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.
With 'tau' you only get the positive half (consider the integer values of x; which are 2, 4, 6 for 'tau'): http://www.wolframalpha.com/input/?i=plot+e^%28pi+*+i+*+x%29
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.
> 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.
This is what it would look like with tau, and it's exactly what you expect: http://www.wolframalpha.com/input/?i=plot+e^%280.5*pi+*+i+*+...
Elegance is not just whether something is "pretty," as in, hey look, integers! It's also whether it has strong meaning.
> 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.
(Yes, we know that Tau doesn't really sound like anything, but this was fun and better than I expected.)
But seems like you could generate nice, non-repeating elevator music by hooking something up to an RNG or PRNG, and never hear the same song in the elevator twice.
Previous discussions:
Plus Tau Day's a fun excuse to eat two pies.
How to derive the value of the integral: Square the integral to make it an integral in two variables, introduce polar coordinates, then change variables.
One could always celebrate 2\pi day. ;)
Yes, pi shows up as the prime counting function, but there it's a function, which clears up the otherwise ambiguous notation. Furthermore these abuses of notation are generally considered a bad thing, something we try to avoid.
As for the intuitiveness of such deep results as Stirling's formula and the even values of the Riemann zeta: this is to concern oneself with the upholstery on the Space Shuttle.
If you want a new pi symbol, might I suggest the variant pi described here: http://en.wikipedia.org/wiki/Pi_%28letter%29 -- though I'm afraid this is all a waste of time and energy.
Slightly different motivation, though, as it seemed to me that pi/2 was really more interesting, and anyway it is really much easier to write multiplication than fractions when you run into a discrepancy. Here the roots of the sine are at even multiples and the roots of the cosine are at odd multiples, and e^(i pi/2) = i is more interesting than the previous incarnations of the Euler formula.
The thing is that the important parts of these formulae are the way they are derived and the structure they represent, not their appearance on a sheet of paper. If you can understand what relates the zeta function at even numbers to the ratio of a circle's radius and circumference, it really does not matter if you choose to represent this relation with a drawing of an elephant.
I am certain my kids will have an easier time remembering tau/4, as I do myself.
The other compelling thing for me came from remembering just how many integrals from 0 to 2pi I wrote over my freshman complex analysis class. A lot. Less notation is always nice; having tau represent the entire circle just makes a lot of sense!
WTF is this? I'm eagerly awaiting an explanation of what you can do with X that you can't do with 2X.
It's a moot anal point. X or 2Y, using one over the other doesn't solve anything.
Far more important step for mathematicians would be to switch from base 10 to something more useful like base 8.
Now, no one is claiming that Tau vs. Pi is even close to the same level of importance. But it makes some things just that much easier.
The fact is that making things easier doesn't necessarily mean improvement.
If you've ever used Stirling's approximation, this is the paper that first points out that it's a divergent series.
Scan of original (it's also on JSTOR): http://www.york.ac.uk/depts/maths/histstat/letter.pdf
With modern typesetting and an explanation: http://www.stat.ucla.edu/history/letter.pdf
(I don't seriously think we should change from pi to tau.)
This statement is never supported (the fan part). Bad BBC, bad!
