I don't have a strong opinion, but from what I've seen I think tau makes the most sense. These things are fun to think about, because it requires that you think about how pi is used in mathematics and contemplate how we organize and use the concepts related to pi. It's a good brain exercise.
But it's hardly worth getting into a fight over.
Edit: OK, e^(itau)=1, which is quite nice. :-/
e^(iτ) = 1
1 = e^0 = e^(iτ) = e^(i2τ) = ...
I have a position on Tau simply because it's fun to think and talk about, and I really don't care if stodgy mathematicians disapprove. :)
But I agree that it's a distraction, and I don't think the radial relationship is that important. I got over my annoyance of typing 2 * pi by thinking about it as the ratio of the diameter to the circumference.
Where else in math do you talk about the diameter?
I'm sorry that my personal idiosyncrasy bothers you so much.
Also notice how many other cases outside of 2d geometry uses sqrt(pi) or sqrt(2pi). The gamma function, normal distribution, stirling approximation etc. are all full of it, and when you rarely need pi or tau, adding a ^2 is much nicer than always having to draw a big ugly sqrt.
I don't mind tau especially, but it is seems the wrong battle to fight. Sqrt(pi) also has mythical powers: http://www.squarerootofpi.com/
I put tau into my math-related code last year, and I have enjoyed the results (which are much the same as before, of course).
All N-sphere volume formulas have whole-powers of 'pi' (or 'tau') as factors
which goes into great detail on surface areas and volumes of hyperspheres. Spoiler alert: it's actually pi/2 = tau/4 that emerges victorious.
pi/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - ...
"No no, we were transmitting on hydrogen * 𝛕!"
Admittedly, the radius does feel like the natural way to define a circle (all points r away from the center), however that is not the only way to define a circle, and we have no other intelligent species to use a reference for how objectively natural that way is.
Here's a similar argument: In mathematics, do we measure angles in radians or diameterans? Why is that?
>that is not the only way to define a circle
Ok, let's hear 'em! What rigorous definition of a circle is more simply expressed in terms of diameter?
The only one I can think of is, "the smallest volume shape that can contain a length D line segment at any angle", but that seems contrived to me. I shutter at using the solution to optimization problem as a definition, except when there is no other choice. What's worst, it seems like a trivial derivative of the radial definition, just with extraneous concepts tacked on.
You could also imagine rotating a line segment about its center. This has a nice sense of symmetry, as we are rotating about the center, not an arbitrary end. This also implies a generalization to off center rotations.
The thing is, tau is really my go-to variable when I need a 2nd constant to compare with t. It is already used as a time constant, a dummy integration variable that substitutes for t... The manifesto has a pre-made counter-argument to this... But I'm not sure it's convincing.
The thing is that we like to say some variation of e^(-tau) a lot, and we also like to say e^(2 pi i) a lot. And often we combine these (rotation and exponential decay), and we get e^(-tau+2pi i). And this would ruin tau notation! It would be e^(-tau'+tau i)...
Before writing The Tau Manifesto, I tried several other candidates before settling on tau. After using it for a while, tau started to "feel right" in ways that other choices didn't. Then, as noted in the manifesto, I found out that I wasn't alone:
Since the original launch of The Tau Manifesto, I have learned that Peter Harremoës independently proposed using τ to “π Is Wrong!” author Bob Palais in 2010, John Fisher proposed τ in a Usenet post in 2004, and Joseph Lindenberg anticipated both the argument and the symbol more than twenty years before!
I like the explanation page for why tau a lot, and I think that page is much more important than actually using tau in real mathematical works. It's a great explanation of how 2*pi is the more natural quantity, and how it comes into play in the different functions. But it's just much easier to deal with factors of 2 than it is to explain alternate notation.
In fact, I think a lot of making mathematical works easy to understand is using commonly accepted notation...
I always favored a variation on ◷, something like ◯̶ or ◯̵ or ○̵. Unfortunately, combined glyphs are representationally very unstable.
There are no available symbols. The only option is to introduce a new one, like ת suggested by samatman, or possibly by using a weird typeface (like \mathfrak in LaTeX).
Incidentally, my friend has pointed out to me that sigma is a much better choice than tau if we're going with existing symbols. Who cares about tau sounding like a "t" for "turn", when sigma actually looks like someone trying to measure the circumference of a circle with the radius of the circle: σ.
Who cares about tau sounding like a "t" for "turn"
Pi comes from perimeter, phi comes from Phidias, etc. There's a long tradition of naming constants using a linguistic root (typically Greek). To my knowledge, there's no precedent for using a symbol based on its visual appearance to a geometric object. (Of course, you could always break with tradition.)
Why not just start in on Chinese characters? There's plenty of those and I'm sure Chinese mathematicians could suggest some likely candidates based on their long mathematical traditions.
Like if we have a function f(x,y), we often write the "total" derivative of f as
df = (df/dx) dx + (df/dy) dy
And the rules for manipulating a differential like this are quite strange (e.g., dx on top cannot cancel the dx on bottom).
There's a language of differential forms that makes this notation a bit more rigorous, but I still think the notation is very misleading and confusing, especially for those who begin to learn calculus.
There's even a gloss: ת in mainstream Kabbalah corresponds to Saturn, the god of time.
Electrical engineers need i for something else, incidentally, and use j for imaginary numbers. Mathematical notation is intentionally flexible.
Indeed, the author of The Pi Manifesto essentially rebutted his own work in March of last year: