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Comment by tankenmate | Hacker News Reader

But as explained in the manifesto, quite often when you see π on its own generally it is a case of 2 * 1/2 π. And by removing the 2 * 1/2 π terminology you're missing the significance of why the 2 * term is there! The best example of this in the article is the λ (nee τ) version of the volume of a n-sphere; the symbolic algebra form makes far more sense.

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Eh, the example I had in mind when I was writing that comment was of the evaluations of the Riemann zeta function, where the actual "factor on the pi" is not all that important because there's other things being mixed in anyways.

I might be misunderstanding you, but if you're referring to things such as ζ(2)=π²/6, then that doesn't strike me as a good example of "nothing to do with circles" and "in those places tau does not do better". This value is a special case of a more general formula that specifically features (2π)^(2n), so on surface tau would seem more natural here [1]. Also, the most intuitive explanation that I have seen for this particular case (the value of ζ(2)) was in a 3Blue1Brown's video, and it did involve circles. [2]

[1] https://en.wikipedia.org/wiki/Particular_values_of_the_Riema...

[2] https://www.youtube.com/watch?v=d-o3eB9sfls

So in what way does tau do worse?

It kinda ruins Euler's identity.

Arguably it makes it more clear what's really going on. With tau, the identity can be formulated either:

- e^(iτ) = 1, meaning that taking a full revolution around the complex plane brings you back to 1, where you started

- e^(iτ/2) = -1, meaning that taking a half revolution takes you to the opposite of where you started, -1.

If you insist on the "contains the additive identity" formulation (which IMO is gratuitous because e^(ix) is all about the complex plane, not regular addition and multiplication), you can do either e^(iτ) + 0 = 1, or e^(iτ) - 1 = 0.

Good points, I was just thinking about it superficially.

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