number, starting with the figures 3.14159265 … (aka pi).
And that is precisely the fundamental error. The point is that we don't really use the diameter to describe circles. We use the radius, or half the diameter. The circle equation uses the radius, the area of a circle uses the radius and the fundamental definition of a circle – "the series of all points in a plane that are at a certain distance from a certain point, the middle"- is based on the radius. If we put that into the circle in our constant equation, we get a new circle constant corresponding to 2π, or 6.28318530717 …, usually indicated by the Greek letter τ (tau). Switch to τ does not make any arbitrary change for the sake of it, it brings one of the most important constants in math into line with how we actually are do math.
Now you may think that this will cause fundamental, seismic changes in mathematics. "How on earth can you replace something as important as pi?" You might ask. But if we are honest, π is not really something that we use in daily mathematics to get you started. Unless you are someone who makes a lot of geometric calculations in your daily life, chances are that you will not actually come across your pi until it is time to complete figures for Pi Day. Of course, it's a good introduction to the idea of irrational numbers, but tau could work just as well for that. And if you do work a lot with π, replacing τ is useful for many reasons, mathematically. Again, I will refer you to The Tau manifesto for the full argument, but I will mention a few here.
Radial angles are a major thing about tau corrections. You may remember that as "those annoying chunks of a circle that are represented by strange fractions of pi from high school mathematics", but with tau it's simple: everything matches where it should be in a fraction. So half the circle (180 degrees) – τ / 2. 1 / 12th? τ / 12. It is a small change, but it makes an angular notation – a frustratingly stupid part of the geometry that requires an elitist thought of remembering angles and conversions through the use of pi – a more welcoming and intuitive prospect for new students .
It also makes circle functions such as sinuses and cosine easier, because it matches one full cycle of the function with one full circle revolution (tau), instead of the seemingly random 2π that you get π as your circle function. As with radial angles, deriving sine and cosine values makes a simple process by simply drawing the function, instead of requiring students to remember that 3π / 2 is the three-quarter point on the wavelength for whatever reason.
Similarly, it simplifies a number of other higher math-like integrals in polar coordinates, the Fourier transformation and the integral formula of Cauchy, since they already work in terms of 2n anyway. Using tau only intersects the middleman. Looking back on years of math and physics notes with the illuminated lens of tau, I cry for my former self and the accumulated hours of unnecessary conversions and complications introduced by pi.
However, they are not only practical goals. Replacing π with τ makes mathematics more elegant in general. And at the heart of it, isn't that what we want to do with math? The universe is huge and almost impossible for us to always fully understand, but by distilling it into a system of logical numbers and symbols, we can create order in the chaos. So why not embrace a circle constant that makes our equations and formulas more beautiful?
Unfortunately, pi is probably too well anchored in traditional mathematics to ever free us from his tyrannical grasp. Maths manuals still embrace the virtues of pi, and learning such a systemic change in how we teach maths is probably a tough fight. (On the other hand, Common Core seems to have somehow succeeded, despite its – in my eyes – incredibly blunt nature, but find out.) And that is a shame, given how much more feeling tau makes like a circle constant for even the more basic functions for which we use pi. But the first step is to stop glorifying pi, so next year I won't celebrate Pi Day – and neither will you.
But everything is not lost for people looking for a fun day to celebrate math: Tau Day (6/28 or 28 June) is today!
Update June 28, 2019, 9AM ET: Updated message for Tau Day 2019.
Correction: Two digits of pi accidentally transposed.