Pi is wrong

[jemidiah]

Pi day was March 14th (3/14). Some people, including me, think pi is the wrong constant to use. There's a movement to replace pi with "tau", where tau = 2*pi. This would also replace pi day with tau day on June 28th (6/28). There's an interesting list of reasons here. Since it's long, I'll pick a few to summarize below.

1. A circle has 2*pi = tau radians. It's convenient to say a quarter turn of a circle is tau/4 rather than pi/2. This also makes converting from radians to degrees cleaner--multiply by 360/tau instead of 180/pi.
2. The radius is more fundamental than the diameter (for instance, the equation of a circle of radius r is x^2 + y^2 = r^2) so we should use tau = circumference / radius instead of pi = circumference / diameter.
3. The identity e^(pi*i) + 1 = 0 can be replaced by e^(tau*i) = 1 + 0.
4. The area of a circle is pi*r^2 = 1/2 tau * r^2. Many other laws include the 1/2: kinetic energy is 1/2 m v^2, spring potential is 1/2 k x^2, distance fallen is 1/2 g t^2, ....
5. Numerous formulas (the *vast* majority of the ones I've come across since learning of tau, actually) are cleaner using tau rather than pi:
   
   • 1/1^(2n) + 1/2^(2n) + 1/3^(2n) + ... = B_n (2 pi)^(2n) / 2*(2n)! = B_n tau^(2n) / 2*(2n)!
   • Fourier transforms, which always annoyingly have that extra 2 floating about.
   • The nth complex roots of 1 are e^(2*pi*i k/n) = e^(tau*i k/n).
   • The volume of an n-dimensional sphere of radius r for n even is: pi^(n/2) / (n/2)! * r^n = (tau/2)^(n/2) / (n/2)! * r^n, or for n odd is: 2^((n+1)/2) pi^((n-1)/2) / n!! * r^n = 2*tau^((n-1)/2) / n!! * r^n. [Random fact: the 7-dimensional sphere of radius 1 has the largest numerical value of surface "area" of any dimension.]
   • The normalization factor for the standard normal distribution includes 1/Sqrt(2*pi) = 1/Sqrt(tau).
   • The BBP formula for pi is easily modified for tau, and the resulting spigot algorithm for the hexademical digits is basically unchanged.
6. A gradual switch can start by simply writing tau = 2*pi whenever it comes up and using tau instead of pi from then on. Conversion between the notations is not at all difficult.
7. The symbol for tau looks very much like the symbol for pi.
8. The symbol pi has numerous standard uses (as the circle constant; as a generic symbol for a function; as a generic symbol for a permutation) as does tau, so people already have to deal with figuring out which usage is meant from context.

I don't have any reasons for continuing to use pi, apart from the fact that everyone knows that symbol. If the world were to forget its knowledge of pi, I think tau would emerge instead as the value everyone knows. Thoughts?