Favorite equation revisited

I wrote about my new favorite equation a couple months ago, but I neglected to mention an insight that has been revealed to me!

It's Tau Day today! Long live Tau! What does this have to do with my equation? Well, my equation was:

[math]\int_{-\infty}^{\infty} e^{-x^2} = \sqrt{\pi}[/math]

(I have no idea why the white bg might not extend all the way.)

But since Tau = 2*pi, this is also expressed as:

[math]\int_{-\infty}^{\infty} e^{-x^2} = \sqrt{\frac{1}{2}\tau}[/math]

I'm a compromiser in the Tau vs. Pi wars: I love the intuitive simplicity of Tau and not having to write factors of 2 all over, yet I also don't like writing 1/2 Tau unless there's some reason to such as in the formula for area of a circle. (A = pi*r^2 = tau/2 * r^2 which is analogous to other integral-derived equations in physics such as kinetic energy where 1/2*m*v^2 = KE.)

I thought my favorite equation was such a place to just use pi. But I was wrong, even by my own reasoning! I want to use Tau when it prompts the intuitions for circles and polar coordinates. If you restate the equation as I have with Tau, it actually illustrates the proof! The sqrt() by itself implies you might want to see what happens if you square the left side, and the tau implies you might want to look at it from a polar coordinate perspective. And indeed when you square the left side, change variables to polar coordinates and integrate over the polar plane, you have your proof.

Enjoy your two pies today!

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Posted on 2011-06-28 by Jach

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