Tau (Constant)

In 2010, Michael Hartl proposed to use the Greek letter tau (τ) to represent the circle constant for which: π = ​^τ⁄₂. He offered two reasons. First, τ is the number of radians in one turn, which allows fractions of a turn to be expressed more directly: for instance, a 3/4 turn would be represented as 3τ/4 rad instead of 3π/2 rad. Second, τ visually resembles π, whose association with the circle constant is unavoidable. Hartl's Tau Manifesto gives many examples of formulas that are asserted to be clearer where τ is used instead of π, such as a tighter association with the geometry of Euler's identity using e^iτ = 1 instead of e^iπ = −1.

Initially, neither of these proposals received widespread acceptance by the mathematical and scientific communities. However, the use of τ has become more widespread, for example:

• In 2012, the educational website Khan Academy began accepting answers expressed in terms of τ.
• The constant τ is made available in the Google calculator, Desmos graphing calculator and in several programming languages such as Python, Raku, Processing, Nim, Rust, Java, and .NET.
• It has also been used in at least one mathematical research article, authored by the τ-promoter Peter Harremoës.

The following table shows how various identities appear if τ = 2π was used instead of π. For a more complete list, see List of formulae involving π.

Formula	Using π	Using τ	Notes
Angle subtended by 1/4 of a circle	π/2 rad	τ/4 rad	τ/4 rad = 1/4 turn
Circumference C of a circle of radius r	C = 2πr	C = τr
Area of a circle	A = πr2	A = 1/2τr2	The area of a sector of angle θ is A = 1/2θr2.
Area of a regular n-gon with unit circumradius	A = n/2 sin 2π/n	A = n/2 sin τ/n
n-ball and n-sphere volume recurrence relation	Vn(r) = r/n Sn−1(r) Sn(r) = 2πr Vn−1(r)	Vn(r) = r/n Sn−1(r) Sn(r) = τr Vn−1(r)	V0(r) = 1 S0(r) = 2
Cauchy's integral formula	$f(a)=\frac{1}{2\pi i}{\oint }_{\gamma }\frac{f(z)}{z-a}dz$	$f(a)=\frac{1}{\tau i}{\oint }_{\gamma }\frac{f(z)}{z-a}dz$
Standard normal distribution	$\phi (x)=\frac{1}{\sqrt{2\pi }}{e}^{-\frac{x^2}{2}}$	$\phi (x)=\frac{1}{\sqrt{\tau }}{e}^{-\frac{x^2}{2}}$
Stirling's approximation	$n!\sim \sqrt{2\pi n}{\left(\frac{n}{e}\right)}^n$	$n!\sim \sqrt{\tau n}{\left(\frac{n}{e}\right)}^n$
Euler's identity	0       eiπ = −1 eiπ + 1 = 0	0     eiτ = 1 eiτ - 1 = 0	For any integer k, eikτ = 1
nth roots of unity	$e^{2\pi i\frac{k}{n}}=\cos \frac{2k\pi }{n}+i\sin \frac{2k\pi }{n}$	$e^{\tau i\frac{k}{n}}=\cos \frac{k\tau }{n}+i\sin \frac{k\tau }{n}$
Planck constant	$h=2\pi \hslash$	$h=\tau \hslash$	ħ is the reduced Planck constant.
Angular frequency	$\omega =2\pi f$	$\omega =\tau f$

References

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