By myself, I tried to make a comparison of the formulae for τ
(and, just for fun, I defined a constant η
/2 = τ
/4 and made formulas based off of that, too).
First off, some definitions, in case some aren't familiar.
n! is the factorial of n (meaning, 1*2*3*...*(n-1)*n).
n!! is the double factorial of n (meaning, 1*3*5*...*(n-2)*n for odd n, and 2*4*6*...*(n-2)*n for even n).
⌊n⌋ is the floor of n (the greatest integer ≤ n)
Also, a sidenote: the π
and r formula is actually traditionally written in terms of the gamma function, which has a weird definition as:
This was supposed to describe circles and spheres with a few more dimensions, so what the heck is that
doing in there? For this analysis, I will look only at formulae that don't require knowledge of calculus to evaluate.
Now for the formulas.
(there are closed-form formulae for each, but I went with the simplest formulae for a result in simplest form)
As you can see, the formulae with D for diameter look very compact, and are pretty easy to actually calculate, with τ
being pretty much equal in this metric, but η
coming out on top (by design). So if you're an engineer, or for some other reason you think diameter is more fundamental than radius, the "simplest" circle constant to use in this formula is η
(though, because it is
diameter, it'd be more compatible in the math to use π
The formulae with r, however, show wildly varied complexities for each. Let's pick apart each one in detail.
In the π
formula, we can see that for even numbers, the coefficient is the reciprocal of successive factorials, and for odd numbers, it is a power of 2 (which happens to be 1 greater than the power of π
...this might suggest something) over successive odd double factorials. This description seems nice and easy...until you think of the following identity for even n:
Because the 2 here has the same power as the π
in the volume formula, this seems to suggest we could unify the even and odd cases with τ
. Indeed, this turns out to be the case.
In the τ
formula, we can see that the coefficient is either 1 or 2 (depending on the parity of n) over n!!. This is even simpler to calculate, and it even demonstrates the recurrence formula more transparently, V(n) = τ
*rē*V(n-2)/n . So if you, like almost all other mathematicians, think radius is more fundamental than diameter, τ
is the best constant to use (and τ
and the radius are compatibly defined, too).
In the η
formula...trying to put it in simplest form is way too complicated. Sure, you could write both cases as 2^n / n!!, but this barely even matters, as the τ
formula has no 2^n to worry about anyways, so the τ
formula is simpler.
Bottom line: the simplest formula for volume of an n-sphere uses either tau and the radius, or eta and the diameter. However, because tau is defined as C/r, and eta is not defined in terms of D, tau is the best constant to use in this context.