Knowledge of the all-important circle constant, the ratio between a circle's circumference to its linear dimension, has been with us for almost 4000 years, and it has almost universally been defined as so:
π = C/D = 3.1415926535…
Pi is a remarkable number — not only does it describe a fundamental law of our universe, but it also is irrational and transcendental (it cannot be expressed as a root of any polynomial with rational coefficients, like e). But suppose that we replace the diameter D with the radius R in our definition of this circle constant:
C/R = 6.283185307…
Exactly 2π. That is what we call tau, or τ. So why does this matter, you ask. It is pretty obvious that C/0.5D = 2π, after all. To find out more about the circle constant, we need to start by looking at its origins.
While the idea of π had been thought about before, the first rigorous method of calculating pi is generally credited to Archimedes, who in 250 BC devised the "polygonal algorithm". He inscribed and circumscribed polygons onto a circle and calculated the perimeters of the large and small polygon, knowing that he could use these values as upper and lower bounds for the circumference of the circle. By then increasing the number of sides of the polygons, and thus making them closer and closer to actual circles, he was able to narrow those bounds until he achieved an accuracy of 99.9% for π, an incredible feat when you consider that back then they had no calculators. Since this formal method was discovered, mathematicians and scientists have stuck to using pi, instead of the equally valid tau, quite simply because nobody ever needed to wonder whether there was an equally (or perhaps more) valid constant that they could use.
So, why is tau better than pi?
To start with, let’s think about the fundamental definition of a circle. A circle is the locus of all points equidistant from the origin. Circles are constructed from the points that are the radius away from the center, not from the points that are the diameter away from the points on the other side. That is our first indication that perhaps there is something wrong with pi. But it doesn't end there.
Radians are generally accepted as the most pure, mathematical measure of angles. We use radians because, while degrees are arbitrary, radians actually have a fundamental relationship with angles. But when you think about the fact that it requires 2π radians to make up a circle, rather than π radians, doesn't that suggest that there is something wrong with them? No! Rather, there is something wrong with pi. When we replace pi with tau, we find ourselves in a much better spot — one tau radians make a circle. And that just makes sense. We don't even have to stop there.
The sine curve graphs the relationship between an angle and the ratio of lengths of the opposite and hypotenuse sides in a right triangle. Analyzing it, we once again realize something horrific. The period of the curve is 2π, not 1π! This tragedy has plagued almost all of us, whether we know it or not, making mathematicians worldwide wade through divisions and multiplications by 2 that really shouldn’t be there.
But wait, what about Euler's Identity, eiπ= -1! We couldn't possibly want to ruin it by making it eiτ/2= -1! Well, the identity itself comes from Euler’s formula:
When we replace π with τ:
Since τ is a full circle on the unit circle, we know that:
cosτ = 1
Mathematics is about elegance, clarity, and truth. Using π unnecessarily complicates mathematics, which in some way makes it less perfect. Whether or not the mathematical society is too steeped in convention to accommodate a change, the point stands. So, if nothing else, when you are confused by trigonometry, the unit circle, or why there are just too many twos on the page, take solace in the fact that it is not your fault.