View source for Tau

{{Short description|Nineteenth letter in the Greek alphabet}}
{{About|the letter of the Greek alphabet|the mathematical constant|Tau (mathematical constant)|other uses}}
{{Distinguish|text=[[Tao]] or the Latin letter [[T]]}}
{{pp-move-indef}}
{{Greek alphabet|letter=tau}}

'''Tau''' ({{IPAc-en|'|t|aʊ|,_|'|t|ɔː|,_|'|t|ɒ}};<ref>{{OED|tau}}<br>{{dict.com|tau}}</ref> uppercase '''Τ''', lowercase '''τ''' or <math>\boldsymbol\tau</math>; {{langx|el|ταυ}} {{IPA|el|taf|}}) is the nineteenth letter of the [[Greek alphabet]], representing the [[voiceless alveolar plosive|voiceless dental or alveolar plosive]] {{IPA|el|t|IPA}}. In the system of [[Greek numerals]], it has a value of 300.

The name in English is pronounced {{IPAc-en|t|aʊ}} or {{IPAc-en|t|ɔː}},<ref>{{cite web|title=Oxford Dictionaries Online|publisher=Oxford University Press|url=http://www.oxforddictionaries.com/definition/english/tau|archive-url=https://web.archive.org/web/20120720081320/http://oxforddictionaries.com/definition/english/tau|url-status=dead|archive-date=July 20, 2012}}</ref> but in [[modern Greek|Greek]] it is {{IPA|el|taf|}}.<ref>{{Cite web|url=https://www.thespruce.com/the-greek-alphabet-1705558|title=The Greek Alphabet|last=Gaifyllia|first=Nancy|date=10 Oct 2016|website=The Spruce|archive-url=https://web.archive.org/web/20171028112456/https://www.thespruce.com/the-greek-alphabet-1705558|archive-date=28 Oct 2017|url-status=live|access-date=28 Oct 2017}}</ref><ref>{{Cite web|url=http://www.eki.ee/wgrs/rom1_el.htm|title=UN Romanization of Greek for Geographical Names (1987)|last=UNGEGN Working Group on Romanization Systems|date=1 Mar 2016|website=Institute of the Estonian Language|archive-url=https://web.archive.org/web/20171018235656/http://www.eki.ee/wgrs/rom1_el.htm|archive-date=18 Oct 2017|url-status=live|access-date=28 Oct 2017}}</ref> This is because the pronunciation of the combination of Greek letters αυ can have the pronunciation of either {{IPA|[ai]}}, {{IPA|[av]}} or {{IPA|[af]}}, depending on what follows and if a [[Diaeresis (diacritic)|diaeresis]] is present on the second vowel (see [[Greek orthography]]).

Tau was derived from the [[Phoenician alphabet|Phoenician letter]] [[taw]] [[Image:Phoenician taw.svg|20px]] (𐤕).<ref>{{Cite web|url=http://www.brighthubeducation.com/history-homework-help/21831-where-did-the-greek-alphabet-come-from/|title=The Greek Alphabet: Where did It Come From & How Did It Become Modern Greek?|last=Panse|first=Sonal|date=1 May 2012|work=Bright Hub Education|access-date=28 Oct 2017|archive-url=https://web.archive.org/web/20161222201818/http://www.brighthubeducation.com/history-homework-help/21831-where-did-the-greek-alphabet-come-from/|archive-date=22 Dec 2016|url-status=live|editor-last=Finn|editor-first=Wendy}}</ref> Letters that arose from tau include Roman [[T]] and Cyrillic [[Te (Cyrillic)|Te]] (Т, т).

== Modern usage ==
{{See also|Greek letters used in mathematics, science, and engineering#Ττ (tau)}}

The lower-case letter τ is used as a symbol for:
* [[Per unit tax|Specific tax]] amount<ref>{{Cite web|url=http://faculty.maxwell.syr.edu/jomcpeak/PPA723/lecture4.htm|title=McPeak, Lecture 4|last=McPeak|first=John|date=10 Jun 2010|website=Syracuse University|archive-url=https://web.archive.org/web/20100610000249/http://faculty.maxwell.syr.edu/jomcpeak/PPA723/lecture4.htm|archive-date=10 Jun 2010|url-status=live|access-date=27 Oct 2017}}</ref>

=== Biology ===
* The expressed period of the [[Free-running sleep|freerunning]] rhythm of an animal, i.e., the length of the daily cycle of an animal when kept in constant light or constant darkness
* The dose interval in [[pharmacokinetics]]<ref>{{Cite journal|last1=MJ|first1=Shelton|last2=MB|first2=Wire|last3=Y|first3=Lou|last4=B|first4=Adamkiewicz|last5=SS|first5=Min|date=Mar 2016|title=Pharmacokinetic and safety evaluation of high-dose combinations of fosamprenavir and ritonavir|pmid=16495253|pmc=1426463|journal=Antimicrobial Agents and Chemotherapy|volume=50|issue=3|pages=928–934|doi=10.1128/AAC.50.3.928-934.2006}}</ref>
* The core variable in [[general tau theory]]
* [[Tau (protein)|Tau]] in biochemistry, a protein associated with [[microtubule]]s and implicated in neurodegenerative diseases such as [[Alzheimer's disease]], some forms of [[frontotemporal lobar degeneration]], and [[chronic traumatic encephalopathy]]<ref>{{Cite journal|last1=C|first1=González|last2=G|first2=Farías|last3=RB|first3=Maccioni|date=1 Nov 1998|title=Modification of tau to an Alzheimer's type protein interferes with its interaction with microtubules|journal=Cell Mol Biol (Noisy-le-grand)|volume=44 | issue = 7 |pages=1117–1127|via=EuropeMC|pmid=9846894}}</ref><ref>{{Cite journal|last1=M|first1=Sjögren|last2=E|first2=Englund|date=2004|title=Negative neurofilament light and tau immunostaining in frontotemporal dementia|pmid=15178951|journal= Dementia and Geriatric Cognitive Disorders|volume=17|issue=4|pages=346–349|doi=10.1159/000077169|s2cid=9306507}}</ref>

=== Mathematics ===
* [[Divisor function]] in number theory, also denoted ''d'' or σ<sub>0</sub><ref>{{Cite web|url=http://mathworld.wolfram.com/DivisorFunction.html|title=Divisor Function|last=Weisstein|first=Eric W.|date=27 Oct 2017|website=MathWorld --A Wolfram Web Resource.|archive-url=https://web.archive.org/web/20170629140345/http://mathworld.wolfram.com/DivisorFunction.html|archive-date=29 Jun 2017|url-status=live|access-date=28 Oct 2017}}</ref><ref group="note" name=":0">The date given on the source is after that of the archive. This is because the original publishing date is unknown, so the latest update date is stated instead.</ref>
* [[Ramanujan tau function]]<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Tau Function |url=https://mathworld.wolfram.com/TauFunction.html |access-date=2025-01-31 |website=mathworld.wolfram.com |language=en |quote=A function τ(n) related to the divisor function σ(n), also sometimes called Ramanujan's tau function.}}</ref><ref>{{Cite web |title=DLMF: §27.14 Unrestricted Partitions ‣ Additive Number Theory ‣ Chapter 27 Functions of Number Theory |url=https://dlmf.nist.gov/27.14#vi |access-date=2025-01-31 |website=dlmf.nist.gov}}</ref>
* [[Golden ratio]] (1.618...), although φ ([[phi]]) is more common<ref>{{Cite web|url=http://mathworld.wolfram.com/GoldenRatio.html|title=Golden Ratio.|last=Weisstein|first=Eric W.|date=27 Oct 2017|website=Mathworld -- A Wolfram Web Resource|archive-url=https://web.archive.org/web/20170822072337/http://mathworld.wolfram.com/GoldenRatio.html|archive-date=22 Aug 2017|url-status=live|access-date=28 Oct 2017}}</ref><ref name=":0" group="note" />
* [[Kendall tau rank correlation coefficient]] in statistics<ref>{{Cite journal|last=Ghent|first=A. W.|date=June 1963|title=Kendall's "Tau" Coefficient as an Index of Similarity in Comparisons of Plant or Animal Communities|url=https://www.cambridge.org/core/journals/canadian-entomologist/article/kendalls-tau-coefficient-as-an-index-of-similarity-in-comparisons-of-plant-or-animal-communities/B791FA0F9D0450699CF9F1AE92D5DA9D|journal=The Canadian Entomologist|volume=95| issue = 6|pages=568–575|via=Cambridge University Press|doi=10.4039/ent95568-6|s2cid=84897435 }}</ref>
* [[Stopping time]] in stochastic processes.<ref>{{Cite web|url=https://almostsure.wordpress.com/2009/11/23/sigma-algebras-at-a-stopping-time/|title=Sigma Algebras at a Stopping Time|last=Lowther|first=George|date=23 Nov 2009|website=Almost Sure at Wordpress|archive-url=https://web.archive.org/web/20161221152426/https://almostsure.wordpress.com/2009/11/23/sigma-algebras-at-a-stopping-time/|archive-date=21 Dec 2016|url-status=live|access-date=28 Oct 2017}}</ref><ref group="note" name=":1">The archived version of this source may take a few minutes to render the [[TeX]] math codes properly.</ref>
* [[Tau (mathematical constant)|Tau]], the ratio of the circumference to the radius of a circle, which is equal to 2[[pi|{{pi}}]] (6.28318...)<ref>{{Cite web|url=https://tauday.com/tau-manifesto|title=The Tau Manifesto|last=Hartl|first=Michael|author-link=Michael Hartl|date=28 Jun 2010|website=Tau Day|archive-url=https://web.archive.org/web/20171007085119/https://tauday.com/tau-manifesto|archive-date=7 Oct 2017|url-status=live|access-date=28 Oct 2017}}</ref><ref>{{Cite web|last=Bartholomew|first=Randyn Charles|date=June 25, 2014|title=Let's Use Tau--It's Easier Than Pi|url=https://www.scientificamerican.com/article/let-s-use-tau-it-s-easier-than-pi/|url-status=live|archive-url=https://web.archive.org/web/20170908232803/https://www.scientificamerican.com/article/let-s-use-tau-it-s-easier-than-pi/|archive-date=September 8, 2017|access-date=2020-08-31|website=Scientific American|language=en}}</ref>
* [[Tau function (disambiguation)|Tau function]]s, several
* [[Torsion of a curve]] in differential geometry<ref>{{Cite web|url=http://mathworld.wolfram.com/Torsion.html|title=Torsion |last=Weisstein|first=Eric W. |website=Wolfram MathWorld|archive-url=https://web.archive.org/web/20170829220240/http://mathworld.wolfram.com/Torsion.html|archive-date=29 Aug 2017|url-status=live|access-date=28 Oct 2017}}</ref><ref name=":0" group="note" />
* [[Translation (geometry)|Translation]] in Euclidean geometry (although the Latin letter [[T]] is used more often)
* The [[Prouhet–Thue–Morse constant]]<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Thue-Morse Sequence |url=https://mathworld.wolfram.com/Thue-MorseSequence.html |access-date=2025-01-31 |website=mathworld.wolfram.com |language=en |quote=The Thue-Morse sequence, also called the Morse-Thue sequence or Prouhet-Thue-Morse sequence (Allouche and Cosnard 2000), is one of a number of related sequences of numbers obtained from the parities of the counts of 1's in the binary representation of the nonnegative integers.}}</ref>

=== Physics ===
* [[Proper time]] in relativity
* [[Shear stress]] in continuum mechanics<ref>{{Citation |last=Elert |first=Glenn |title=Special Symbols |date=2023 |work=The Physics Hypertextbook|quote= τ shear stress |url=https://physics.info/symbols/ |access-date=2025-02-01 |publisher=hypertextbook |language=en}}</ref>
* The lifetime of a [[spontaneous emission]] process
* [[Tau (particle)|Tau]], an elementary particle in particle physics
* Tau in [[astronomy]] is a measure of [[optical depth]], or how much sunlight cannot penetrate the atmosphere
* In the physical sciences, tau is sometimes used as [[time]] variable, to avoid confusing t as temperature
* [[Time constant]] (also [[relaxation time]]) of any system, such as an [[RC circuit]]<ref>{{Citation |last=Elert |first=Glenn |title=Special Symbols |date=2023 |work=The Physics Hypertextbook|quote= τ time constant |url=https://physics.info/symbols/ |access-date=2025-02-01 |publisher=hypertextbook |language=en}}</ref>
* [[Torque]], the rotational force in mechanics<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Tau |url=https://mathworld.wolfram.com/Tau.html |access-date=2025-02-01 |website=mathworld.wolfram.com |language=en |quote=The symbol also has a number of other different meanings in physics, for example the tau particle, tau neutrino, as a symbol for torque, etc.}}</ref><ref>{{Citation |last=Elert |first=Glenn |title=Special Symbols |date=2023 |work=The Physics Hypertextbook |quote= '''τ''', τ torque|url=https://physics.info/symbols/ |access-date=2025-02-01 |publisher=hypertextbook |language=en}}</ref>
* The symbol for [[tortuosity]] in hydrogeology
* In thermal physics, the symbol for the fundamental temperature <math>\tau = k_B T</math>

== Symbolism ==
* In ancient times, tau was used as a symbol for [[life]] or [[resurrection]], whereas the eighth letter of the Greek alphabet, [[theta]], was considered the symbol of [[death]].{{Citation needed|date=January 2024}}
* In Biblical times, the [[taw]] was put on men to distinguish those who lamented sin, although newer versions of the Bible have replaced the ancient term ''taw'' with ''mark'' (Ezekiel 9:4) or ''signature'' (Job 31:35). Its original sound value is a [[voiceless alveolar plosive]], IPA /t/{{Citation needed|date=January 2024}}
* The symbolism of the [[cross]] was connected not only to the letter [[chi (letter)|chi]] but also to tau, the equivalent of the last letter in the [[Phoenician alphabet|Phoenician]] and Old Hebrew alphabets, and which was originally cruciform in shape; see [[Cross of Tau]].{{Citation needed|date=January 2024}}
* An essay written around 160 AD, attributed to [[Lucian]], a mock legal prosecution called ''[[The Consonants at Law - Sigma vs. Tau, in the Court of the Seven Vowels]]'', contains a reference to the cross attribution. [[Sigma (letter)|Sigma]] petitions the court to sentence Tau to death by crucifixion, saying:
*: ''Men weep, and bewail their lot, and curse Cadmus with many curses for introducing ''Tau'' into the family of letters; they say it was his body that tyrants took for a model, his shape that they imitated, when they set up structures on which men are crucified. ''Stauros'' (cross) the vile engine is called, and it derives its vile name from him. Now, with all these crimes upon him, does he not deserve death, nay, many deaths? For my part I know none bad enough but that supplied by his own shape — that shape which he gave to the [[gibbet]] named ''stauros'' after him by men''
* [[Tau Cross|Tau]] is usually considered as the symbol of [[Franciscan]] orders due to [[Saint Francis of Assisi|St. Francis']] love for it, symbol of the redemption and of the [[Cross]]. Almost all Franciscan churches have painted a tau with two crossing arms, both with [[stigmata]], the one of Jesus and the other of Francis; members of the [[Secular Franciscan Order]] usually wear a wooden τ in a string with three knots around the neck
* The title and symbol of "Tau" is used by [[neo-Gnostic]] bishops as it has some symbolism in many of the modern branches of Gnosticism.

==Unicode==
For the Greek and Coptic letter tau:<ref>Unicode: "[https://www.unicode.org/charts/PDF/U0370.pdf Greek and Coptic (Range: 0370-03FF)]".</ref>

* {{unichar|03A4|html=}}
* {{unichar|03C4|html=}}
* {{unichar|2CA6|html=}}
* {{unichar|2CA7|html=}}
* {{unichar|1D6BB|html=}}{{efn|The {{sc|mathematical}} characters are used only in math. Stylized Greek text should be encoded using the normal Greek letters, with markup and formatting to indicate text style.}}
* {{unichar|1D6D5|html=}}
* {{unichar|1D6F5|html=}}
* {{unichar|1D70F|html=}}
* {{unichar|1D72F|html=}}
* {{unichar|1D749|html=}}
* {{unichar|1D769|html=}}
* {{unichar|1D783|html=}}
* {{unichar|1D7A3|html=}}
* {{unichar|1D7BD|html=}}
{{notelist}}

== See also ==

* [[Greek letters used in mathematics, science, and engineering]]

==Notes==
{{Reflist|group=note}}

== References ==
{{Wiktionary|Τ|τ}}
{{Reflist}}

{{Authority control}}

[[Category:Greek letters]]
[[Category:Cross symbols]]