Tomorrow we are celebrating the day of another mathematical constant! Huzzah!
A while back I wrote a post on the basics of the circle constant π (see A Slice of π). Now we shall meet its archrival, τ (the Greek letter tau). The ongoing war of π vs. τ has continued for centuries and its toll has been devastating. Cities wiped out, millions of human lives lost…
Okay, I admit that’s not true. However, there is indeed a small proportion of mathematicians who believe that τ is the way forward. Some people have cheekily named them tauists. But we shall see why τ is not an entirely unreasonable candidate to replace π with.
τ is another circle constant is equal to 2π. Literally. It’s that simple. In accordance with a tradition that started three months ago, I shall list τ to a hundred decimal places for your pleasure:
You may wonder why there is any point in arguing about this at all. Nevertheless, mathematicians strive to perfect their world of equations and formulas, and deciding whether to use π and τ is one such problem that needs to be resolved.
The primary argument for τ is how the circumference of a circle is calculated. A circle is defined as the set of points a distance r (the radius) from a point (the centre). The radius of a circle is what defines the circle, not the diameter, as there are many shapes with a constant diameter that aren’t circles. Therefore it would make sense for the constant to be the ratio of the circumference to the radius, not the ratio of the circumference to the diameter:
In addition, one τ represents a full rotation in terms of angles. Therefore a half turn would be τ/2 (or π), and quarter turn would be τ/4 (or π/2).
On the other hand, using τ causes more complexities in defining the area of a circle:
From this equation, the area of a unit circle (a circle with radius 1) is exactly equal to π. This is the beauty of π, as areas are a very significant aspect of geometry. You can see the benefit of π when you look at fractions of a circle: a half circle is π/2 (or τ/4); a quarter circle is π/4 (or τ/8).
How about outside of the geometry of circles? τ has benefits in writing the famous Euler’s identity. Traditionally it is written like this:
However with τ:
You can see that using τ produces a more elegant formula. There are arguments against this on the basis that the identity using τ is simply a derivative of the identity using π since τ is just a multiple of π, and there would be equally elegant forms with other multiples of π.
The various advantages of π and τ are scattered around pure mathematics. 2π is the period of the majority of trigonometric functions, and exists in many equations, such as the Gaussian distribution (the recognisable bell-shaped curve) and the Fourier transform.
However, it is easy to cherry-pick equations to argue for either side. π by itself appears in the integrals of regularly used functions, the gamma function, the area of an ellipse and the combined interior angles in a polygon.
For most mathematicians, the choice is to continue using π as if this debate never existed. Arguments for τ are simply not convincing enough to be worth the effort to change traditions that have lasted for over two millennia. Nonetheless, this debate is potentially attracting a number of non-mathematicians to the field, and getting more people involved in mathematics is most certainly a plus.