Gracious Living

tauism
December 11, 2010, 19:32
Filed under: Math, personal | Tags: , , ,

I’ve been really busy with finals and haven’t had time to finish a proper post, but wow is this amazing.  The suggestion (not new, but very eloquently expressed) is to stop using $\pi$, the ratio of a circle’s circumference to its diameter, and start using $\tau=2\pi$, the ratio of its circumference to its radius.  (Spelled “tau”, rhymes with “cow”: do you know your Greek alphabet?)  Pretty much every important use of $\pi$ is actually a use of $2\pi$: the circumference formula, integration in polar coordinates or around a circle, finding roots of unity, but also Gaussian/normal distribution stuff, Fourier transforms, and zeta function identities.  Where there are exceptions, the insistence on using $\pi$ instead of $2\pi$ obscures the nature of the equation: writing the area of a circle as $\frac{1}{2}\tau r^2$ instead of $\pi r^2$ highlights the fact that it’s the output of an integral, and though $e^{i\pi}+1=0$ looks beautiful, it obscures its own meaning: complex exponentiation by $\pi$ corresponds to a rotation by 180 degrees or $\pi$ radians.  If you write $e^{i\tau}=1$ (or, as the author suggests, $e^{i\tau}=1+0$ to preserve the “beauty”), you immediately see what the formula’s getting at: $x\mapsto e^{ix}$ is a periodic function with period $\tau$ corresponding to a rotation of the complex plane.

And if that weren’t enough, look at this piece of beauty:

See how easy things could be?

Although $\pi$ has been around for a long time, changes in terminology do happen and math moves right along. I don’t know what we would do if we still had to do group theory with Galois’ original wordings of “substitutions” and “arrangements.” So I think that among mathematicians, this could catch on pretty easily.  It’s easy to spread the meme, too — when you’re presenting something, just start out with “let $\tau=2\pi$” and watch the daylight glimmer on people’s faces.

But I’m not so sure that this could catch on in schools, which is where it is most needed.  I’m a math major and I still take a few seconds to remember those godforsaken radian angle measures.  It’s annoying and breaks one’s train of thought.  Why do we expect school-age kids to have to jump through this hoop every time they do a trig calculation, which is pretty much all they’re graded on? (another stupid thing about education, but I digress.)  All it teaches them is that math is about memorization of arbitrary things.  With $\tau$, there’s nothing to memorize: one-third of the circle is $\tau/3$.

I just feel that there’s considerable inertia in changing math curricula, especially over something as basic as this.  The only reason we learn math at all is because Eisenhower and Kennedy decided to put a man on the moon.  And because Egypt was troubled by the horrible asp.  Putting $\tau$ into the curriculum would require schools to pay for new versions of these stupid things called “curriculum planners” and “textbooks.”  Quite frankly, that’s not what American schools need to be spending their money on, and it’s by no means even the biggest problem with math education.  The author of the page I linked suggests that teachers introduce $\tau$ to teach kids critical thinking and show them that the best way to do something isn’t always the way you’re first taught.  This sounds awesome — an actually interesting debate in a math classroom?! — but it seems like a teacher that did that would meet interference from higher up, on the grounds that it stifles kids’ ability to communicate effectively (all math education being about communication, to people as diverse as cashiers and accountants).

On the other hand, the textbook-publishing lobby would love an excuse to publish a new edition.  In addition, if $\tau$ becomes accepted in the math community, it’s only a matter of time before education starts changing as well.  Especially if the curriculum started to change so that it had more to do with actual math… wouldn’t that be a treat.