Tau (2π)

Circles are defined by their radius, not their diameter, so the argument runs that the natural circle constant is circumference over radius: τ = 2π = 6.28318530717959. Under that convention a full turn is τ radians, a quarter turn is τ/4, and the endless factors of 2π in Fourier analysis, the Gaussian integral and quantum mechanics collapse into single symbols. The angle fractions in particular become obvious to a student rather than something to be memorised.

Bob Palais made the case in a 2001 Mathematical Intelligencer opinion piece titled "π Is Wrong!", and Michael Hartl's Tau Manifesto of 2010 gave it the symbol and the following it has today, complete with Tau Day on 28 June. The counter-argument is not that τ is worse but that π is entrenched beyond dislodging, and that a handful of formulas — the area of a circle among them — are tidier with π. Nothing mathematical is at stake; it is a genuine and unusually good-natured argument about notation.