View source for Turn (angle)

{{short description|Unit of plane angle where a full circle equals 1}}
{{Redirect2|360 degrees|360°}}
{{Use dmy dates|date=August 2019|cs1-dates=y}}
{{Use list-defined references|date=July 2022}}
{{Infobox unit
| name     = Turn
| othernames = Revolution, Cycles
| image = angle-fractions.png
| caption = {{longitem|Counterclockwise [[rotation]]s about the center point starting from the right, where a complete rotation corresponds to an angle of rotation of 1&nbsp;turn.}}
| standard = 
| quantity = [[Plane angle]]
| symbol   = tr
| symbol2  = pla
| symbol3  = rev
| symbol4  = cyc
| units1   = [[radian]]s
| inunits1 = {{math|2''π''}}&nbsp;rad<br/>≈ {{val|6.283185307|end=...|u=rad}}
| units3   = [[milliradian]]s
| inunits3 = {{math|2000''π''}}&nbsp;mrad<br/>≈ {{val|6283.185307|end=...|u=mrad}}
| units4   = [[Degree (angle)|degree]]s
| inunits4 = 360°
| units5   = [[gradian]]s
| inunits5 = 400<sup>g</sup>
}}

The '''turn''' (symbol '''tr''' or '''pla''') is a unit of [[plane angle]] measurement that is the measure of a [[complete angle]]—the angle [[Subtended angle|subtended]] by a complete [[circle]] at its center. One turn is equal to {{math|2[[Pi|''π'']]}}&nbsp;[[radian]]s, 360&nbsp;[[degree (angle)|degrees]] or 400&nbsp;[[gradian]]s. As an [[angular unit]], one turn also corresponds to one '''cycle''' (symbol '''cyc''' or '''c''')<ref name="Fitzpatrick_2021" /> or to one '''revolution''' (symbol '''rev''' or '''r''').<ref name="IET_2016" /> Common related [[Frequency#Unit|units of frequency]] are ''[[cycles per second]]'' (cps) and ''[[revolutions per minute]]'' (rpm).{{efn|The angular unit terms "cycles" and "revolutions" are also used, ambiguously, as shorter versions of the related frequency units.{{cn|date=July 2023}}}} The angular unit of the turn is useful in connection with, among other things, [[electromagnetic coil]]s (e.g., [[transformer]]s), rotating objects, and the [[winding number]] of curves. 
Divisions of a turn include the half-turn and quarter-turn, spanning a [[Angle#Individual_angles|straight angle]] and a [[right angle]], respectively; [[metric prefixes]] can also be used as in, e.g., centiturns (ctr), milliturns (mtr), etc.

Because one turn is <math>2\pi</math> radians, some have proposed [[#Tau proposals|representing {{math|2''π''}} with a single letter]]. In 2010, [[Michael Hartl]] proposed using the Greek letter <math>\tau</math> ([[tau]]), equal to the ratio of a circle's [[circumference]] to its [[radius]] (<math>2\pi</math>) and corresponding to one turn, for greater conceptual simplicity when stating angles in radians.<ref name="Hartl_2019" /> This proposal did not initially gain widespread acceptance in the mathematical community,<ref name="Telegraph_2011" /> but the constant has become more widespread,<ref name="McMillan_2020" /> having been added to several major programming languages and calculators.

In the [[International System of Quantities|ISQ]], an arbitrary "number of turns" (also known as "number of revolutions" or "number of cycles") is formalized as a [[dimensionless quantity]] called '''''rotation''''', defined as the [[ratio]] of a given angle and a full turn. It is represented by the symbol ''N''. {{xref|(See [[#In the ISQ/SI|below]] for the formula.)}}

== Unit symbols ==

There are several unit symbols for the turn.

=== EU and Switzerland ===
The German standard [[DIN 1315]] (March 1974) proposed the unit symbol "pla" (from Latin: {{lang|la|plenus angulus}} 'full angle') for turns.<ref name="German_2013"/><ref name="Kurzweil_1999"/> Covered in {{ill|DIN 1301-1|de}} (October 2010)<!-- in a table "allgemein anwendbare Einheiten außerhalb des SI" -->, the so-called {{lang|de|Vollwinkel}} ('full angle') is not an [[SI unit]]. However, it is a [[legal unit of measurement]] in the EU<ref name="EWG_1980"/><ref name="EG_2009"/> and Switzerland.<ref name="Einheitenverordnung_1994"/>

=== Calculators ===
The scientific calculators [[HP&nbsp;39gII]] and [[HP&nbsp;Prime]] support the unit symbol "tr" for turns since 2011 and 2013, respectively. Support for "tr" was also added to [[newRPL]] for the [[HP&nbsp;50g]] in 2016, and for the [[hp&nbsp;39g+]], [[HP&nbsp;49g+]], [[HP&nbsp;39gs]], and [[HP&nbsp;40gs]] in 2017.<ref name="Lapilli_2016"/><ref name="Lapilli_2018"/> An angular mode <var>TURN</var> was suggested for the [[WP&nbsp;43S]] as well,<ref name="Paul_2016"/> but the calculator instead implements "MUL{{pi}}" (''[[multiples of π|multiples of {{pi}}]]'') as mode and unit since 2019.<ref name="Bonin_2019_OG"/><ref name="Bonin_2019_RG"/>

== Divisions ==
{{see also|Angle#Units}}

Many angle units are defined as a division of the turn. For example, the [[Degree (angle)|degree]] is defined such that one turn is 360 degrees. 

Using [[metric prefix]]es, the turn can be divided in 100 centiturns or {{val|1000}} milliturns, with each milliturn corresponding to an [[angle]] of 0.36°, which can also be written as [[Minute and second of arc|21′&nbsp;36″]].<ref name="Hoyle_1962" /><ref name="Klein_2012" /> A [[protractor]] divided in centiturns is normally called a "[[percentage]] protractor". While percentage protractors have existed since 1922,<ref name="Croxton_1992" /> the terms centiturns, milliturns and microturns<!-- ca. 1.3" --> were introduced much later by the British astronomer [[Fred Hoyle]] in 1962.<ref name="Hoyle_1962" /><ref name="Klein_2012" /> Some measurement devices for artillery and [[satellite watching]] carry milliturn scales.<ref name="Schiffner_1965" /><ref name="Hayes_1975" />

[[Binary angular measurement|Binary fractions of a turn]] are also used. Sailors have traditionally divided a turn into 32 [[points of the compass|compass points]], which implicitly have an angular separation of 1/32&nbsp;turn. The ''binary degree'', also known as the ''[[binary radian]]'' (or ''brad''), is {{sfrac|1|256}}&nbsp;turn.<ref name="Savage_2007" /> The binary degree is used in computing so that an angle can be represented to the maximum possible precision in a single [[byte]]. Other measures of angle used in computing may be based on dividing one whole turn into {{math|2<sup>''n''</sup>}} equal parts for other values of {{mvar|n}}.<ref name="Hargreaves_2010" />

== Proposals for a single letter to represent 2''π'' ==
{{anchor|Tau proposal|Tau proposals}} <!-- These are used in many redirects -->
{{see also|Pi#Adoption of the symbol π}}

[[File:Circle_radians_tau.gif|thumb|200px|right|An arc of a circle with the same length as the radius of that circle corresponds to an angle of 1&nbsp;radian. A full circle corresponds to a full turn, or approximately 6.28&nbsp;radians, which is expressed here using the Greek letter [[tau]] ({{mvar|&tau;}}).]]
[[File:Tau-angles.svg|thumb|right|Some special angles in radians, stated in terms of {{tau}}.]]
[[File:Degree-Radian Conversion tau.svg|right|thumb|A comparison of angles expressed in degrees and radians.]]

The number {{math|2{{pi}}}} (approximately 6.28) is the ratio of a circle's [[circumference]] to its [[radius]], and the number of radians in one turn.

The meaning of the symbol <math>\pi</math> was not originally fixed to the ratio of the circumference and the diameter. In 1697, [[David Gregory (mathematician)|David Gregory]] used {{math|{{sfrac|''π''|''ρ''}}}} (pi over rho) to denote the [[perimeter]] of a circle (i.e., the [[circumference]]) divided by its radius.<ref name="Beckmann_1989" /><ref name="Schwartzman_1994" /> However, earlier in 1647, [[William Oughtred]] had used {{math|{{sfrac|''δ''|''π''}}}} (delta over pi) for the ratio of the diameter to perimeter. The first use of the symbol {{pi}} on its own with its present meaning (of perimeter divided by diameter) was in 1706 by the [[Wales|Welsh]] mathematician [[William Jones (mathematician)|William Jones]].<ref name=jones/><ref name="Veling_2001"/>

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 {{pi}}.<ref>{{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}}.]"</ref><ref>{{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}}</ref> Euler would later use the letter {{pi}} for the 3.14... constant in his 1736 ''[[Mechanica]]''<ref>{{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"</ref> and 1748 ''[[Introductio in analysin infinitorum]],''<ref>{{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 (1707–1783) |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}}</ref> 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 {{pi}} for one-fourth of the circumference of a unit circle, or 1.57... .<ref>{{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}}</ref><ref>{{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}}</ref> 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;<ref>{{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}}}}</ref> afterward, {{pi}} was standardized as being equal to 3.14... .<ref name=":0">{{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}}</ref><ref name="Euler_1746"/>

Several people have independently proposed using {{math|1={{tau}} = 2{{pi}}}}, including:<ref>{{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}}</ref>

* Joseph Lindenburg ({{circa}} 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 {{pi}}, 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 (<math>\pi\!\;\!\!\!\pi = 2\pi</math>).<ref name="Palais_2001" />

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 (Greek)|psi]]: <math>\psi = 2 \pi</math>.<ref>{{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}}</ref>

The same year, Thomas Colignatus proposed the uppercase Greek letter [[theta]], Θ, to represent 2{{pi}}.<ref name="Cool_2008" />
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.<ref name="Mann-Janzen-Ali-Scourboutakos-Guleria_2015" /> It has also been proposed to use the wheel symbol, teth, to represent the value 2{{pi}}, 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 2{{pi}}.<ref name="Mann-Chen-Aylward-Jorritsma-Mann-Defaz-Pierce-Lam-Stairs-Hermandez-Li-Xiang-Kanaan_2019" />

In 2010, [[Michael Hartl]] proposed to use the Greek letter [[tau]] to represent the circle constant: {{math|1=''&tau;'' = 2''&pi;''}}. 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 {{sfrac|3|4}}&nbsp;turn would be represented as {{math|{{sfrac|3''τ''|4}}}}&nbsp;rad instead of {{math|{{sfrac|3''π''|2}}}}&nbsp;rad. As for the choice of notation, he offered two reasons. First, {{mvar|&tau;}} is the number of radians in one ''turn'', and both {{mvar|&tau;}} and ''turn'' begin with a {{IPAc-en|t}} sound. Second, {{mvar|&tau;}} visually resembles {{pi}}, whose association with the circle constant is unavoidable. Hartl's ''Tau Manifesto''{{efn|Original version,<ref name="Hartl_2010" /> current version<ref name="Hartl_2019" />}} gives many examples of formulas that are asserted to be clearer where {{math|1=''&tau;''}} is used instead of {{math|1=''&pi;''}}.<ref name="Aron_2011" /><ref name="Landau_2011" /><ref name="Bartholomew_2014" /> For example, Hartl asserts that replacing [[Euler's identity]] {{math|1=''e''<sup>''iπ''</sup> = −1}} by {{math|1=''e''<sup>''iτ''</sup> = 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 {{mvar|&tau;}}, {{math|1=''A'' = {{sfrac|1|2}}{{tau}}''r''<sup>2</sup>}}, contains a natural factor of {{sfrac|1|2}} arising from [[Integral|integration]].

Initially, this proposal did not receive significant acceptance by the mathematical and scientific communities.<ref name="Telegraph_2011" /> However, the use of {{math|''&tau;''}} has become more widespread.<ref name="McMillan_2020" />

The following table shows how various identities appear when {{math|1=''τ'' = 2''π''}} is used instead of {{pi}}.<ref name="Abbott_2012" /><ref name="Palais_2001" /> For a more complete list, see ''[[List of formulae involving π|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;" |<math>{\color{orangered}\frac{\pi}{2}} \text{ rad}</math>
| style="text-align: center; padding: 0.5% 2em 0.5% 0.5em;" |<math>{\color{orangered}\frac{\tau}{4}} \text{ rad}</math>
|{{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;" |<math>C = {\color{orangered}2 \pi} r</math>
| style="text-align: center; padding: 0.5% 2em 0.5% 0.5em;" |<math>C = {\color{orangered}\tau} r</math>
| The length of an [[Circular arc|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;" |<math>A = {\color{orangered}\pi}r^2</math>
| style="text-align: center; padding: 0.5% 2em 0.5% 0.5em;" |<math>A = {\color{orangered}\frac{1}{2} \tau} r^2</math>
| The area of a [[Circular sector|sector]] of angle {{math|''θ''}} is {{math|1=''A'' = {{sfrac|1|2}}''θr''<sup>2</sup>}}.
|-
| style="text-align: center; padding-left: 0.fem; padding-right: 0.5em;" | Area of a regular [[Polygon|{{mvar|n}}-gon]] with unit circumradius
| style="text-align: center; border-style: solid none solid none; padding-left: 0.5em;" |<math>A = \frac{n}{2} \sin \frac{{\color{orangered}2 \pi}}{n}</math>
| style="text-align: center; padding: 0.5% 2em 0.5% 0.5em;" |<math>A = \frac{n}{2} \sin \frac{{\color{orangered}\tau}}{n}</math>
|
|-
| style="text-align: center; padding-left: 0.fem; padding-right: 0.5em;" | [[n-sphere#Volume and surface area|{{mvar|n}}-ball and {{math|''n''}}-sphere volume recurrence relation]]
| style="text-align: center; border-style: solid none solid none; padding-left: 0.5em;" | <math>V_n(r) = \frac{r}{n} S_{n-1}(r)</math>
<math>S_n(r) = {\color{orangered} 2 \pi} r V_{n-1}(r)</math>
| style="text-align: center; padding: 0.5% 2em 0.5% 0.5em;" | <math>V_n(r) = \frac{r}{n} S_{n-1}(r)</math>
<math>S_n(r) = {\color{orangered}\tau}rV_{n-1}(r)</math>
|{{math|1=''V''<sub>0</sub>(''r'') = 1}}{{br}} {{math|1=''S''<sub>0</sub>(''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;" |<math>f(a) = \frac{1}{{\color{orangered}2\pi} i} \oint_\gamma \frac{f(z)}{z-a}\, dz</math>
| style="text-align: center; padding: 0.5% 2em 0.5% 0.5em;" |<math>f(a) = \frac{1}{{\color{orangered}\tau} i} \oint_\gamma \frac{f(z)}{z-a}\, dz</math>
| <math>\gamma</math> is the boundary of a disk containing <math>a</math> 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;" |<math>\varphi(x) = \frac{1}{\sqrt{{\color{orangered}2\pi}}}e^{-\frac{x^2}{2}}</math>
| style="text-align: center; padding: 0.5% 2em 0.5% 0.5em;" |<math>\varphi(x) = \frac{1}{\sqrt{{\color{orangered}\tau}}}e^{-\frac{x^2}{2}}</math>
|
|-
| 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;" |<math>n! \sim \sqrt{{\color{orangered}2 \pi} n}\left(\frac{n}{e}\right)^n</math>
| style="text-align: center; padding: 0.5% 2em 0.5% 0.5em;" |<math>n! \sim \sqrt{{\color{orangered}\tau} n}\left(\frac{n}{e}\right)^n</math>
|
|-
| style="text-align: center; padding-left: 0.fem; padding-right: 0.5em;" | [[Root of unity|{{mvar|n}}th roots of unity]]
| style="text-align: center; border-style: solid none solid none; padding-left: 0.5em;" |<math>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}</math>
| style="text-align: center; padding: 0.5% 2em 0.5% 0.5em;" |<math>e^{{\color{orangered}\tau} i \frac{k}{n}} = \cos\frac{k {\color{orangered}\tau}}{n} + i \sin\frac{k {\color{orangered}\tau}}{n}</math>
|
|-
| 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;" |<math>h = {\color{orangered}2 \pi} \hbar</math>
| style="text-align: center; padding: 0.5% 2em 0.5% 0.5em;" |<math>h = {\color{orangered}\tau} \hbar</math>
| {{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;" |<math>\omega = {\color{orangered}2 \pi} f</math>
| style="text-align: center; padding: 0.5% 2em 0.5% 0.5em;" |<math>\omega = {\color{orangered}\tau} f</math>
|
|-
|}

=== In culture ===

{{tau}} has made numerous appearances in culture. It is celebrated annually on June 28, known as Tau Day.<ref>{{cite web |last1=Hartl |first1=Michael |title=Tau Day |url=https://tauday.com/ |access-date=1 November 2024}}</ref> {{tau}} has been covered in videos by [[Vi Hart]],<ref>{{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}}</ref><ref>{{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}}</ref><ref>{{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}}</ref> ''[[Numberphile]]'',<ref>{{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}}</ref><ref>{{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}}</ref><ref>{{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}}</ref> ''[[SciShow]]'',<ref>{{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}}</ref> [[Steve Mould]],<ref>{{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}}</ref><ref>{{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}}</ref><ref>{{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}}</ref> [[Khan Academy]],<ref>{{cite AV media |people=Khan, Sal |date=2011-07-11 |title=Tau versus pi <nowiki>|</nowiki> Graphs of trig functions <nowiki>|</nowiki> Trigonometry <nowiki>|</nowiki> Khan Academy |language=English |url=https://www.youtube.com/watch?v=1jDDfkKKgmc |access-date=2024-11-24}}</ref> and [[3Blue1Brown]],<ref>{{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}}</ref><ref>{{cite AV media |people=Sanderson, Grant |date=2019-07-07 |title=e^(iπ) in 3.14 minutes, using dynamics <nowiki>|</nowiki> DE5 |language=English |url=https://www.youtube.com/watch?v=v0YEaeIClKY |access-date=2024-11-24 |time=3:08}}</ref> and it has appeared in the comics ''[[xkcd]]'',<ref>{{cite web |last1=Munroe |first1=Randall |title=Pi vs. Tau |url=https://www.xkcd.com/1292/ |website=xkcd |access-date=1 November 2024}}</ref><ref>{{cite web |last1=Munroe |first1=Randall |title=Symbols |url=https://www.xkcd.com/2520/ |website=xkcd |access-date=1 November 2024}}</ref> ''[[Saturday Morning Breakfast Cereal]]'',<ref>{{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}}</ref><ref>{{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}}</ref><ref>{{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}}</ref> and ''[[Sally Forth (Greg Howard comic strip)|Sally Forth]]''.<ref>{{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}}</ref> 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.<ref>{{cite web |title=Fun & Culture – MIT Facts |url=https://facts.mit.edu/fun-culture/ |website=Massachusetts Institute of Technology |access-date=2 November 2024}}</ref>

== Unit conversion ==
[[File:2pi-unrolled.gif|400px|thumb|right|The [[circumference]] of the [[unit circle]] (whose [[radius]] is one) is {{math|2''π''}}.]]

One turn is equal to {{math|2''π''}} (≈&nbsp;{{val|6.283185307179586}})<ref name="OEIS2C_A019692" /> [[radian]]s, 360 [[degree (angle)|degrees]], or 400 [[gradian]]s.

{| class="wikitable" style="text-align:center;"
|+ Conversion of common angles
|-
! [[Turn (angle)|Turn]]s
! colspan="2" | [[Radian]]s
! [[Degree (angle)|Degree]]s
! [[Gradian]]s
|-
| 0 turn
| colspan="2" | 0 rad
| 0°
| 0<sup>g</sup>
|-
| {{sfrac|1|72}} turn
| {{sfrac|{{tau}}|72}} rad
| {{sfrac|{{pi}}|36}} rad
| 5°
| {{sfrac|5|5|9}}<sup>g</sup>
|-
| {{sfrac|1|24}} turn
| {{sfrac|{{tau}}|24}} rad
| {{sfrac|{{pi}}|12}} rad
| 15°
| {{sfrac|16|2|3}}<sup>g</sup>
|-
| {{sfrac|1|16}} turn
| {{sfrac|{{tau}}|16}} rad
| {{sfrac|{{pi}}|8}} rad
| 22.5°
| 25<sup>g</sup>
|-
| {{sfrac|1|12}} turn
| {{sfrac|{{tau}}|12}} rad
| {{sfrac|{{pi}}|6}} rad
| 30°
| {{sfrac|33|1|3}}<sup>g</sup>
|-
| {{sfrac|1|10}} turn
| {{sfrac|{{tau}}|10}} rad
| {{sfrac|{{pi}}|5}} rad
| 36°
| 40<sup>g</sup>
|-
| {{sfrac|1|8}} turn
| {{sfrac|{{tau}}|8}} rad
| {{sfrac|{{pi}}|4}} rad
| 45°
| 50<sup>g</sup>
|-
| {{sfrac|1|2{{pi}}}} turn
| colspan="2" | 1 rad
| {{circa}} 57.3°
| {{circa}} 63.7<sup>g</sup>
|-
| {{sfrac|1|6}} turn
| {{sfrac|{{tau}}|6}} rad
| {{sfrac|{{pi}}|3}} rad
| 60°
| {{sfrac|66|2|3}}<sup>g</sup>
|-
| {{sfrac|1|5}} turn
| {{sfrac|{{tau}}|5}} rad
| {{sfrac|2{{pi}}|5}} rad
| 72°
| 80<sup>g</sup>
|-
| {{sfrac|1|4}} turn
| {{sfrac|{{tau}}|4}} rad
| {{sfrac|{{pi}}|2}} rad
| 90°
| 100<sup>g</sup>
|-
| {{sfrac|1|3}} turn
| {{sfrac|{{tau}}|3}} rad
| {{sfrac|2{{pi}}|3}} rad
| 120°
| {{sfrac|133|1|3}}<sup>g</sup>
|-
| {{sfrac|2|5}} turn
| {{sfrac|2{{tau}}|5}} rad
| {{sfrac|4{{pi}}|5}} rad
| 144°
| 160<sup>g</sup>
|-
| {{sfrac|1|2}} turn
| {{sfrac|{{tau}}|2}} rad
| {{pi}} rad
| 180°
| 200<sup>g</sup>
|-
| {{sfrac|3|4}} turn
| {{sfrac|3{{tau}}|4}} rad
| {{sfrac|3{{pi}}|2}} rad
| 270°
| 300<sup>g</sup>
|-
| 1 turn
| {{tau}} rad
| 2{{pi}} rad
| 360°
| 400<sup>g</sup>
|}


== In the ISQ/SI ==
{{anchor|In_the_ISQ/SI}}
{{Infobox physical quantity
| name = Rotation
| othernames = number of revolutions, number of cycles, number of turns, number of rotations
| width =
| background =
| image = 
| caption = 
| unit = [[Unitless]]
| otherunits =
| symbols = ''N''
| baseunits =
| dimension = [[Dimension one|1]]
| extensive =
| intensive =
| conserved =
| transformsas =
| derivations =
}}

In the [[International System of Quantities]] (ISQ), '''rotation''' (symbol '''''N''''') is a [[physical quantity]] defined as '''number of revolutions''':<ref name="ISO80000-3_2019" />

<blockquote>''N'' is the number (not necessarily an integer) of revolutions, for example, of a rotating body about a given axis. Its value is given by:<!-- Difference from the original formula present in the following formula is intentional. See [https://en.wikipedia.org/wiki/Talk:Turn_(angle)#Way_forward]. -->
: <math>N = \frac{\varphi}{2 \pi \text{ rad}}</math>
where {{varphi}} denotes the measure of [[rotational displacement]].</blockquote>

The above definition is part of the ISQ, formalized in the international standard [[ISO 80000-3]] (Space and time),<ref name="ISO80000-3_2019" /> and adopted in the [[International System of Units]] (SI).<ref name="SIBrochure_9" /><ref name="NISTGuide_2009" /> 

Rotation count or number of revolutions is a [[quantity of dimension one]], resulting from a ratio of angular displacement.
It can be negative and also greater than 1 in modulus.
The relationship between quantity rotation, ''N'', and unit turns, tr, can be expressed as:
: <math>N = \frac \varphi \text{tr} = \{ \varphi \}_\text{tr}</math>
where <nowiki>{</nowiki>{{varphi}}<nowiki>}</nowiki><sub>tr</sub> is the numerical value of the angle {{varphi}} in units of turns (see ''{{slink|Physical quantity#Components}}'').

In the ISQ/SI, rotation is used to derive [[rotational frequency]] (the [[Derivative|rate of change]] of rotation with respect to time), denoted by {{mvar|n}}:
: <math>n = \frac{\mathrm{d}N}{\mathrm{d}t}</math>

The SI unit of rotational frequency is the [[reciprocal second]] (s<sup>−1</sup>). Common related [[Frequency#Unit|units of frequency]] are ''[[hertz]]'' (Hz), ''[[cycles per second]]'' (cps), and ''[[revolutions per minute]]'' (rpm).

{{Infobox unit
| name          = Revolution
| othernames    = Cycle
| standard      = 
| quantity      = [[Rotation (quantity)|Rotation]]
| symbol        = rev
| symbol2       = r
| symbol3       = cyc
| symbol4       = c
| units1        = [[Base unit (measurement)|Base unit]]s
| inunits1      = [[One (unit)|1]]
}}

{{anchor|Rotational unit}}
The superseded version ISO 80000-3:2006 defined "revolution" as a special name for the [[dimensionless unit]] "one",{{efn|"The special name revolution, symbol r, for this unit [name 'one', symbol '1'] is widely used in specifications on rotating machines."<ref name="ISO 80000-3:2006">{{cite web | title=ISO 80000-3:2006 | website=ISO | date=2001-08-31 | url=https://www.iso.org/standard/31888.html | access-date=2023-04-25}}</ref>}}
which also received other special names, such as the radian.{{efn|"Measurement units of quantities of dimension one are numbers. In some cases, these measurement units are given special names, e.g. radian..."<ref name="ISO 80000-3:2006"/>}}
Despite their [[dimensional homogeneity]], these two specially named dimensionless units are applicable for non-comparable [[kind of quantity|kinds of quantity]]: rotation and angle, respectively.<ref name="ISO 80000-1">{{cite web |title=ISO 80000-1:2009(en) Quantities and units — Part 1: General |url=https://www.iso.org/obp/ui/#iso:std:iso:80000:-1:ed-1:v1:en |access-date=2023-05-12 |website=iso.org}}</ref>
"Cycle" is also mentioned in ISO 80000-3, in the definition of ''[[period (physics)|period]]''.{{efn|"3-14) period duration, period: duration (item 3‑9) of one cycle of a periodic event"<ref name="ISO80000-3_2019"/>}}

== In programming languages ==

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_Sharp_(programming_language)|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_(programming_language)|Crystal]] || [https://crystal-lang.org/api/1.13.2/Math.html#constant-summary TAU] || 0.36.0 || 2021
|-
| [[Eiffel_(programming_language)|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
|-
| [[Godot_(game_engine)#GDScript|GDScript]] || [https://docs.godotengine.org/en/stable/classes/class_@gdscript.html#class-gdscript-constant-tau TAU] || Godot 3.0 || 2018
|-
| [[Java_(programming_language)|Java]] || [https://docs.oracle.com/en/java/javase/21/docs/api/java.base/java/lang/Math.html#TAU Math.TAU] || 19 || 2022
|-
| [[Nim_(programming_language)|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_(programming_language)|Python]] || [https://docs.python.org/3/library/math.html#math.tau math.tau] || 3.6 || 2016
|-
| [[Raku_(programming_language)|Raku]] || [https://docs.raku.org/language/terms#term_tau tau] and [https://docs.raku.org/language/terms#term_%CF%84 τ] ||  || 
|-
| [[Rust_(programming_language)|Rust]] || [https://doc.rust-lang.org/stable/std/f64/consts/constant.TAU.html std::f64::consts::TAU] || 1.47.0 || 2020
|-
| [[Zig_(programming_language)|Zig]] || [https://github.com/ziglang/zig/blob/master/lib/std/math.zig#L18 std.math.tau] || 0.6.0 || 2019
|}

== See also ==
* [[Ampere-turn]]
* [[Hertz]] (modern) or [[Cycle per second]] (older)
* [[Angle of rotation]]
* [[Revolutions per minute]]
* [[Repeating circle]]
* [[Spat (angular unit)]] – the [[solid angle]] counterpart of the turn, equivalent to {{math|4''π''}}&nbsp;[[steradian]]s.
* [[Unit interval]]
* ''[[Divine Proportions: Rational Trigonometry to Universal Geometry]]''
* [[Modulo operation]]
* [[Twist (rational trigonometry)|Twist (mathematics)]]

== Notes ==
{{notelist}}

== References ==
{{reflist|refs=

<ref name=jones>{{cite book |last=Jones |first=William |author-link=William Jones (mathematician) |year=1706 |title=Synopsis Palmariorum Matheseos |place=London |publisher=J. Wale |url=https://archive.org/details/SynopsisPalmariorumMatheseosOrANewIntroductionToTheMathematics/page/n283/ |pages=[https://archive.org/details/SynopsisPalmariorumMatheseosOrANewIntroductionToTheMathematics/page/n261/ 243], [https://archive.org/details/SynopsisPalmariorumMatheseosOrANewIntroductionToTheMathematics/page/n283/ 263] |quote-page=263 |quote=There are various other ways of finding the ''Lengths'', or ''Areas'' of particular ''Curve Lines'' or ''Planes'', which may very much facilitate the Practice; as for instance, in the ''Circle'', the Diameter is to Circumference as 1 to {{br}}<math>
\overline{\tfrac{16}5 - \tfrac4{239}}
- \tfrac13\overline{\tfrac{16}{5^3} - \tfrac4{239^3}}
+ \tfrac15\overline{\tfrac{16}{5^5} - \tfrac4{239^5}}
-,\, \&c. =</math>{{br}}{{math|1=3.14159, &''c.'' = ''π''}}. This ''Series'' (among others for the same purpose, and drawn from the same Principle) I receiv'd from the Excellent Analyst, and my much Esteem'd Friend Mr. ''[[John Machin]]''; and by means thereof, ''[[Ludolph van Ceulen|Van Ceulen]]''{{'}}s Number, or that in Art. 64.38. may be Examin'd with all desireable Ease and Dispatch.
}} {{pb}} Reprinted in {{cite book |last=Smith |first=David Eugene |year=1929 |title=A Source Book in Mathematics |publisher=McGraw–Hill |chapter=William Jones: The First Use of {{mvar|π}} for the Circle Ratio |chapter-url=https://archive.org/details/sourcebookinmath1929smit/page/346/ |pages=346–347 }}</ref>

<ref name="Savage_2007">{{cite web |title=ooPIC Programmer's Guide - Chapter 15: URCP |work=[[ooPIC]] Manual & Technical Specifications - ooPIC Compiler Ver 6.0 |orig-date=1997 |date=2007 |publisher=Savage Innovations, LLC |url=http://www.oopic.com/pgchap15.htm |access-date=2019-08-05 |url-status=dead |archive-url=https://web.archive.org/web/20080628051746/http://www.oopic.com/pgchap15.htm |archive-date=2008-06-28}}</ref>

<ref name="Hargreaves_2010">{{cite web |title=Angles, integers, and modulo arithmetic |author-first=Shawn |author-last=Hargreaves |author-link=:pl:Shawn Hargreaves |publisher=blogs.msdn.com |url=http://blogs.msdn.com/shawnhar/archive/2010/01/04/angles-integers-and-modulo-arithmetic.aspx |access-date=2019-08-05 |url-status=live |archive-url=https://web.archive.org/web/20190630223817/http://www.shawnhargreaves.com/blogindex.html |archive-date=2019-06-30}}</ref>

<ref name="Beckmann_1989">{{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}}</ref>

<ref name="Schwartzman_1994">{{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]}}</ref>

<ref name="Veling_2001">{{cite web |title=Pi through the ages |date=2001 |author-first=Anne |author-last=Veling |website=veling.nl |url=http://www.veling.nl/anne/templars/Pi_through_the_ages.html |url-status=dead |archive-url=https://web.archive.org/web/20090702234950/http://www.veling.nl/anne/templars/Pi_through_the_ages.html |archive-date=2009-07-02}}</ref>

<ref name="Croxton_1992">{{cite journal |author-first=Frederick E. |author-last=Croxton |date=1922 |title=A Percentage Protractor - Designed for Use in the Construction of Circle Charts or "Pie Diagrams" |series=Short Note |journal=[[Journal of the American Statistical Association]] |volume=18 |issue=137 |pages=108–109 |doi=10.1080/01621459.1922.10502455}}</ref>

<ref name="Hoyle_1962">{{cite book |author-first=Fred |author-last=Hoyle |author-link=Fred Hoyle |editor-first=M. H. |editor-last=Chandler |title=Astronomy |url=https://archive.org/details/astronom00hoyl |url-access=registration |publisher=[[Macdonald & Co. (Publishers) Ltd.]] / Rathbone Books Limited |location=London, UK |date=1962 |edition=1 |lccn=62065943 |oclc=7419446}} (320 pages)</ref>

<ref name="Klein_2012">{{cite book |author-first=Herbert Arthur |author-last=Klein |title=The Science of Measurement: A Historical Survey (The World of Measurements: Masterpieces, Mysteries and Muddles of Metrology) |chapter=Chapter 8: Keeping Track of Time |edition=corrected reprint of original |date=2012 |orig-date=1988, 1974 |lccn=88-25858 |publisher=[[Dover Publications, Inc.]] / [[Courier Corporation]] (originally by [[Simon & Schuster, Inc.]]) |series=Dover Books on Mathematics |isbn=978-0-48614497-9 |page=102 |chapter-url=https://books.google.com/books?id=CrmuSiCFyikC&pg=PA102 |access-date=2019-08-06}} (736 pages)</ref>

<ref name="Schiffner_1965">{{cite journal |title=Bestimmung von Satellitenbahnen |language=de |author-first=Friedrich |author-last=Schiffner |editor-first=Maria Emma |editor-last=Wähnl |editor-link=:de:Maria Emma Wähnl |journal=[[Astronomische Mitteilungen der Urania-Sternwarte Wien]] |publisher=[[Volksbildungshaus Wiener Urania]] |location=Wien, Austria |volume=8 |issue= |date=1965}}</ref>

<ref name="Hayes_1975">{{cite book |title=Trackers of the Skies |author-first=Eugene Nelson |author-last=Hayes |series=History of the Smithsonian Satellite-tracking Program |publisher=[[Academic Press]] / Howard A. Doyle Publishing Company |location=Cambridge, Massachusetts, USA |date=1975 |orig-date=1968 |url=https://siris-sihistory.si.edu/ipac20/ipac.jsp?&profile=all&source=~!sichronology&uri=full=3100001~!3190~!0#focus}}</ref>

<ref name="EWG_1980">{{cite web |title=Richtlinie 80/181/EWG - Richtlinie des Rates vom 20. Dezember 1979 zur Angleichung der Rechtsvorschriften der Mitgliedstaaten über die Einheiten im Meßwesen und zur Aufhebung der Richtlinie 71/354/EWG |language=de |date=1980-02-15 |url=https://eur-lex.europa.eu/legal-content/DE/TXT/PDF/?uri=CELEX:01980L0181-20090527 |access-date=2019-08-06 |url-status=live |archive-url=https://web.archive.org/web/20190622210052/https://eur-lex.europa.eu/legal-content/DE/TXT/PDF/?uri=CELEX:01980L0181-20090527 |archive-date=2019-06-22}}</ref>

<ref name="EG_2009">{{cite web |title=Richtlinie 2009/3/EG des Europäischen Parlaments und des Rates vom 11. März 2009 zur Änderung der Richtlinie 80/181/EWG des Rates zur Angleichung der Rechtsvorschriften der Mitgliedstaaten über die Einheiten im Messwesen (Text von Bedeutung für den EWR) |language=de |date=2009-03-11 |url=https://eur-lex.europa.eu/legal-content/DE/TXT/PDF/?uri=CELEX:32009L0003&from=EN |access-date=2019-08-06 |url-status=live |archive-url=https://web.archive.org/web/20190806184426/https://eur-lex.europa.eu/legal-content/DE/TXT/PDF/?uri=CELEX:32009L0003&from=EN |archive-date=2019-08-06}}</ref>

<ref name="Einheitenverordnung_1994">{{cite book |title=Einheitenverordnung |chapter=Art. 15 Einheiten in Form von nichtdezimalen Vielfachen oder Teilen von SI-Einheiten |id=941.202 |date=1994-11-23 |language=de-ch |publisher=[[Schweizerischer Bundesrat]]<!-- |work=Der Bundesrat - Das Portal der Schweizer Regierung--> |chapter-url=http://www.admin.ch/opc/de/classified-compilation/19940345/ |access-date=2013-01-01 |url-status=live |archive-url=https://web.archive.org/web/20190510122902/https://www.admin.ch/opc/de/classified-compilation/19940345/ |archive-date=2019-05-10}}</ref>

<ref name="German_2013">{{cite book |title=Handbuch SI-Einheiten: Definition, Realisierung, Bewahrung und Weitergabe der SI-Einheiten, Grundlagen der Präzisionsmeßtechnik |author-first1=Sigmar |author-last1=German |author-first2=Peter |author-last2=Drath |publisher=[[Friedrich Vieweg & Sohn Verlagsgesellschaft mbH]], reprint: [[Springer-Verlag]] |language=de |date=2013-03-13 |orig-date=1979 |edition=1 |isbn=978-3-32283606-9 |id=978-3-528-08441-7, 978-3-32283606-9 |page=421 |url=https://books.google.com/books?id=63qcBgAAQBAJ&pg=PA421 |access-date=2015-08-14}}</ref>

<ref name="Kurzweil_1999">{{cite book |title=Das Vieweg Einheiten-Lexikon: Formeln und Begriffe aus Physik, Chemie und Technik |author-first=Peter |author-last=Kurzweil |language=de |publisher=Vieweg, reprint: [[Springer-Verlag]] |edition=1 |date=2013-03-09 |orig-date=1999 |isbn=978-3-32292920-4 |id=978-3-322-92921-1 |doi=10.1007/978-3-322-92920-4 |page=403 |url=https://books.google.com/books?id=2zecBgAAQBAJ |access-date=2015-08-14}}</ref>

<ref name="Lapilli_2016">{{cite web |title=RE: newRPL: Handling of units |author-first=Claudio Daniel |author-last=Lapilli |date=2016-05-11 |work=HP Museum |url=http://www.hpmuseum.org/forum/thread-4783-post-55836.html#pid55836 |access-date=2019-08-05 |url-status=live |archive-url=https://web.archive.org/web/20170810012742/http://www.hpmuseum.org/forum/thread-4783-post-55836.html |archive-date=2017-08-10}}</ref>

<ref name="Lapilli_2018">{{cite book |title=newRPL User Manual |chapter=Chapter 3: Units - Available Units - Angles |author-first=Claudio Daniel |author-last=Lapilli |date=2018-10-25 |chapter-url=https://newrpl.wiki.hpgcc3.org/doku.php?id=manual:chapter3:units#available-units |access-date=2019-08-07 |url-status=live |archive-url=https://web.archive.org/web/20190806225910/https://newrpl.wiki.hpgcc3.org/doku.php?id=manual:chapter3:units#available-units |archive-date=2019-08-06}}</ref>

<ref name="OEIS2C_A019692">Sequence {{OEIS2C|A019692}}</ref>

<ref name="Palais_2001">{{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}}</ref>

<ref name="Hartl_2010">{{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}}</ref>

<ref name="Hartl_2019">{{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}}</ref>

<ref name="Aron_2011">{{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}}</ref>

<ref name="Landau_2011">{{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=live |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}}</ref>

<ref name="Bartholomew_2014">{{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}}</ref>

<ref name="Telegraph_2011">{{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}}</ref>

<ref name="Paul_2016">{{cite web |title=RE: WP-32S in 2016? |date=2016-01-12 |orig-date=2016-01-11 |author-first=Matthias R. |author-last=Paul |work=HP Museum |url=https://www.hpmuseum.org/forum/thread-5427-post-48945.html#pid48945 |access-date=2019-08-05 |url-status=live |archive-url=https://web.archive.org/web/20190805163709/https://www.hpmuseum.org/forum/thread-5427-post-48945.html |archive-date=2019-08-05 |quote=[…] I'd like to see a TURN mode being implemented as well. TURN mode works exactly like DEG, RAD and GRAD (including having a full set of angle unit conversion functions like on the [[WP 34S]]), except for that a full circle doesn't equal 360 degree, 6.2831... rad or 400 gon, but 1 turn. (I […] found it to be really convenient in engineering/programming, where you often have to convert to/from other unit representations […] But I think it can also be useful for educational purposes. […]) Having the angle of a full circle normalized to 1 allows for easier [[Rule of three (mathematics)|conversions]] to/from a whole bunch of other angle units […]}}</ref>

<ref name="Bonin_2019_OG">{{cite book |title=WP&nbsp;43S Owner's Manual |date=2019 |orig-date=2015 |author-last=Bonin |author-first=Walter |isbn=978-1-72950098-9 |edition=draft |version=0.12 |pages=72, 118–119, 311 |url=https://gitlab.com/wpcalculators/wp43/-/raw/master/docs/OwnersManual.pdf |access-date=2019-08-05 |url-status=live |archive-url=https://web.archive.org/web/20230718192232/https://gitlab.com/rpncalculators/wp43/-/raw/master/docs/OwnersManual.pdf |archive-date=2023-07-18}} [https://gitlab.com/Over_score/wp43s] [https://gitlab.com/wpcalculators/wp43] (314 pages)</ref>

<ref name="Bonin_2019_RG">{{cite book |title=WP&nbsp;43S Reference Manual |date=2019 |orig-date=2015 |author-last=Bonin |author-first=Walter |isbn=978-1-72950106-1 |edition=draft |version=0.12 |pages=iii, 54, 97, 128, 144, 193, 195 |url=https://gitlab.com/wpcalculators/wp43/-/raw/master/docs/ReferenceManual.pdf |access-date=2019-08-05 |url-status=live |archive-url=https://web.archive.org/web/20230718192332/https://gitlab.com/rpncalculators/wp43/-/raw/master/docs/ReferenceManual.pdf |archive-date=2023-07-18}} [https://gitlab.com/Over_score/wp43s] [https://gitlab.com/wpcalculators/wp43] (271 pages)</ref>

<ref name="Euler_1746">{{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.}}</ref>

<ref name="Cool_2008">{{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)</ref>

<ref name="Mann-Janzen-Ali-Scourboutakos-Guleria_2015">{{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}}</ref>

<ref name="Mann-Chen-Aylward-Jorritsma-Mann-Defaz-Pierce-Lam-Stairs-Hermandez-Li-Xiang-Kanaan_2019">{{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}}</ref>

<ref name="McMillan_2020">{{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}}</ref>

<ref name="Abbott_2012">{{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}}</ref>

<ref name="Fitzpatrick_2021">{{cite book |author-last=Fitzpatrick |author-first=Richard |title=Newtonian Dynamics: An Introduction |publisher=[[CRC Press]] |date=2021 |isbn=978-1-000-50953-3 |url=https://books.google.com/books?id=rRpSEAAAQBAJ&pg=PA116 |access-date=2023-04-25 |page=116}}</ref>

<ref name="IET_2016">{{cite book |title=Units & Symbols for Electrical & Electronic Engineers |date=2016 |publisher=[[Institution of Engineering and Technology]] |publication-place=London, UK |url=https://www.theiet.org/media/4173/units-and-symbols.pdf |access-date=2023-07-18 |url-status=live |archive-url=https://web.archive.org/web/20230718183635/https://www.theiet.org/media/4173/units-and-symbols.pdf |archive-date=2023-07-18}} (1+iii+32+1 pages)</ref>

<ref name="ISO80000-3_2019">{{cite web |title=ISO 80000-3:2019 Quantities and units — Part 3: Space and time |publisher=[[International Organization for Standardization]] |date=2019 |edition=2 |url=https://www.iso.org/standard/64974.html |access-date=2019-10-23}} [https://www.iso.org/obp/ui/#iso:std:iso:80000:-3:ed-2:v1:en] (11 pages)</ref>

<ref name="SIBrochure_9">{{SIbrochure9th}}</ref>

<ref name="NISTGuide_2009">{{cite web |title=The NIST Guide for the Use of the International System of Units, Special Publication 811 |author-first1=Ambler |author-last1=Thompson |author-first2=Barry N. |author-last2=Taylor |edition=2008 |publisher=[[National Institute of Standards and Technology]] |date=2020-03-04 |orig-date=2009-07-02 |ref={{sfnref|NIST|2009}} |url=https://www.nist.gov/pml/special-publication-811 |access-date=2023-07-17}} [https://web.archive.org/web/20230515201622/https://nvlpubs.nist.gov/nistpubs/Legacy/SP/nistspecialpublication811e2008.pdf]</ref>

}} <!-- END reflist -->

== External links ==
* [https://tauday.com/tau-manifesto ''The Tau Manifesto'']

{{DEFAULTSORT:Turn (Geometry)}}
[[Category:Units of plane angle]]
[[Category:Mathematical concepts]]
[[Category:Angle]]
[[Category:1 (number)]]