I'm a Tauist myself, but it should be pointed out the first one is a legitimately good argument. (Local introduction of a constant is easy, though.)
I don't expect to wake up one day and everybody suddenly agrees "Yes, tau is the winner!" I expect that either things will peter out, or tau will just gradually start showing up in real papers and stuff. Unfortunately, since K-12 mathematical curricula seem to have gotten stuck in 1920, switching the "official curricula" to tau is well down on my list of things that needs to happen to K-12 math education and at the current rate even if formal mathematics did just wake up tomorrow and decide tau was the way to go, it would be at least 50 years before that penetrated back down.
I've dozens of subtle little reasons, but I think that one shows it off best and is easiest to understand.
I was going to add that zero crossings for sine waves (I am into sound synthesis) are at integer multiples of pi, but that's just a funny way of stating the above.
I keep seeing this in my comments so I might as well respond for posterity, especially since this is not a sound mathematical argument for pi.
> zero crossings for sine waves are at integer multiples of pi
This is actually a strong argument for tau.
The sin wave measures the height of a circle at the angle given in radians. "Integer multiples of pi" don't immediately show you that there are two very different zeros: one going up, and one going down. Using tau shows you that explicitly: on half turns around the circle sin(tau/2), you're at 0 going down; on whole turns sin(tau) you're going up. You (literally) "come full circle" with integer multiples of tau—those 0s are equivalent.
The argument is the same with e^(i * pi * x). See Section 2.3 on tauday.com and the chart under "Eulerian Identities." Each integer increment corresponds to a rotation in the complex plane. The reason it's on the real line at 2, 4, 6 is because it takes two rotations to get back to the real line.
At 1 rotation (i), you're fully imaginary; at 2, fully real, but negative; 3, fully imaginary again, but negative; 4, you're back where you started, real and positive.
I'm sorry but your comment came off as condescending.
The graph I linked coincides with my needs, it was specially crafted to demonstrate that, not to show off some silly notion of elegance with no ounce of application. Sorry for not making that clear.
As for: Elegance is not just whether something is "pretty," as in, hey look, integers! It's also whether it has strong meaning. -- please try applying that value to your opinions about giving up pi for tau. I'm not the one guilty of that kind of thinking, you are.
> The mathematical world is as full of lonely pi's, as it is of 2*pi's. Now we need to move to tau/2 and tau, only to get a pi-manifesto in a couple of decades.
I don't expect to wake up one day and everybody suddenly agrees "Yes, tau is the winner!" I expect that either things will peter out, or tau will just gradually start showing up in real papers and stuff. Unfortunately, since K-12 mathematical curricula seem to have gotten stuck in 1920, switching the "official curricula" to tau is well down on my list of things that needs to happen to K-12 math education and at the current rate even if formal mathematics did just wake up tomorrow and decide tau was the way to go, it would be at least 50 years before that penetrated back down.