Tau (mathematics)

File:Circle_radians_tau.gif ({{mvar|τ}}).]]

{{about|the proposed mathematical constant|other uses|Tau#Mathematics}}

The number {{Mathematics|{{tau}}}} ({{IPAc-en|'|t|aʊ|,_|'|t|ɔː|,_|'|t|ɒ|audio=LL-Q1860 (eng)-Flame, not lame-Tau.wav}}; 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 2{{pi}}.

{{Mathematics|{{tau}}}} and {{Mathematics|{{pi}}}} are both circle constants relating the circumference of a circle to its linear dimension: the radius in the case of {{Mathematics|{{tau}}}}; the diameter in the case of {{Mathematics|{{pi}}}}.

While {{Mathematics|{{pi}}}} is used almost exclusively in mainstream mathematical education and practice, it has been proposed, most notably by Michael Hartl in 2010, that {{Mathematics|{{tau}}}} should be used instead. Hartl and other proponents argue that {{Mathematics|{{tau}}}} 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 {{Mathematics|{{tau}}}} over {{Mathematics|{{pi}}}} are trivial and that given the ubiquity and historical significance of {{Mathematics|{{pi}}}} a change is unlikely to occur.

The proposal did not initially gain widespread acceptance in the mathematical community, but awareness of {{Mathematics|{{tau}}}} has become more widespread, having been added to several major programming languages and calculators.

Fundamentals

= Definition =

{{Mathematics|{{tau}}}} is commonly defined as the ratio of a circle's circumference ${C}$ to its radius ${r}$ : $\tau = \frac{C}{r}$ 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 distance around the circle is the circumference, and the ratio $\frac{C}{r}$ is constant regardless of the circle's size. Thus, {{Mathematics|{{tau}}}} denotes the fixed relationship between the circumference of any circle and the fundamental defining property of that circle, the radius.

= Units of angle =

File:Tau-angles.svg

When radians are used as the unit of angular measure there are {{tau}} radians in one full turn of a circle, and the radian angle is aligned with the proportion of a full turn around the circle: $\frac{\tau}{8}$ rad is an eighth of a turn; $\frac{3\tau}{4}$ rad is three-quarters of a turn.

= Relationship to {{Mathematics|{{pi}}}} =

As {{Mathematics|{{tau}}}} is exactly equal to 2{{pi}} it shares many of the properties of {{pi}} including being both an irrational and transcendental number.

History

The proposal to use the Greek letter {{tau}} as a circle constant representing 2{{pi}} dates to Michael Hartl's 2010 publication, The Tau Manifesto{{efn|Original version, current version}}, although the symbol had been independently suggested earlier by Joseph Lindenburg ({{circa}}1990), John Fisher (2004) and Peter Harremoës (2010).{{cite AV media |people=sudgylacmoe; Hartl, Michael |date=28 June 2023 |title=The Tau Manifesto - With Michael Hartl |type=YouTube video |language=English |url=https://www.youtube.com/watch?v=kMtgV18Iew8 |access-date=24 July 2024 |time=18:35 |time-caption= Information shown at}}

Hartl offered two reasons for the choice of notation. First, {{mvar|τ}} is the number of radians in one turn, and both {{mvar|τ}} and turn begin with a {{IPAc-en|t}} sound. Second, {{mvar|τ}} visually resembles {{pi}}, 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 2{{pi}}, together with varying suggestions for it's name and symbol.

In 2001, Robert Palais of the University of Utah proposed that {{Math|{{pi}}}} was "wrong" as the fundamental circle constant arguing instead that 2{{pi}} was the proper value. His proposal used a "π with three legs" symbol to denote the constant ( $\pi\!\;\!\!\!\pi = 2\pi$ ), and referred to angles as fractions of a "turn" ( $\tfrac 1 4 \pi\!\;\!\!\!\pi = \tfrac 1 4\,\mathrm{turn}$ ). 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.{{Cite web |title=Pi Is Wrong! |url=https://www.math.utah.edu/~palais/pi.html |access-date=2025-04-26 |website=www.math.utah.edu}}

In 2008, Robert P. Crease proposed defining a constant as the ratio of circumference to radius, a idea supported by John Horton Conway. Crease used the Greek letter psi: $\psi = 2 \pi$ .{{cite magazine |last=Crease |first=Robert |date=2008-02-01 |title=Constant failure |url=https://physicsworld.com/a/constant-failure/ |magazine=Physics World |publisher=Institute of Physics |access-date=2024-08-03}}

The same year, Thomas Colignatus proposed the uppercase Greek letter theta, Θ, to represent 2{{pi}} 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 {{Mathematics|{{pi}}}} to represent 6.28 =

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

Early works in circle geometry used the letter {{pi}} to designate the perimeter (i.e., circumference) in different fractional representations of circle constants and in 1697 David Gregory used {{math|{{sfrac|π|ρ}}}} (pi over rho) to denote the perimeter divided by the radius (6.28...).

Subsequently {{pi}} came to be used as a single symbol to represent the ratios in whole. Leonhard Euler initially used the single letter {{pi}} was to denote the constant 6.28... in his 1727 Essay Explaining the Properties of Air.{{Cite journal |last=Euler |first=Leonhard |date=1727 |title=Tentamen explicationis phaenomenorum aeris|url=http://eulerarchive.maa.org/docs/originals/E007.pdf#page=5 |journal=Commentarii Academiae Scientiarum Imperialis Petropolitana |language=la |volume=2 |page=351 |id=[http://eulerarchive.maa.org/pages/E007.html E007] |quote=Sumatur pro ratione radii ad peripheriem, {{math|I : π}} |access-date=15 October 2017|archive-url=https://web.archive.org/web/20160401072718/http://eulerarchive.maa.org/docs/originals/E007.pdf#page=5|archive-date=1 April 2016|url-status=live}} [http://www.17centurymaths.com/contents/euler/e007tr.pdf#page=3 English translation by Ian Bruce] {{Webarchive|url=https://web.archive.org/web/20160610172054/http://www.17centurymaths.com/contents/euler/e007tr.pdf#page=3 |date=10 June 2016 }}: "{{mvar|π}} is taken for the ratio of the radius to the periphery [note that in this work, Euler's {{pi}} is double our {{pi}}.]"{{Cite book |url=https://books.google.com/books?id=3C1iHFBXVEcC&pg=PA139 |title=Lettres inédites d'Euler à d'Alembert |last=Euler |first=Leonhard |series=Bullettino di Bibliografia e di Storia delle Scienze Matematiche e Fisiche |year=1747 |editor-last=Henry|editor-first=Charles |volume=19 |publication-date=1886 |page=139 |language=fr |id=[http://eulerarchive.maa.org/pages/E858.html E858] |quote=Car, soit π la circonference d'un cercle, dout le rayon est {{math|{{=}} 1}}}} English translation in {{Cite journal |last=Cajori |first=Florian |date=1913 |title=History of the Exponential and Logarithmic Concepts |jstor=2973441 |journal=The American Mathematical Monthly |volume=20 |issue=3 |pages=75–84 |doi=10.2307/2973441 |quote=Letting {{pi}} be the circumference (!) of a circle of unit radius}} Euler would later use the letter {{pi}} for 3.14... in his 1736 Mechanica{{Cite book|last=Euler|first=Leonhard|title=Mechanica sive motus scientia analytice exposita. (cum tabulis)|date=1736|publisher=Academiae scientiarum Petropoli|volume=1|page=113|language=la|chapter=Ch. 3 Prop. 34 Cor. 1|id=[http://eulerarchive.maa.org/pages/E015.html E015]|quote=Denotet {{math|1 : π}} rationem diametri ad peripheriam|chapter-url=https://books.google.com/books?id=jgdTAAAAcAAJ&pg=PA113}} [http://www.17centurymaths.com/contents/euler/mechvol1/ch3a.pdf#page=26 English translation by Ian Bruce] {{Webarchive|url=https://web.archive.org/web/20160610183753/http://www.17centurymaths.com/contents/euler/mechvol1/ch3a.pdf#page=26|date=10 June 2016}} : "Let {{math|1 : π}} denote the ratio of the diameter to the circumference" and 1748 Introductio in analysin infinitorum,{{Cite book |url=http://gallica.bnf.fr/ark:/12148/bpt6k69587/f155 |title=Leonhardi Euleri opera omnia. 1, Opera mathematica. Volumen VIII, Leonhardi Euleri introductio in analysin infinitorum. Tomus primus / ediderunt Adolf Krazer et Ferdinand Rudio |last=Euler |first=Leonhard |date=1922 |publisher=B.G. Teubneri |location=Lipsae |pages=133–134 |language=la |id=[http://eulerarchive.maa.org/pages/E101.html E101]|access-date=15 October 2017|archive-url=https://web.archive.org/web/20171016022758/http://gallica.bnf.fr/ark:/12148/bpt6k69587/f155|archive-date=16 October 2017|url-status=live}} 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 {{pi}} for one-fourth of the circumference of a unit circle, or 1.57... .{{cite book |last=Euler |first=Leonhard |date=1736 |title=Mechanica sive motus scientia analytice exposita |url=https://scholarlycommons.pacific.edu/euler-works/15/ |page=185 |access-date=2025-02-12}}{{cite AV media |people=Sanderson, Grant |date=2018-03-14 |title=How pi was almost 6.283185... |language=English |url=https://www.youtube.com/watch?v=bcPTiiiYDs8 |access-date=2025-02-11 |time=2:29}} Usage of the letter {{pi}}, sometimes for 3.14... and other times for 6.28..., became widespread, with the definition varying as late as 1761;{{Cite book |url=https://books.google.com/books?id=P-hEAAAAcAAJ&pg=PA374 |title=Cursus Mathematicus: Elementorum Analyseos Infinitorum Elementorum Analyseos Infinitorvm |last=Segner |first=Johann Andreas von |date=1761 |publisher=Renger |page=374 |language=la |quote=Si autem {{pi}} notet peripheriam circuli, cuius diameter eſt {{math|2}}}} afterward, {{pi}} was standardized as being equal to 3.14... .{{Cite web |date=2024-03-14 |title=Pi |url=https://www.britannica.com/science/pi-mathematics |access-date=2024-03-26 |website=Encyclopaedia Brittanica |language=en}}

Notion using {{tau}}

Proponents argue that while use of {{tau}} in place of 2{{pi}} does not change any of the underlying mathematics, it does lead to simpler and more intuitive notation in many areas. Michael Hartl's Tau Manifesto{{efn|Original version, current version}} gives many examples of formulas that are asserted to be clearer where {{math|1=τ}} is used instead of {{math|1=π}}.

= Units of angle =

Hartl and Robert Palais have argued that {{tau}} 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, {{math|{{sfrac|3τ|4}}}} rad can be easily interpreted as {{sfrac|3|4}}⁠ of a turn around the unit circle in contrast with the numerically equal ⁠{{math|{{sfrac|3π|2}}}}⁠ rad, 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 {{math|1={{sfrac|τ|4}} = {{sfrac|π|2}} ≈ 1.57}}, which he denotes with the Greek letter η (eta), should be seen as the fundamental circle constant.{{Cite web |last=Butler |first=David |title=Pi, Tau and Eta |url=https://www.adelaide.edu.au/mathslearning/news/list/2011/06/08/pi-tau-and-eta}}{{Cite AV media |url=https://www.youtube.com/watch?v=1qpVdwizdvI |title=Pi may be wrong, but so is Tau! |date=2011-06-28 |last=David Butler |access-date=2025-05-01 |via=YouTube}}

= Trigonometric Functions =

Hartl has argued that the periodic trigonometric functions are simplified using {{tau}} as it aligns the function argument (radians) with the function period: sin θ repeats with period {{math|1=T = τ}} rad, reaches a maximum at {{math|1={{sfrac|T|4}}={{sfrac|τ|4}}}} rad and a minimum at {{math|1={{sfrac|3T|4}}={{sfrac|3τ|4}}}} rad.

= Area of a circle =

Critics have argued that the formula for the area of a circle is more complicated when restated as {{math|1=A = {{sfrac|1|2}}{{tau}}r²}}. Hartl and others respond that the {{sfrac|1|2}} 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 {{math|1=τ}} is that Euler's identity, {{math|1=e^iπ + 1 = 0}}, sometimes claimed to be "the most beautiful theorem in mathematics"{{Cite web |last=Peshin |first=Akash |date=2017-12-24 |title=Euler's Identity: 'The Most Beautiful Theorem In Mathematics' |url=https://www.scienceabc.com/nature/eulers-identity-beautiful-theorem-mathematics.html |access-date=2025-04-27 |website=ScienceABC |language=en-US}} is made less elegant rendered as {{math|1=e^iτ/2 + 1 = 0}}.{{Cite web |last=Colin |date=2011-06-28 |title=Tau versus Pi |url=https://www.piglets.org/blog/2011/06/28/tau-versus-pi/ |access-date=2025-04-27 |website=Proving the Obviously Untrue |language=en-GB}} Hartl has asserted that {{math|1=e^iτ = 1}} (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 n^th roots of unity, {{math|1=ⁿ√1 = e^iτk/n (k = 1,2,..,n)}}, which he maintained is preferable and more economical than Euler’s.

= Comparison of identities =

The following table shows how various identities appear when {{math|1=τ = 2π}} is used instead of {{pi}}. For a more complete list, see List of formulae involving {{pi}}.

class="wikitable" style="border: none;" \|+
Formula	Using {{pi}}	Using {{math\|1=τ}}	Notes
style="text-align: center; padding-left: 0.fem; padding-right: 0.5em;" \| Angle subtended by {{sfrac\|1\|4}} of a circle \| style="text-align: center; border-style: solid none solid none; padding-left: 0.5em;" \| ${\color{orangered}\frac{\pi}{2}} \text{ rad}$ \| style="text-align: center; padding: 0.5% 2em 0.5% 0.5em;" \| ${\color{orangered}\frac{\tau}{4}} \text{ rad}$ \|{{math\|1={{sfrac\|τ\|4}} rad = {{sfrac\|1\|4}} turn}}
style="text-align: center; padding-right: 0.5em;" \| Circumference of a circle \| style="text-align: center; border-style: solid none solid none; padding-left: 0.5em;" \| $C = {\color{orangered}2 \pi} r$ \| style="text-align: center; padding: 0.5% 2em 0.5% 0.5em;" \| $C = {\color{orangered}\tau} r$ \| The length of an arc of angle {{math\|θ}} is {{math\|1=L = θr}}.
style="text-align: center; padding-left: 0.fem; padding-right: 0.5em;" \| Area of a circle \| style="text-align: center; border-style: solid none solid none; padding-left: 0.5em;" \| $A = {\color{orangered}\pi}r^2$ \| style="text-align: center; padding: 0.5% 2em 0.5% 0.5em;" \| $A = {\color{orangered}\frac{1}{2} \tau} r^2$ \| The area of a sector of angle {{math\|θ}} is {{math\|1=A = {{sfrac\|1\|2}}θr²}}.
style="text-align: center; padding-left: 0.fem; padding-right: 0.5em;" \| Area of a regular Polygon with unit circumradius \| style="text-align: center; border-style: solid none solid none; padding-left: 0.5em;" \| $A = \frac{n}{2} \sin \frac{{\color{orangered}2 \pi}}{n}$ \| style="text-align: center; padding: 0.5% 2em 0.5% 0.5em;" \| $A = \frac{n}{2} \sin \frac{{\color{orangered}\tau}}{n}$ \|
style="text-align: center; padding-left: 0.fem; padding-right: 0.5em;" \| n-sphere#Volume and surface area \| style="text-align: center; border-style: solid none solid none; padding-left: 0.5em;" \| $V_n(r) = \frac{r}{n} S_{n-1}(r)$ $S_n(r) = {\color{orangered} 2 \pi} r V_{n-1}(r)$ \| style="text-align: center; padding: 0.5% 2em 0.5% 0.5em;" \| $V_n(r) = \frac{r}{n} S_{n-1}(r)$ $S_n(r) = {\color{orangered}\tau}rV_{n-1}(r)$ \|{{math\|1=V₀(r) = 1}}{{br}} {{math\|1=S₀(r) = 2}}
style="text-align: center; padding-left: 0.fem; padding-right: 0.5em;" \| Cauchy's integral formula \| style="text-align: center; border-style: solid none solid none; padding-left: 0.5em;" \| $f(a) = \frac{1}{{\color{orangered}2\pi} i} \oint_\gamma \frac{f(z)}{z-a}\, dz$ \| style="text-align: center; padding: 0.5% 2em 0.5% 0.5em;" \| $f(a) = \frac{1}{{\color{orangered}\tau} i} \oint_\gamma \frac{f(z)}{z-a}\, dz$ \| $\gamma$ is the boundary of a disk containing $a$ in the complex plane.
style="text-align: center; padding-left: 0.fem; padding-right: 0.5em;" \| Standard normal distribution \| style="text-align: center; border-style: solid none solid none; padding-left: 0.5em;" \| $\varphi(x) = \frac{1}{\sqrt{{\color{orangered}2\pi}}}e^{-\frac{x^2}{2}}$ \| style="text-align: center; padding: 0.5% 2em 0.5% 0.5em;" \| $\varphi(x) = \frac{1}{\sqrt{{\color{orangered}\tau}}}e^{-\frac{x^2}{2}}$ \|
style="text-align: center; padding-left: 0.fem; padding-right: 0.5em;" \| Stirling's approximation \| style="text-align: center; border-style: solid none solid none; padding-left: 0.5em;" \| $n! \sim \sqrt{{\color{orangered}2 \pi} n}\left(\frac{n}{e}\right)^n$ \| style="text-align: center; padding: 0.5% 2em 0.5% 0.5em;" \| $n! \sim \sqrt{{\color{orangered}\tau} n}\left(\frac{n}{e}\right)^n$ \|
style="text-align: center; padding-left: 0.fem; padding-right: 0.5em;" \| Root of unity \| style="text-align: center; border-style: solid none solid none; padding-left: 0.5em;" \| $e^{{\color{orangered}2 \pi} i \frac{k}{n}} = \cos\frac{{\color{orangered}2} k {\color{orangered}\pi}}{n} + i \sin\frac{{\color{orangered}2} k {\color{orangered}\pi}}{n}$ \| style="text-align: center; padding: 0.5% 2em 0.5% 0.5em;" \| $e^{{\color{orangered}\tau} i \frac{k}{n}} = \cos\frac{k {\color{orangered}\tau}}{n} + i \sin\frac{k {\color{orangered}\tau}}{n}$ \|
style="text-align: center; padding-left: 0.fem; padding-right: 0.5em;" \| Planck constant \| style="text-align: center; border-style: solid none solid none; padding-left: 0.5em;" \| $h = {\color{orangered}2 \pi} \hbar$ \| style="text-align: center; padding: 0.5% 2em 0.5% 0.5em;" \| $h = {\color{orangered}\tau} \hbar$ \| {{math\|ħ}} is the reduced Planck constant.
style="text-align: center; padding-left: 0.fem; padding-right: 0.5em;" \| Angular frequency \| style="text-align: center; border-style: solid none solid none; padding-left: 0.5em;" \| $\omega = {\color{orangered}2 \pi} f$ \| style="text-align: center; padding: 0.5% 2em 0.5% 0.5em;" \| $\omega = {\color{orangered}\tau} f$ \|
style="text-align: center; padding-left: 0.fem; padding-right: 0.5em;" \| Riemann's functional equation \| style="text-align: center; border-style: solid none solid none; padding-left: 0.5em;" \| $\zeta(s) = {\color{orangered}2^s \pi^{s-1}}\ \sin\left( s{\color{orangered}\frac{\pi}{2}} \right)\ \Gamma(1-s)\ \zeta(1-s)$ \| style="text-align: center; padding: 0.5% 2em 0.5% 0.5em;" \| $\zeta(s) = {\color{orangered}2\tau^{s-1}}\ \sin\left( s{\color{orangered}\frac{\tau}{4}} \right)\ \Gamma(1-s)\ \zeta(1-s)$ \| $2^s (\frac{\tau}{2})^{s-1}$ reduces to $2 \tau^{s-1}$

class="wikitable" style="border: none;"

Formula

Using {{pi}}

Using {{math|1=τ}}

Notes

style="text-align: center; padding-left: 0.fem; padding-right: 0.5em;" | Angle subtended by {{sfrac|1|4}} of a circle

| style="text-align: center; border-style: solid none solid none; padding-left: 0.5em;" | ${\color{orangered}\frac{\pi}{2}} \text{ rad}$

| style="text-align: center; padding: 0.5% 2em 0.5% 0.5em;" | ${\color{orangered}\frac{\tau}{4}} \text{ rad}$

|{{math|1={{sfrac|τ|4}} rad = {{sfrac|1|4}} turn}}

style="text-align: center; padding-right: 0.5em;" | Circumference of a circle

| style="text-align: center; border-style: solid none solid none; padding-left: 0.5em;" | $C = {\color{orangered}2 \pi} r$

| style="text-align: center; padding: 0.5% 2em 0.5% 0.5em;" | $C = {\color{orangered}\tau} r$

| The length of an arc of angle {{math|θ}} is {{math|1=L = θr}}.

style="text-align: center; padding-left: 0.fem; padding-right: 0.5em;" | Area of a circle

| style="text-align: center; border-style: solid none solid none; padding-left: 0.5em;" | $A = {\color{orangered}\pi}r^2$

| style="text-align: center; padding: 0.5% 2em 0.5% 0.5em;" | $A = {\color{orangered}\frac{1}{2} \tau} r^2$

| The area of a sector of angle {{math|θ}} is {{math|1=A = {{sfrac|1|2}}θr²}}.

style="text-align: center; padding-left: 0.fem; padding-right: 0.5em;" | Area of a regular Polygon with unit circumradius

| style="text-align: center; border-style: solid none solid none; padding-left: 0.5em;" | $A = \frac{n}{2} \sin \frac{{\color{orangered}2 \pi}}{n}$

| style="text-align: center; padding: 0.5% 2em 0.5% 0.5em;" | $A = \frac{n}{2} \sin \frac{{\color{orangered}\tau}}{n}$

style="text-align: center; padding-left: 0.fem; padding-right: 0.5em;" | n-sphere#Volume and surface area

| style="text-align: center; border-style: solid none solid none; padding-left: 0.5em;" | $V_n(r) = \frac{r}{n} S_{n-1}(r)$

$S_n(r) = {\color{orangered} 2 \pi} r V_{n-1}(r)$

| style="text-align: center; padding: 0.5% 2em 0.5% 0.5em;" | $V_n(r) = \frac{r}{n} S_{n-1}(r)$

$S_n(r) = {\color{orangered}\tau}rV_{n-1}(r)$

|{{math|1=V₀(r) = 1}}{{br}} {{math|1=S₀(r) = 2}}

style="text-align: center; padding-left: 0.fem; padding-right: 0.5em;" | Cauchy's integral formula

| style="text-align: center; border-style: solid none solid none; padding-left: 0.5em;" | $f(a) = \frac{1}{{\color{orangered}2\pi} i} \oint_\gamma \frac{f(z)}{z-a}\, dz$

| style="text-align: center; padding: 0.5% 2em 0.5% 0.5em;" | $f(a) = \frac{1}{{\color{orangered}\tau} i} \oint_\gamma \frac{f(z)}{z-a}\, dz$

| $\gamma$ is the boundary of a disk containing $a$ in the complex plane.

style="text-align: center; padding-left: 0.fem; padding-right: 0.5em;" | Standard normal distribution

| style="text-align: center; border-style: solid none solid none; padding-left: 0.5em;" | $\varphi(x) = \frac{1}{\sqrt{{\color{orangered}2\pi}}}e^{-\frac{x^2}{2}}$

| style="text-align: center; padding: 0.5% 2em 0.5% 0.5em;" | $\varphi(x) = \frac{1}{\sqrt{{\color{orangered}\tau}}}e^{-\frac{x^2}{2}}$

style="text-align: center; padding-left: 0.fem; padding-right: 0.5em;" | Stirling's approximation

| style="text-align: center; border-style: solid none solid none; padding-left: 0.5em;" | $n! \sim \sqrt{{\color{orangered}2 \pi} n}\left(\frac{n}{e}\right)^n$

| style="text-align: center; padding: 0.5% 2em 0.5% 0.5em;" | $n! \sim \sqrt{{\color{orangered}\tau} n}\left(\frac{n}{e}\right)^n$

style="text-align: center; padding-left: 0.fem; padding-right: 0.5em;" | Root of unity

| style="text-align: center; border-style: solid none solid none; padding-left: 0.5em;" | $e^{{\color{orangered}2 \pi} i \frac{k}{n}} = \cos\frac{{\color{orangered}2} k {\color{orangered}\pi}}{n} + i \sin\frac{{\color{orangered}2} k {\color{orangered}\pi}}{n}$

| style="text-align: center; padding: 0.5% 2em 0.5% 0.5em;" | $e^{{\color{orangered}\tau} i \frac{k}{n}} = \cos\frac{k {\color{orangered}\tau}}{n} + i \sin\frac{k {\color{orangered}\tau}}{n}$

style="text-align: center; padding-left: 0.fem; padding-right: 0.5em;" | Planck constant

| style="text-align: center; border-style: solid none solid none; padding-left: 0.5em;" | $h = {\color{orangered}2 \pi} \hbar$

| style="text-align: center; padding: 0.5% 2em 0.5% 0.5em;" | $h = {\color{orangered}\tau} \hbar$

| {{math|ħ}} is the reduced Planck constant.

style="text-align: center; padding-left: 0.fem; padding-right: 0.5em;" | Angular frequency

| style="text-align: center; border-style: solid none solid none; padding-left: 0.5em;" | $\omega = {\color{orangered}2 \pi} f$

| style="text-align: center; padding: 0.5% 2em 0.5% 0.5em;" | $\omega = {\color{orangered}\tau} f$

style="text-align: center; padding-left: 0.fem; padding-right: 0.5em;" | Riemann's functional equation

| style="text-align: center; border-style: solid none solid none; padding-left: 0.5em;" | $\zeta(s) = {\color{orangered}2^s \pi^{s-1}}\ \sin\left( s{\color{orangered}\frac{\pi}{2}} \right)\ \Gamma(1-s)\ \zeta(1-s)$

| style="text-align: center; padding: 0.5% 2em 0.5% 0.5em;" | $\zeta(s) = {\color{orangered}2\tau^{s-1}}\ \sin\left( s{\color{orangered}\frac{\tau}{4}} \right)\ \Gamma(1-s)\ \zeta(1-s)$

| $2^s (\frac{\tau}{2})^{s-1}$ reduces to $2 \tau^{s-1}$

In culture

{{tau}} has made numerous appearances in culture. It is celebrated annually on June 28, known as Tau Day.{{cite web |last1=Hartl |first1=Michael |title=Tau Day |url=https://tauday.com/ |access-date=1 November 2024}} Supporters of {{tau}} are called tauists. {{tau}} has been covered in videos by Vi Hart,{{cite web |last1=Hart |first1=Vi |title=Pi is (still) Wrong. |url=https://www.youtube.com/watch?v=jG7vhMMXagQ |website=YouTube |date=14 March 2011 |access-date=1 November 2024}}{{cite web |last1=Hart |first1=Vi |title=A Song About A Circle Constant |url=https://www.youtube.com/watch?v=FtxmFlMLYRI |website=YouTube |date=28 June 2012 |access-date=1 November 2024}}{{cite web |last1=Hart |first1=Vi |title=360 Video for Tau Day |url=https://www.youtube.com/watch?v=S3xOB-Bigc8 |website=YouTube |date=28 June 2015 |access-date=1 November 2024}} Numberphile,{{cite web |last1=Haran |first1=Brady |last2=Moriarty |first2=Phil |title=Tau replaces Pi - Numberphile |url=https://www.youtube.com/watch?v=83ofi_L6eAo |website=YouTube |date=9 November 2012 |access-date=1 November 2024}}{{cite web |last1=Haran |first1=Brady |last2=Moriarty |first2=Phil |title=Tau of Phi - Numberphile |url=https://www.youtube.com/watch?v=aiibxmqXV9M |website=YouTube |date=19 November 2012 |access-date=1 November 2024}}{{cite web |last1=Haran |first1=Brady |last2=Mould |first2=Steve |last3=Parker |first3=Matthew |title=Tau vs Pi Smackdown - Numberphile |url=https://www.youtube.com/watch?v=ZPv1UV0rD8U |website=YouTube |date=14 December 2012 |access-date=1 November 2024}} SciShow,{{cite web |last1=Hofmeister |first1=Caitlin |title=Happy Tau Day! |url=https://www.youtube.com/watch?v=QwPo0Y7BcEE |website=YouTube |date=26 June 2015 |access-date=1 November 2024}} Steve Mould,{{cite AV media |people=Mould, Steve |date=2018-11-06 |title=Stand-up comedy routine about bad science |language=English |url=https://www.youtube.com/watch?v=C91gKuxutTU |access-date=2024-11-17 |time=10:31}}{{cite AV media |people=Mould, Steve |date=2023-11-06 |title=A cast saw on human skin |language=English |url=https://www.youtube.com/watch?v=Bx1AiQdMQro |access-date=2024-11-13 |time=7:22}}{{cite AV media |people=Mould, Steve |date=2024-03-14 |title=world record calculation of tau by hand |language=English |url=https://www.youtube.com/shorts/N8KduziZ8Vc |access-date=2024-11-13}} Khan Academy,{{cite AV media |people=Khan, Sal |date=2011-07-11 |title=Tau versus pi | Graphs of trig functions | Trigonometry | Khan Academy |language=English |url=https://www.youtube.com/watch?v=1jDDfkKKgmc |access-date=2024-11-24}} and 3Blue1Brown,{{cite AV media |people=Sanderson, Grant |date=2018-03-14 |title=How pi was almost 6.283185... |language=English |url=https://www.youtube.com/watch?v=bcPTiiiYDs8 |access-date=2024-11-24}}{{cite AV media |people=Sanderson, Grant |date=2019-07-07 |title=e^(iπ) in 3.14 minutes, using dynamics | DE5 |language=English |url=https://www.youtube.com/watch?v=v0YEaeIClKY |access-date=2024-11-24 |time=3:08}} and it has appeared in the comics xkcd,{{cite web |last1=Munroe |first1=Randall |title=Pi vs. Tau |url=https://www.xkcd.com/1292/ |website=xkcd |access-date=1 November 2024}}{{cite web |last1=Munroe |first1=Randall |title=Symbols |url=https://www.xkcd.com/2520/ |website=xkcd |access-date=1 November 2024}} Saturday Morning Breakfast Cereal,{{cite web |last1=Weinersmith |first1=Zachary |title=Fresh |url=https://www.smbc-comics.com/index.php?id=3134 |website=Saturday Morning Breakfast Cereal |access-date=2 November 2024}}{{cite web |last1=Weinersmith |first1=Zachary |title=Better than Pi |url=https://www.smbc-comics.com/comic/better-than-pi |website=Saturday Morning Breakfast Cereal |access-date=2 November 2024}}{{cite web |last1=Weinersmith |first1=Zachary |title=Social |url=https://www.smbc-comics.com/comic/social |website=Saturday Morning Breakfast Cereal |access-date=2 November 2024}} and Sally Forth.{{cite web |last1=Marciuliano |first1=Francesco |title=Sally Forth Comic Strip 2018-10-13 |url=https://comicskingdom.com/sally-forth/2018-10-13 |website=Comics Kingdom |access-date=13 November 2024}} The Massachusetts Institute of Technology usually announces admissions on March 14 at 6:28{{nbsp}}p.m., which is on Pi Day at Tau Time.{{cite web |title=Fun & Culture – MIT Facts |url=https://facts.mit.edu/fun-culture/ |website=Massachusetts Institute of Technology |access-date=2 November 2024}} Peter Harremoës has used {{mvar|τ}} 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.

class="wikitable" style="text-align:right; font-size:small" ! Language	Identifiers	First Version	Year Released
C# / .NET	[https://learn.microsoft.com/en-us/dotnet/api/system.math.tau System.Math.Tau] and [https://learn.microsoft.com/en-us/dotnet/api/system.mathf.tau System.MathF.Tau]	5.0	2020
Crystal	[https://crystal-lang.org/api/1.13.2/Math.html#constant-summary TAU]	0.36.0	2021
Eiffel	[https://wiki.liberty-eiffel.org/index.php/Versions_history#Curtiss_(2022.dev,_to_be_named_after_Glenn_Curtiss) math_constants.Tau]	Curtiss	Not yet released
GDScript	[https://docs.godotengine.org/en/stable/classes/class_@gdscript.html#class-gdscript-constant-tau TAU]	Godot 3.0	2018
Java	[https://docs.oracle.com/en/java/javase/21/docs/api/java.base/java/lang/Math.html#TAU Math.TAU]	19	2022
Nim	[https://nim-lang.org/docs/math.html#TAU TAU]	0.14.0	2016
Processing	[https://processing.org/reference/TAU.html TAU] and [https://processing.org/reference/TWO_PI.html TWO_PI]	2.0	2013
Python	[https://docs.python.org/3/library/math.html#math.tau math.tau]	3.6	2016
Raku	[https://docs.raku.org/language/terms#term_tau tau] and [https://docs.raku.org/language/terms#term_%CF%84 τ]
Rust	[https://doc.rust-lang.org/stable/std/f64/consts/constant.TAU.html std::f64::consts::TAU]	1.47.0	2020
Zig	[https://github.com/ziglang/zig/blob/master/lib/std/math.zig#L18 std.math.tau]	0.6.0	2019

The constant {{mvar|τ}} is made available in the Google calculator, Desmos graphing calculator, and the iPhone's Convert Angle option expresses the turn as {{mvar|τ}}.

Notes

References

{{reflist|refs=

{{cite book |author-first=Petr |author-last=Beckmann |author-link=Petr Beckmann |title=A History of Pi |title-link=A History of Pi |publisher=Barnes & Noble Publishing |date=1989 |orig-date=1970}}

{{cite book |author-first=Steven |author-last=Schwartzman |title=The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English |url=https://archive.org/details/wordsofmathemati0000schw |url-access=registration |publisher=The Mathematical Association of America |date=1994 |isbn=978-0-88385511-9 |page=[https://archive.org/details/wordsofmathemati0000schw/page/165 165]}}

{{cite journal |author-first=Robert |author-last=Palais |date=2001 |title=Pi is Wrong |journal=The Mathematical Intelligencer |publisher=Springer-Verlag |location=New York, USA |volume=23 |issue=3 |pages=7–8 |doi=10.1007/bf03026846 |s2cid=120965049 |url=http://www.math.utah.edu/%7Epalais/pi.pdf |access-date=2019-08-05 |url-status=live |archive-url=https://web.archive.org/web/20190718051848/http://www.math.utah.edu/~palais/pi.pdf |archive-date=2019-07-18}}

{{cite web |title=The Tau Manifesto |author-first=Michael |author-last=Hartl |date=2010-03-14 |url=https://hexnet.org/files/documents/tau-manifesto.pdf |access-date=2019-08-05 |url-status=live |archive-url=https://web.archive.org/web/20190718051848/http://www.math.utah.edu/~palais/pi.pdf |archive-date=2019-07-18}}

{{cite web |title=The Tau Manifesto |author-first=Michael |author-last=Hartl |date=2019-03-14 |orig-date=2010-03-14 |url=http://tauday.com/tau-manifesto |access-date=2013-09-14 |url-status=live |archive-url=https://web.archive.org/web/20190628230418/https://tauday.com/tau-manifesto |archive-date=2019-06-28}}

{{cite journal |author-first=Jacob |author-last=Aron |date=2011-01-08 |title=Michael Hartl: It's time to kill off pi |series=Interview |journal=New Scientist |volume=209 |issue=2794 |doi=10.1016/S0262-4079(11)60036-5 |bibcode=2011NewSc.209...23A |page=23}}

{{cite web |author-first=Elizabeth |author-last=Landau |author-link=Elizabeth Landau |date=2011-03-14 |title=On Pi Day, is 'pi' under attack? |work=cnn.com |publisher=CNN |url=http://edition.cnn.com/2011/TECH/innovation/03/14/pi.tau.math/index.html |access-date=2019-08-05 |url-status=dead |archive-url=https://web.archive.org/web/20181219142051/http://edition.cnn.com/2011/TECH/innovation/03/14/pi.tau.math/index.html |archive-date=2018-12-19}}

{{cite news |title=Let's Use Tau--It's Easier Than Pi - A growing movement argues that killing pi would make mathematics simpler, easier and even more beautiful |author-first=Randyn Charles |author-last=Bartholomew |date=2014-06-25 |work=Scientific American |url=http://www.scientificamerican.com/article/let-s-use-tau-it-s-easier-than-pi/ |access-date=2015-03-20 |url-status=live |archive-url=https://web.archive.org/web/20190618184747/https://www.scientificamerican.com/article/let-s-use-tau-it-s-easier-than-pi/ |archive-date=2019-06-18}}

{{cite journal |title=Life of pi in no danger – Experts cold-shoulder campaign to replace with tau |journal=Telegraph India |date=2011-06-30 |url=http://www.telegraphindia.com/1110630/jsp/nation/story_14178997.jsp |access-date=2019-08-05 |url-status=dead |archive-url=https://web.archive.org/web/20130713084345/http://www.telegraphindia.com/1110630/jsp/nation/story_14178997.jsp |archive-date=2013-07-13}}

{{cite journal |author-last=Harremoës |author-first=Peter |title=Bounds on tail probabilities for negative binomial distributions |journal=Kybernetika |date=2017 |volume=52 |issue=6 |doi=10.14736/kyb-2016-6-0943 |arxiv=1601.05179 |pages=943–966 |s2cid=119126029}}

{{cite web |title=Supported Functions |url=https://help.desmos.com/hc/en-us/articles/212235786-Supported-Functions |access-date=2023-03-21 |website=help.desmos.com |url-status=live |archive-url=https://web.archive.org/web/20230326032414/https://help.desmos.com/hc/en-us/articles/212235786-Supported-Functions |archive-date=2023-03-26}}

{{cite news |last1=Naumovski |first1=Jovana |title=iOS 16 Has a Hidden Unit Converter for Temperatures, Time Zones, Distance, and Other Measurements |url=https://ios.gadgethacks.com/how-to/ios-16-has-hidden-unit-converter-for-temperatures-time-zones-distance-and-other-measurements-0385095/ |access-date=21 October 2024 |work=Gadget Hacks |date=2022-08-05}}

{{cite book |author-last=Euler |author-first=Leonhard |author-link=Leonhard Euler |date=1746 |title=Nova theoria lucis et colorum. Opuscula varii argumenti |language=la |page=200|publisher=sumtibus Ambr. Haude & Jo. Carol. Speneri, bibliop. |url=https://archive.org/details/bub_gb_V4XtNi8BGl4C/page/n217/mode/2up|quote=unde constat punctum B per datum tantum spatium de loco fuo naturali depelli, ad quam maximam distantiam pertinget, elapso tempore t=π/m denotante π angulum 180°, quo fit cos(mt)=- 1 & B b=2α. |trans-quote=from which it is clear that the point B is pushed by a given distance from its natural position, and it will reach the maximum distance after the elapsed time t=π/m, π denoting an angle of 180°, which becomes cos(mt)=- 1 & B b=2α.}}

{{cite web |title=Trig rerigged. Trigonometry reconsidered. Measuring angles in 'unit meter around' and using the unit radius functions Xur and Yur |author-last=Cool |author-first=Thomas "Colignatus" |date=2008-07-18 |orig-date=2008-04-08, 2008-05-06 |url=http://thomascool.eu/Papers/Math/TrigRerigged.pdf |access-date=2023-07-18 |archive-url=https://web.archive.org/web/20230718192531/http://thomascool.eu/Papers/Math/TrigRerigged.pdf |archive-date=2023-07-18}} (18 pages)

{{cite journal |title=Integral Kinematics (Time-Integrals of Distance, Energy, etc.) and Integral Kinesiology |author-last1=Mann |author-first1=Steve |author-link1=Steve Mann (inventor) |author-last2=Janzen |author-first2=Ryan E. |author-last3=Ali |author-first3=Mir Adnan |author-last4=Scourboutakos |author-first4=Pete |author-last5=Guleria |author-first5=Nitin |date=22–24 October 2014 |journal=Proceedings of the 2014 IEEE GEM |location=Toronto, Ontario, Canada |s2cid=6462220 |pages=627–629 |url=https://www.researchgate.net/publication/306262297 |access-date=2023-07-18}}

{{cite journal |title=Eye Itself as a Camera: Sensors, Integrity, and Trust |author-last1=Mann |author-first1=Steve |author-link1=Steve Mann (inventor) |author-last2=Chen |author-first2=Hongyu |author-last3=Aylward |author-first3=Graeme |author-last4=Jorritsma |author-first4=Megan |author-last5=Mann |author-first5=Christina |author-last6=Defaz Poveda |author-first6=Diego David |author-last7=Pierce |author-first7=Cayden |author-last8=Lam |author-first8=Derek |author-last9=Stairs |author-first9=Jeremy |author-last10=Hermandez |author-first10=Jesse |author-last11=Li |author-first11=Qiushi |author-last12=Xiang |author-first12=Yi Xin |author-last13=Kanaan |author-first13=Georges |date=June 2019 |type=Keynote |journal=The 5th ACM Workshop on Wearable Systems and Applications |doi=10.1145/3325424.3330210 |s2cid=189926593 |pages=1–2 |url=https://www.researchgate.net/publication/333834426 |access-date=2023-07-18}}

{{cite news |author-last=McMillan |author-first=Robert |date=2020-03-13 |title=For Math Fans, Nothing Can Spoil Pi Day—Except Maybe Tau Day |language=en-US |work=Wall Street Journal |issn=0099-9660 |url=https://www.wsj.com/articles/for-math-fans-nothing-can-spoil-pi-dayexcept-maybe-tau-day-11584123031 |url-access=subscription |access-date=2020-05-21}}

{{cite journal |author-last=Abbott |author-first=Stephen |title=My Conversion to Tauism |journal=Math Horizons |date=April 2012 |volume=19 |issue=4 |page=34 |doi=10.4169/mathhorizons.19.4.34 |s2cid=126179022 |url=http://www.maa.org/sites/default/files/pdf/Mathhorizons/apr12_aftermath.pdf |url-status=live |archive-url=https://web.archive.org/web/20130928095819/http://www.maa.org/sites/default/files/pdf/Mathhorizons/apr12_aftermath.pdf |archive-date=2013-09-28}}

}}

External links

[https://tauday.com/tau-manifesto The Tau Manifesto]

Category:Units of plane angle

Category:Mathematical concepts

Category:Angle

Category:1 (number)