One of the things that really gets to me about this whole debate is insistence on both sides (however jocular it may be) that they are

right and the other side is

wrong. It is about context. There are times when [tex]\tau[/tex] is clearly superior, and other times when [tex]\pi[/tex] is better. There are convincing arguments in favor of both---your argument regarding the area of a unit circle is pretty good.

On the other hand, I find your argument regarding squaring the circle to be a draw, at best (one that I honestly had not thought about before). Squaring the circle cannot be done with compass and straightedge. Using [tex]\tau[/tex], terms involving [tex]\sqrt{2}[/tex] pop out. That should raise eyebrows regarding the impossibility of the construction. The [tex]\sqrt{\pi}[/tex] terms are interesting, and should also raise questions, but they don't really have the same pizazz.

I also find the argument that [tex]\tau[/tex] is overloaded to be a very weak argument. [tex]\pi[/tex] is also overloaded. Big [tex]\Pi[/tex] notation for products could be confusing, but more importantly, lowercase [tex]\pi[/tex] is used for osmotic pressure, and I've seen it used to express long term probabilities for Markov processes. I'm pretty sure that I have also seen it in economics. People make the argument that Hartl and Palais are showing their true colors as pure mathematicians, but it seems to me that the anti-[tex]\tau[/tex] side has lost track of the various uses of [tex]\pi[/tex].

Both letters are a bit overloaded---which is why I kind of like Palais' three-legged [tex]\pi[/tex].

Of course, at the end of the day, as I said, both sides have something going for them. The real question we should be asking is not "Which is better?" but "In what contexts is which better?" To someone immersed in the study of mathematics or its applications, it doesn't really matter which is used---we are sophisticated enough to use the symbol that best expresses the idea that we wish to express. On the other hand, what is a boy or girl to do when encountering trigonometry for the first time? In the US, the better part of a semester of high school is spent learning trigonometry in the context of the unit circle. I've had much better luck using [tex]\tau[/tex] than I ever had with [tex]\pi[/tex] (though, admittedly, my samples of each are small).

tl;dr version: Both symbols have pros and cons. [tex]\tau[/tex] is easier for beginning students. Advanced students and professionals can fend for themselves.

xander

EDIT: I should also note that I have started to add "Let [tex]\tau\equiv 2\pi[/tex]," to the beginning of most of my papers, despite the fact that I have this on my arm:

Clearly, I like both. ;)

FURTHER EDIT: The Russian letter т may be a good substitute for [tex]\tau[/tex], as well. We already use Hebrew, why not Russian as well? The cursive version looks like this: