As for the volume of a sphere, etc., I don’t know why you’d say π appears all by itself in contrast to τ there. It’s no more intrinsically 4/3 π r[sup]3[/sup] than it is 2/3 τ r[sup]3[/sup]. Indeed, the very fact that we give these formulas in terms of the radius rather than the diameter shows the primary role the radius plays in mathematics, lending support again to the greater relevance of ratios to radius (as in τ) than of ratios to diameter.
When you leave the SDMB but do not have enough escape velocity, you eventually circle back, but to land safely when come back, bring pie.
Right. I’d read that Wikipedia page, though it also says that the 5 trillion digits have yet to be “verified”. (Computed independently by a second party, I assume.)
Hmmm. So how would you feel about making the circle’s circumference the primary yardstick — expressing areas, volumes, radii, solid angles, etcetera, in terms of that? Then we’d probably prefer a new constant defined as 1/π or 1/τ, to keep things tidy.
(This is not a serious proposal, in case that’s not obvious.)
Well, I’d like to reply, but since you say the proposal is non-serious, I’m not sure what you actually mean to get at with it.
That having been said, I’ll say the thoughts it brings to mind anyway. First, as for measuring angles, I think there are two most natural and commonly useful ways in mathematics to measure the angle of a circular arc: the ratio of its length to the radius (i.e., in radians) and the ratio of its length to the circumference (i.e., in revolutions). Absolutely, there are cases in which one would most naturally express angles in terms of the circumference. Hell, I think revolutions are, if anything, a more fundamental unit of angular measure than radians. τ’s purpose is simply to serve as the conversion factor between revolutions and radians. Of course, one could convert the other way with 1/τ; that’s just the flip-side of the same ratio. The point is, this conversion is the important, mathematically ubiquitous one, rather than the conversion between half-revolutions and radians which is given by π.
As for the rest of it (volumes, etc.), we tend to talk an awful lot about “unit circles”, “unit spheres”, etc., in the sense of having radius 1, and not so much about circles/spheres of circumference 1. And almost never about circles/spheres of diameter 1. And the reasons we do this are, whenever they apply, the same reasons we would most naturally want to express quantities in terms of the radius rather than other things.
Quoth glowacks:
Gamma(1/2) = sqrt(pi), or the more cumbersome sqrt(tau)/sqrt(2) .
True, although the presence of the sqrt already imputes some cumbersomeness. Put another way, what one has is (Gamma(1/2))[sup]2[/sup] = 1/2 τ. Why the 1/2 on the right? Well, given the 1/2 on the left, it’s not such an out-of-kind intrusion.
But, anyway, in explanation, the usual way to determine the value of Gamma(1/2) is by recognizing it as the Gaussian integral, and the usual way to determine the value of the Gaussian integral is by recognizing its square as the volume of a surface of revolution with unit height and unit mean squared radius, which is to say as the area of a unit radius circle. So the presence of the 1/2 here is the same as that of the 1/2 previously explained in determining the the area of a circle in terms of its radius and circumference, which is the same 1/2 as in the area of a triangle in terms of its base and height.
In general, for a unit radius ball in N dimensions, its (N-dimensional) “volume” will be 1/N * its ((N-1)-dimensional) “surface area” (because each infinitesimal patch of surface area contributes a pyramidal wedge of volume, and the mean cross-sectional “area” of an N-dimensional pyramidal wedge is 1/N * the base area (because x[sup]N-1[/sup] integrates to 1/N x[sup]N[/sup])). The presence of this same 1/N in some form or another throughout mathematics in general, accordingly, is unavoidable. The best we can do is make it most clearly visible as that 1/N, rather than hoping to cancel it out in some particular dimensions while only obfuscating it in other dimensions (to use the expression π rather than 1/2 τ with the intention of avoiding the 1/2 which arises in the 2-dimensional case is to get fixated on the arbitrary particular case N = 2, without doing anything for, and in fact somewhat ruffling, the expressions at other values of N).
[It’s also worth noting that τ gives not only the circumference of the unit circle in two dimensions, but is also, invariantly, the ratio of the “surface area” of S[sub]N+1[/sub] to the “interior volume” of D[sub]N[/sub] for any N, where D[sub]N[/sub] is the N-dimensional ball of unit radius, and S[sub]N[/sub] is the N-dimensional boundary of D[sub]N+1[/sub]. Thus, for example, we also have that τ is the ratio between the surface area of a 3d unit sphere and the length of a 1d unit radius interval, the ratio between the surface volume of a 4d unit hypersphere and the area of a 2d unit circle, etc. This combined with the previous observation recursively determines the interior volumes and surface areas of hyperspheres in any number of dimensions; again, τ is playing a primary role here.]
[And, one final remark: is there anyone who doesn’t find the generalized factorial (aka, the Pi function) more natural than the Gamma function (which is shifted over by one)? Amusingly, in terms of factorial, what we have is (1/2)! = 1/2 (1/2 τ)[sup]1/2[/sup], halves all over the place]
I’m here to eat my words. :smack: Perhaps, as an EE, I was simply too close to see the forest for the trees, but I’m convinced, especially after reading the τ Manifesto and Pi is Wrong. And there’s also a good discussion here which deftly addresses the area of a circle requiring τ/2 rather than simply π. I guess I was just too used (30+ years as an engineer) to always thinking it’s natural for 2π to represent a complete revolution or cycle and a single π for a half turn. It makes much more sense (now that I’ve had time to digest all this) that everything really would be simpler if we used τ to represent a complete rotation and (gasp!) τ/2 for a half turn! This is even more evident when considering other fractions of a circle, e.g. a quarter turn would be τ/4 rather than π/2. Three quarters would be 3τ/4 instead of 3π/2. :smack:
Probably little chance of it ever happening at this point; about as likely as converting the world to a Base 12 number system or the U.S. to the metric system. Man oh man, forget about the embarrassment of broadcasting Gilligan’s Island to the cosmos (it’s out to 45 light-years by now). Wait until they get a look at our mathematics. They’re going to think we’re all idiots. 
Of course, one of the chief differences between pi and pie is that if you have an infinite amount of pie, it will almost certainly repeat on you.
So pie is rational.
Especially since the definition of the Gamma function has that (n-1) and (n-2) in it, instead of the simpler (n) and (n-1). I really wonder, sometimes, why it was defined that way in the first place: I mean, I can at least see the historical rationale behind pi rather than tau, but I’ve never seen any reason for defining Gamma the way it is.
I believe Gauss originally introduced the Pi function, but Legendre switched over to the Gamma function and his influence on French mathematicians happened, in the shuffle of history, to become the dominant one. It’s not known why Legendre used the Gamma function instead, but one theory is that he found it more aesthetically pleasing to place the pole at 0 than at -1.
(The first pole, that is. You know what I mean.)