Why we have to get rid of pi for the sake of good math

(via io9)
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I believe you've mentioned that tau reveals connections that pi does not. Could you provide an example of this?

The canonical example involves radian angle measure. For example, a right angle is a quarter turn of a circle, and its measure is tau over four, or one-quarter tau. Using pi, the same angle is pi over two, or one-half pi, which obscures the natural relationship between angle measure and the circle constant. As discussed in The Tau Manifesto, using tau also helps reveal the relationship between complex exponentiation and rotations in the complex plane. The geometric meaning of Euler's identity, for instance, is much clearer when written in terms of tau.

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And what about counterexamples where pi seems more useful? For instance, the area of a circle formula (A = πr^2) seems far more elegant with pi than with tau.

The formula for circular area is actually The Tau Manifesto's coup de grâce. You need to read the manifesto to get the full impact of the argument, but the short version is that the area of a circle has a natural factor of a half that disappears when using pi.

Incidentally, all counterexamples I know of are addressed in The Tau Manifesto. When they hear about the basic idea of tau, some people (without reading The Tau Manifesto) object that "tau ruins Euler's identity" or "the formula for circular area is better with pi". When questions about software are answered in the software's documentation, computer programmers are notorious for responding with "RTFM", which stands for "Read The *ahem* Fine Manual". In this vein, I urge potential correspondents to Read The Fine Manifesto before voicing their objections.

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