Tau

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

Not to be confused with Tao or the Latin letter T.

Tau ( /ˈtaʊ,ˈtɔː,ˈtɒ/ ^ⓘ ; ^[1] uppercase Τ, lowercase τ or ${\displaystyle {\boldsymbol {\tau }}}$; Greek : ταυ [taf] ) is the nineteenth letter of the Greek alphabet, representing the voiceless dental or alveolar plosive IPA: [t] . In the system of Greek numerals, it has a value of 300.

The name in English is pronounced /taʊ/ or /tɔː/ , ^[2] but in Greek it is [taf] . ^{[3] [4]} This is because the pronunciation of the combination of Greek letters αυ can have the pronunciation of either [ai], [av] or [af], depending on what follows and if a diaeresis is present on the second vowel (see Greek orthography).

Tau was derived from the Phoenician letter taw (𐤕). ^[5] Letters that arose from tau include Roman T and 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:

• Specific tax amount ^[6]

Biology

• The expressed period of the 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 ^[7]
• The core variable in general tau theory
• Tau in biochemistry, a protein associated with microtubules and implicated in neurodegenerative diseases such as Alzheimer's disease, some forms of frontotemporal lobar degeneration, and chronic traumatic encephalopathy ^{[8] [9]}

Mathematics

• Divisor function in number theory, also denoted d or σ₀ ^{[10] [note 1]}
• Ramanujan tau function ^{[11] [12]}
• Golden ratio (1.618...), although φ (phi) is more common ^{[13] [note 1]}
• Kendall tau rank correlation coefficient in statistics ^[14]
• Stopping time in stochastic processes. ^{[15] [note 2]}
• Tau, the ratio of the circumference to the radius of a circle, which is equal to 2π (6.28318...) Tau is also used to calculate how many rads are in a circle. ^{[16] [17]}
• Tau functions, several
• Torsion of a curve in differential geometry ^{[18] [note 1]}
• Translation in Euclidean geometry (although the Latin letter T is used more often)
• The Prouhet–Thue–Morse constant ^[19]

Physics

• Proper time in relativity
• Shear stress in continuum mechanics ^[20]
• The lifetime of a spontaneous emission process
• 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 ^[21]
• Torque, the rotational force in mechanics ^{[22] [23]}
• The symbol for tortuosity in hydrogeology
• In thermal physics, the symbol for the fundamental temperature ${\displaystyle \tau =k_{B}T}$

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 ]}
• 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 ]}
• The symbolism of the cross was connected not only to the letter chi but also to tau, the equivalent of the last letter in the Phoenician and Old Hebrew alphabets, and which was originally cruciform in shape; see Cross of Tau.^{[ citation needed ]}
• 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 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 is usually considered as the symbol of Franciscan orders due to 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: ^[24]

• U+03A4ΤGREEK CAPITAL LETTER TAU (Τ)
• U+03C4τGREEK SMALL LETTER TAU (τ)
• U+2CA6ⲦCOPTIC CAPITAL LETTER TAU
• U+2CA7ⲧCOPTIC SMALL LETTER TAU
• U+1D6BB𝚻MATHEMATICAL BOLD CAPITAL TAU ^[a]
• U+1D6D5𝛕MATHEMATICAL BOLD SMALL TAU
• U+1D6F5𝛵MATHEMATICAL ITALIC CAPITAL TAU
• U+1D70F𝜏MATHEMATICAL ITALIC SMALL TAU
• U+1D72F𝜯MATHEMATICAL BOLD ITALIC CAPITAL TAU
• U+1D749𝝉MATHEMATICAL BOLD ITALIC SMALL TAU
• U+1D769𝝩MATHEMATICAL SANS-SERIF BOLD CAPITAL TAU
• U+1D783𝞃MATHEMATICAL SANS-SERIF BOLD SMALL TAU
• U+1D7A3𝞣MATHEMATICAL SANS-SERIF BOLD ITALIC CAPITAL TAU
• U+1D7BD𝞽MATHEMATICAL SANS-SERIF BOLD ITALIC SMALL TAU

1. The 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.

See also

• Greek letters used in mathematics, science, and engineering

Notes

1. 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.
2. The archived version of this source may take a few minutes to render the TeX math codes properly.

References

1. "tau" . Oxford English Dictionary (Online ed.). Oxford University Press.(Subscription or participating institution membership required.)
   "tau". Dictionary.com Unabridged (Online). n.d.
2. "Oxford Dictionaries Online". Oxford University Press. Archived from the original on July 20, 2012.
3. Gaifyllia, Nancy (10 Oct 2016). "The Greek Alphabet". The Spruce. Archived from the original on 28 Oct 2017. Retrieved 28 Oct 2017.
4. UNGEGN Working Group on Romanization Systems (1 Mar 2016). "UN Romanization of Greek for Geographical Names (1987)". Institute of the Estonian Language. Archived from the original on 18 Oct 2017. Retrieved 28 Oct 2017.
5. Panse, Sonal (1 May 2012). Finn, Wendy (ed.). "The Greek Alphabet: Where did It Come From & How Did It Become Modern Greek?". Bright Hub Education. Archived from the original on 22 Dec 2016. Retrieved 28 Oct 2017.
6. McPeak, John (10 Jun 2010). "McPeak, Lecture 4". Syracuse University. Archived from the original on 10 Jun 2010. Retrieved 27 Oct 2017.
7. MJ, Shelton; MB, Wire; Y, Lou; B, Adamkiewicz; SS, Min (Mar 2016). "Pharmacokinetic and safety evaluation of high-dose combinations of fosamprenavir and ritonavir". Antimicrobial Agents and Chemotherapy. 50 (3): 928–934. doi:10.1128/AAC.50.3.928-934.2006. PMC   1426463 . PMID   16495253.
8. C, González; G, Farías; RB, Maccioni (1 Nov 1998). "Modification of tau to an Alzheimer's type protein interferes with its interaction with microtubules". Cell Mol Biol (Noisy-le-grand). 44 (7): 1117–1127. PMID   9846894 – via EuropeMC.
9. M, Sjögren; E, Englund (2004). "Negative neurofilament light and tau immunostaining in frontotemporal dementia". Dementia and Geriatric Cognitive Disorders. 17 (4): 346–349. doi:10.1159/000077169. PMID   15178951. S2CID   9306507.
10. Weisstein, Eric W. (27 Oct 2017). "Divisor Function". MathWorld --A Wolfram Web Resource. Archived from the original on 29 Jun 2017. Retrieved 28 Oct 2017.
11. Weisstein, Eric W. "Tau Function". mathworld.wolfram.com. Retrieved 2025-01-31. A function τ(n) related to the divisor function σ(n), also sometimes called Ramanujan's tau function.
12. "DLMF: §27.14 Unrestricted Partitions ‣ Additive Number Theory ‣ Chapter 27 Functions of Number Theory". dlmf.nist.gov. Retrieved 2025-01-31.
13. Weisstein, Eric W. (27 Oct 2017). "Golden Ratio". Mathworld -- A Wolfram Web Resource. Archived from the original on 22 Aug 2017. Retrieved 28 Oct 2017.
14. Ghent, A. W. (June 1963). "Kendall's "Tau" Coefficient as an Index of Similarity in Comparisons of Plant or Animal Communities". The Canadian Entomologist. 95 (6): 568–575. doi:10.4039/ent95568-6. S2CID   84897435 – via Cambridge University Press.
15. Lowther, George (23 Nov 2009). "Sigma Algebras at a Stopping Time". Almost Sure at Wordpress. Archived from the original on 21 Dec 2016. Retrieved 28 Oct 2017.
16. Hartl, Michael (28 Jun 2010). "The Tau Manifesto". Tau Day. Archived from the original on 7 Oct 2017. Retrieved 28 Oct 2017.
17. Bartholomew, Randyn Charles (June 25, 2014). "Let's Use Tau--It's Easier Than Pi". Scientific American. Archived from the original on September 8, 2017. Retrieved 2020-08-31.
18. Weisstein, Eric W. "Torsion". Wolfram MathWorld. Archived from the original on 29 Aug 2017. Retrieved 28 Oct 2017.
19. Weisstein, Eric W. "Thue-Morse Sequence". mathworld.wolfram.com. Retrieved 2025-01-31. 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.
20. Elert, Glenn (2023), "Special Symbols", The Physics Hypertextbook, hypertextbook, retrieved 2025-02-01, τ shear stress
21. Elert, Glenn (2023), "Special Symbols", The Physics Hypertextbook, hypertextbook, retrieved 2025-02-01, τ time constant
22. Weisstein, Eric W. "Tau". mathworld.wolfram.com. Retrieved 2025-02-01. 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.
23. Elert, Glenn (2023), "Special Symbols", The Physics Hypertextbook, hypertextbook, retrieved 2025-02-01, τ, τ torque
24. Unicode: "Greek and Coptic (Range: 0370-03FF)".