Tau (mathematics)

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 (Template:Math).

A comparison of angles expressed in degrees and radians

The number Template:Math (Template:IPAc-en; spelled out as tau) is a mathematical constant that is the ratio of a circle's circumference to its radius. It is exactly equal to 2[[pi|Template:Math]] and approximately equal to 6.283185.

Template:Math and [[Pi|Template:Math]] are both circle constants relating the circumference of a circle to its linear dimension: the radius in the case of Template:Math; the diameter in the case of Template:Math.

While Template:Math is used almost exclusively in mainstream mathematical education and practice, it has been proposed, most notably by Michael Hartl in 2010, that Template:Math should be used instead. Hartl and other proponents argue that Template:Math is the more natural circle constant and its use leads to conceptually simpler and more intuitive mathematical notation.

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

The proposal did not initially gain widespread acceptance in the mathematical community, but awareness of Template:Math has become more widespread, including having been added to several major programming languages and calculators.

Fundamentals

The number Template:Math is commonly defined as the ratio of the circumference Template:Math to the radius Template:Math of a circle:$$\tau =\frac{C}{r}.$$ Here, the circumference is the distance around the circle. A circle is defined as a closed curve formed by the set of all points in a plane that are a given distance from a fixed point, where the given distance is called the radius. The ratio Template:Math is constant regardless of the circle's size, designating Template:Math as the fixed ratio between the circumference and the radius of any circle.

The ratio of a circle's circumference Template:Math to its radius Template:Math can be defined through the number pi Template:Math: $$\pi =\frac{C}{2r},$$ implying that Template:Math equals Template:Math. Accordingly, the number Template:Math shares many of the properties of Template:Math, including being a transcendental number and hence also irrational.

Some special angles in radians, stated in terms of Template:Math

When radians are used as the unit of angular measure there are Template:Math radians in one full turn of a circle, and the radian angle is aligned with the proportion of a full turn around the circle: Template:Math is an eighth of a turn; Template:Math is three-quarters of a turn.

Digits of Template:Math

The first 51 decimal digits of Template:Math are: Template:Gaps

In 1424, Jamshid al-Kashi computed Template:Math to 9 sexagesimal (base 60) digits. The first 16 sexagesimal digits of Template:Math are: Template:Gaps

Numerical definitions

Like Template:Math, Template:Math can also be defined analytically, in terms of integrals, series or trigonometric functions. Template:Math can be defined as the smallest positive real number Template:Math such that Template:Math, or the period length of the sine and cosine functions. The sine and cosine can be defined independently of geometry using Taylor series.

Template:Math can be defined as the integral Template:Tmath, which is half the area of a circle of radius 2, or as the sum of an infinite series, such as $$\tau =8\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n}{2n+1}=8-\frac{8}{3}+\frac{8}{5}-\frac{8}{7}+\frac{8}{9}\mathrm{...}$$ More series definitions can be found at [[List of formulae involving π|List of formulae involving Template:Math]].

History

The proposal to use the Greek letter Template:Math as a circle constant representing Template:Math dates to Michael Hartl's 2010 publication, The Tau Manifesto,Template:Efn although the symbol had been independently suggested earlier by Joseph Lindenburg (Template:Circa), John Fisher (2004) and Peter Harremoës (2010).

Hartl offered two reasons for the choice of notation. First, Template:Math is the number of radians in one turn, and both Template:Math and turn begin with a Template:IPAc-en sound. Second, Template:Math visually resembles Template:Math, whose association with the circle constant is unavoidable.

Earlier proposals

There had been a number of earlier proposals for a new circle constant equal to Template:Math, together with varying suggestions for its name and symbol.

In 2001, Dr. Bob Palais of the University of Utah asserted that Template:Math is "wrong" as the fundamental circle constant arguing instead that Template:Math was the proper value. He proposed using a "Template:Math with three legs" symbol to denote the constant (Template:Tmath), and referred to angles as fractions of a "turn" (Template:Tmath). Palais stated that the word "turn" served as both the name of the new constant and a reference to the ordinary language meaning of turn.

In 2008, Robert P. Crease proposed defining a constant as the ratio of circumference to radius, an idea supported by John Horton Conway. Crease used the Greek letter psi: Template:Nowrap

The same year, Thomas Colignatus proposed the uppercase Greek letter theta, Template:Math, to represent Template:Math due to its visual resemblance of a circle. For a similar reason another proposal suggested the Phoenician and Hebrew letter teth, 𐤈 or ט, (from which the letter theta was derived), due to its connection with wheels and circles in ancient cultures.

Use of the symbol π to represent 6.28

The meaning of the symbol Template:Math was not originally defined as the ratio of circumference to diameter, and at times was used in representations of the constant 6.28... .

Early works in circle geometry used the letter Template:Math to designate the perimeter (i.e., circumference) in different fractional representations of circle constants and in 1697 David Gregory used Template:Math (pi over rho) to denote the perimeter divided by the radius (6.28...).

Subsequently Template:Math came to be used as a single symbol to represent the ratios in whole. Leonhard Euler initially used the single letter Template:Math to denote the constant 6.28... in his 1727 Essay Explaining the Properties of Air. Euler would later use the letter Template:Math for 3.14... in his 1736 Mechanica and 1748 Introductio in analysin infinitorum, though defined as half the circumference of a circle of radius 1 rather than the ratio of circumference to diameter. Elsewhere in Mechanica, Euler instead used the letter Template:Math for one-fourth of the circumference of a unit circle, or 1.57... . Usage of the letter Template:Math, sometimes for 3.14... and other times for 6.28..., became widespread, with the definition varying as late as 1761; afterward, Template:Math was standardized as being equal to 3.14... .

Notation using τ

Proponents argue that while use of Template:Math in place of Template:Math does not change any of the underlying mathematics, it does lead to simpler and more intuitive notation in many areas. Michael Hartl's Tau ManifestoTemplate:Efn gives many examples of formulas that are asserted to be clearer where Template:Math is used instead of Template:Math.

Units of angle

Hartl and Robert Palais have argued that Template:Math allows radian angles to be expressed more directly and in a way that makes clear the link between the radian measure and rotation around the unit circle. For instance, Template:Math can be easily interpreted as Template:Math of a turn around the unit circle in contrast with the same angle written as Template:Math, where the meaning could be obscured, particularly for children and students of mathematics.

Critics have responded that a full rotation is not necessarily the correct or fundamental reference measure for angles and two other possibilities, the right angle and straight angle, each have historical precedent. Euclid used the right angle as the basic unit of angle, and David Butler has suggested that Template:Math, which he denotes with the Greek letter Template:Math (eta), should be seen as the fundamental circle constant.

Trigonometric functions

Plots of common trigonometric functions (sin, cos, tan, cot, sec, csc) with x-axis running from −τ through +τ

Hartl has argued that the periodic trigonometric functions are simplified when using Template:Math, as it aligns the function argument with the function period: Template:Math repeats with period Template:Math, reaches a maximum at Template:Math and a minimum at Template:Math.

Area of a circle

Critics have argued that the formula for the area of a circle is more complicated when restated as Template:Math. Hartl and others respond that the Template:Sfrac factor is meaningful, arising from either integration or geometric proofs for the area of a circle as half the circumference times the radius.

Euler's identity

A common criticism of Template:Math is that Euler's identity, Template:Math, sometimes claimed to be "the most beautiful theorem in mathematics" is made less elegant rendered as Template:Math. Hartl has asserted that Template:Math (which he also called "Euler's identity") is more fundamental and meaningful. John Conway noted that Euler's identity is a specific case of the general formula of the Template:Mathth roots of unity, Template:Math, which he maintained is preferable and more economical than Euler's identity.

Comparison of identities

The following table shows how various identities appear when Template:Math is used instead of Template:Math. For a more complete list, see [[List of formulae involving π|List of formulae involving Template:Math]].

Formula	Using Template:Math	Using Template:Math	Notes
Angle subtended by Template:Sfrac of a circle	$\frac{\pi }{2}\text{ rad}$	$\frac{\tau }{4}\text{ rad}$	Template:Math
Circumference of a circle	$C=2\pi r$	$C=\tau r$	The length of an arc of angle Template:Math is Template:Math.
Area of a circle	$A=\pi r^2$	$A=\frac{1}{2}\tau r^2$	The area of a sector of angle Template:Math is Template:Math.
Area of a regular [[Polygon|Template:Mvar-gon]] with unit circumradius	$A=\frac{n}{2}\sin \frac{2\pi }{n}$	$A=\frac{n}{2}\sin \frac{\tau }{n}$
[[n-sphere#Volume and surface area|Template:Mvar-ball and Template:Math-sphere volume recurrence relation]]	$V_n(r)=\frac{r}{n}{S}_{n-1}(r)$ $S_n(r)=2\pi r{V}_{n-1}(r)$	$V_n(r)=\frac{r}{n}{S}_{n-1}(r)$ $S_n(r)=\tau r{V}_{n-1}(r)$	Template:MathTemplate:Br Template:Math
Cauchy's integral formula	$f(a)=\frac{1}{2\pi i}{\oint }_{\gamma }\frac{f(z)}{z-a}dz$	$f(a)=\frac{1}{\tau i}{\oint }_{\gamma }\frac{f(z)}{z-a}dz$	Template:Math is the boundary of a disk containing Template:Math in the complex plane.
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	$n!\sim \sqrt{2\pi n}{\left(\frac{n}{e}\right)}^n$	$n!\sim \sqrt{\tau n}{\left(\frac{n}{e}\right)}^n$
[[Root of unity|Template:Mvarth 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	$h=2\pi \hslash$	$h=\tau \hslash$	Template:Math is the reduced Planck constant.
Angular frequency	$\omega =2\pi f$	$\omega =\tau f$
Riemann's functional equation	$$\zeta (s)=2^s{\pi }^{s-1}\sin \left(s\frac{\pi }{2}\right)\mathrm{\Gamma }(1-s)\zeta (1-s)$$	$$\zeta (s)=2{\tau }^{s-1}\sin \left(s\frac{\tau }{4}\right)\mathrm{\Gamma }(1-s)\zeta (1-s)$$	$2^s(\frac{\tau }{2}{)}^{s-1}$ reduces to $2{\tau }^{s-1}$

In culture

Template:Math has made numerous appearances in culture. It is celebrated annually on June 28, known as Tau Day. Supporters of Template:Math are called tauists. Template:Math 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 6:28Template:Nbspp.m., which is on Pi Day at Tau Time. Peter Harremoës has used Template:Math in a mathematical research article which was granted Editor's award of the year.

In programming languages and calculators

The following table documents various programming languages that have implemented the circle constant for converting between turns and radians. All of the languages below support the name "Tau" in some casing, but Processing also supports "TWO_PI" and Raku also supports the symbol "τ" for accessing the same value.

Support for the circle constant in various programming languages

Language	Identifiers	First Version	Year Released
C# / .NET	System.Math.Tau and System.MathF.Tau	5.0	2020
Crystal	TAU	0.36.0	2021
Eiffel	math_constants.Tau	Curtiss	Not yet released
Erlang	math:tau/0	OTP 26.0	2023
GDScript	TAU	Godot 3.0	2018
Java	Math.TAU	19	2022
Nim	TAU	0.14.0	2016
Processing	TAU and TWO_PI	2.0	2013
Python	math.tau	3.6	2016
Raku	tau and τ
Rust	core::f64::consts::TAU	1.47.0	2020
Zig	std.math.tau	0.6.0	2019

The constant Template:Math is made available in the Google calculator, Desmos graphing calculator, and the iPhone's Convert Angle option expresses the turn as Template:Math.

See also

• Turn (angle) represents Template:Math radians
• [[Pi|Template:Math]] represents Template:Math/2

External links

• The Tau Manifesto