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This page is about the mathematical constant. For the Greek letter, see Tau (letter). For other uses, see Tau (disambiguation).

The number 𝜏 (/ˈtaʊ, ˈtɔː, ˈtɒ/ ^ⓘ; spelled out as tau) is a mathematical constant that is the ratio of a circle's circumference to its radius. It is approximately equal to 6.28 and exactly equal to 2π.

𝜏 and π are both circle constants relating the circumference of a circle to its linear dimension - the radius in the case of 𝜏 and diameter in the case of π.

While π is used almost exclusively in mainstream mathematical education and practice, it has been proposed that 𝜏 should be used instead. Proponents argue that 𝜏 is the more natural circle constant and it's use leads to conceptually simpler and more intuitive mathematical notation (ref).

Critics have responded that the benefits of using 𝜏 over π are trivial and that given the ubiquity and historical significance of π a change is unlikely to occur.

Fundamentals

Definition

𝜏 is defined as the ratio of a circle's circumference to its radius:$$\tau ={\frac {C}{r}}$$

Relationship to π

As 𝜏 is exactly equal to 2π it shares many of the properties of π including being both an irrational and transcendental number.

History of proposals for a single letter to represent 2π

See also: Pi § Adoption of the symbol π

An arc of a circle with the same length as the radius of that circle corresponds to an angle of 1 radian. A full circle corresponds to a full turn, or approximately 6.28 radians, which is expressed here using the Greek letter tau (τ).

Some special angles in radians, stated in terms of 𝜏.

A comparison of angles expressed in degrees and radians.

The meaning of the symbol $\pi$ was not originally fixed to the ratio of the circumference and the diameter. In 1697, David Gregory used ⁠π/ρ⁠ (pi over rho) to denote the perimeter of a circle (i.e., the circumference) divided by its radius. However, earlier in 1647, William Oughtred had used ⁠δ/π⁠ (delta over pi) for the ratio of the diameter to perimeter. The first use of the symbol π on its own with its present meaning (of perimeter divided by diameter) was in 1706 by the Welsh mathematician William Jones.

The first known usage of a single letter to denote the 6.28... constant was in Leonhard Euler's 1727 Essay Explaining the Properties of Air, where it was denoted by the letter π. Euler would later use the letter π for the 3.14... constant in his 1736 Mechanica and 1748 Introductio in analysin infinitorum, though defined as half the circumference of a circle of radius 1—a unit circle—rather than the ratio of circumference to diameter. Elsewhere in Mechanica, Euler instead used the letter π for one-fourth of the circumference of a unit circle, or 1.57... . Usage of the letter π, sometimes for 3.14... and other times for 6.28..., became widespread, with the definition varying as late as 1761; afterward, π was standardized as being equal to 3.14... .

Several people have independently proposed using 𝜏 = 2π, including:

• Joseph Lindenburg (c. 1990)
• John Fisher (2004)
• Peter Harremoës (2010)
• Michael Hartl (2010)

In 2001, Robert Palais proposed using the number of radians in a turn as the fundamental circle constant instead of π, which amounts to the number of radians in half a turn, in order to make mathematics simpler and more intuitive. His proposal used a "π with three legs" symbol to denote the constant ($\pi \!\;\!\!\!\pi =2\pi$).

In 2008, Robert P. Crease proposed the idea of defining a constant as the ratio of circumference to radius, a proposal supported by John Horton Conway. Crease used the Greek letter psi: $\psi =2\pi$.

The same year, Thomas Colignatus proposed the uppercase Greek letter theta, Θ, to represent 2π. The Greek letter theta derives from the Phoenician and Hebrew letter teth, 𐤈 or ט, and it has been observed that the older version of the symbol, which means wheel, resembles a wheel with four spokes. It has also been proposed to use the wheel symbol, teth, to represent the value 2π, and more recently a connection has been made among other ancient cultures on the existence of a wheel, sun, circle, or disk symbol—i.e. other variations of teth—as representation for 2π.

In 2010, Michael Hartl proposed to use the Greek letter tau to represent the circle constant: τ = 2π. He offered several reasons for the choice of constant, primarily that it 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. As for the choice of notation, he offered two reasons. First, τ is the number of radians in one turn, and both τ and turn begin with a /t/ sound. 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. For example, Hartl asserts that replacing Euler's identity e^iπ = −1 by e^iτ = 1 (which Hartl also calls "Euler's identity") is more fundamental and meaningful. He also claims that the formula for circular area in terms of τ, A = ⁠1/2⁠𝜏r², contains a natural factor of ⁠1/2⁠ arising from integration.

Initially, this proposal did not receive significant acceptance by the mathematical and scientific communities. However, the use of τ has become more widespread.

The following table shows how various identities appear when τ = 2π is 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	${\color {orangered}{\frac {\pi }{2}}}{\text{ rad}}$	${\color {orangered}{\frac {\tau }{4}}}{\text{ rad}}$	⁠τ/4⁠ rad = ⁠1/4⁠ turn
Circumference of a circle	$C={\color {orangered}2\pi }r$	$C={\color {orangered}\tau }r$	The length of an arc of angle θ is L = θr.
Area of a circle	$A={\color {orangered}\pi }r^{2}$	$A={\color {orangered}{\frac {1}{2}}\tau }r^{2}$	The area of a sector of angle θ is A = ⁠1/2⁠θr2.
Area of a regular n-gon with unit circumradius	$A={\frac {n}{2}}\sin {\frac {\color {orangered}2\pi }{n}}$	$A={\frac {n}{2}}\sin {\frac {\color {orangered}\tau }{n}}$
n-ball and n-sphere volume recurrence relation	$V_{n}(r)={\frac {r}{n}}S_{n-1}(r)$ $S_{n}(r)={\color {orangered}2\pi }rV_{n-1}(r)$	$V_{n}(r)={\frac {r}{n}}S_{n-1}(r)$ $S_{n}(r)={\color {orangered}\tau }rV_{n-1}(r)$	V0(r) = 1 S0(r) = 2
Cauchy's integral formula	$f(a)={\frac {1}{{\color {orangered}2\pi }i}}\oint _{\gamma }{\frac {f(z)}{z-a}}\,dz$	$f(a)={\frac {1}{{\color {orangered}\tau }i}}\oint _{\gamma }{\frac {f(z)}{z-a}}\,dz$	$\gamma$ is the boundary of a disk containing $a$ in the complex plane.
Standard normal distribution	$\varphi (x)={\frac {1}{\sqrt {\color {orangered}2\pi }}}e^{-{\frac {x^{2}}{2}}}$	$\varphi (x)={\frac {1}{\sqrt {\color {orangered}\tau }}}e^{-{\frac {x^{2}}{2}}}$
Stirling's approximation	$n!\sim {\sqrt {{\color {orangered}2\pi }n}}\left({\frac {n}{e}}\right)^{n}$	$n!\sim {\sqrt {{\color {orangered}\tau }n}}\left({\frac {n}{e}}\right)^{n}$
nth roots of unity	$e^{{\color {orangered}2\pi }i{\frac {k}{n}}}=\cos {\frac {{\color {orangered}2}k{\color {orangered}\pi }}{n}}+i\sin {\frac {{\color {orangered}2}k{\color {orangered}\pi }}{n}}$	$e^{{\color {orangered}\tau }i{\frac {k}{n}}}=\cos {\frac {k{\color {orangered}\tau }}{n}}+i\sin {\frac {k{\color {orangered}\tau }}{n}}$
Planck constant	$h={\color {orangered}2\pi }\hbar$	$h={\color {orangered}\tau }\hbar$	ħ is the reduced Planck constant.
Angular frequency	$\omega ={\color {orangered}2\pi }f$	$\omega ={\color {orangered}\tau }f$
Riemann's functional equation	$$\zeta (s)={\color {orangered}2^{s}\pi ^{s-1}}\ \sin \left(s{\color {orangered}{\frac {\pi }{2}}}\right)\ \Gamma (1-s)\ \zeta (1-s)$$	$$\zeta (s)={\color {orangered}2\tau ^{s-1}}\ \sin \left(s{\color {orangered}{\frac {\tau }{4}}}\right)\ \Gamma (1-s)\ \zeta (1-s)$$	$2^{s}({\frac {\tau }{2}})^{s-1}$ reduces to $2\tau ^{s-1}$

In culture

𝜏 has made numerous appearances in culture. It is celebrated annually on June 28, known as Tau Day. 𝜏 has been covered in videos by Vi Hart, Numberphile, SciShow, Steve Mould, Khan Academy, and 3Blue1Brown, and it has appeared in the comics xkcd, Saturday Morning Breakfast Cereal, and Sally Forth. The Massachusetts Institute of Technology usually announces admissions on March 14 at 6:28 p.m., which is on Pi Day at Tau Time.