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 (/ˈtɔː, ˈtɒ, ˈtaʊ/; uppercase Τ, lowercase τ or ${\boldsymbol {\tau }}$; 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/, but in Greek it is [taf]. 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). Letters that arose from tau include Roman T and Cyrillic Te (Т, т).

The letter occupies the Unicode slots U+03C4 (lowercase) and U+03A4 (uppercase). In HTML, they can be produced with named entities (τ and Τ), decimal references (τ and Τ), or hexadecimal references (τ and Τ).

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

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
• 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

Mathematics

• Divisor function in number theory, also denoted d or σ₀
• Golden ratio (1.618...), although φ (phi) is more common
• Kendall tau rank correlation coefficient in statistics
• Stopping time in stochastic processes.
• Tau, the ratio of the circumference to the radius of a circle, which is equal to 2π (6.28318...)
• Tau functions, several
• Torsion of a curve in differential geometry
• Translation in Euclidean geometry (although the Latin letter T is used more often)
• The Prouhet–Thue–Morse constant

Physics

• Proper time in relativity
• Shear stress in continuum mechanics
• 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
• Torque, the rotational force in mechanics
• The symbol for tortuosity in hydrogeology
• In thermal physics, the symbol for the fundamental temperature $\tau =k_{B}T$

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.
• 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/
• 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.
• 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.

Character encodings

For the Greek and Coptic letter tau:

Character information

Preview	Τ		τ		Ⲧ		ⲧ
Unicode name	GREEK CAPITAL LETTER TAU		GREEK SMALL LETTER TAU		COPTIC CAPITAL LETTER TAU		COPTIC SMALL LETTER TAU
Encodings	decimal	hex	dec	hex	dec	hex	dec	hex
Unicode	932	U+03A4	964	U+03C4	11430	U+2CA6	11431	U+2CA7
UTF-8	206 164	CE A4	207 132	CF 84	226 178 166	E2 B2 A6	226 178 167	E2 B2 A7
Numeric character reference	Τ	Τ	τ	τ	Ⲧ	Ⲧ	ⲧ	ⲧ
Named character reference	Τ		τ
DOS Greek	147	93	171	AB
DOS Greek-2	208	D0	238	EE
Windows 1253	212	D4	244	F4
TeX			\tau

For the mathematical letter tau:

Character information

Preview	𝚻		𝛕		𝛵		𝜏		𝜯		𝝉
Unicode name	MATHEMATICAL BOLD CAPITAL TAU		MATHEMATICAL BOLD SMALL TAU		MATHEMATICAL ITALIC CAPITAL TAU		MATHEMATICAL ITALIC SMALL TAU		MATHEMATICAL BOLD ITALIC CAPITAL TAU		MATHEMATICAL BOLD ITALIC SMALL TAU
Encodings	decimal	hex	dec	hex	dec	hex	dec	hex	dec	hex	dec	hex
Unicode	120507	U+1D6BB	120533	U+1D6D5	120565	U+1D6F5	120591	U+1D70F	120623	U+1D72F	120649	U+1D749
UTF-8	240 157 154 187	F0 9D 9A BB	240 157 155 149	F0 9D 9B 95	240 157 155 181	F0 9D 9B B5	240 157 156 143	F0 9D 9C 8F	240 157 156 175	F0 9D 9C AF	240 157 157 137	F0 9D 9D 89
UTF-16	55349 57019	D835 DEBB	55349 57045	D835 DED5	55349 57077	D835 DEF5	55349 57103	D835 DF0F	55349 57135	D835 DF2F	55349 57161	D835 DF49
Numeric character reference	𝚻	𝚻	𝛕	𝛕	𝛵	𝛵	𝜏	𝜏	𝜯	𝜯	𝝉	𝝉

Character information

Preview	𝝩		𝞃		𝞣		𝞽
Unicode name	MATHEMATICAL SANS-SERIF BOLD CAPITAL TAU		MATHEMATICAL SANS-SERIF BOLD SMALL TAU		MATHEMATICAL SANS-SERIF BOLD ITALIC CAPITAL TAU		MATHEMATICAL SANS-SERIF BOLD ITALIC SMALL TAU
Encodings	decimal	hex	dec	hex	dec	hex	dec	hex
Unicode	120681	U+1D769	120707	U+1D783	120739	U+1D7A3	120765	U+1D7BD
UTF-8	240 157 157 169	F0 9D 9D A9	240 157 158 131	F0 9D 9E 83	240 157 158 163	F0 9D 9E A3	240 157 158 189	F0 9D 9E BD
UTF-16	55349 57193	D835 DF69	55349 57219	D835 DF83	55349 57251	D835 DFA3	55349 57277	D835 DFBD
Numeric character reference	𝝩	𝝩	𝞃	𝞃	𝞣	𝞣	𝞽	𝞽

These characters are used only as mathematical symbols. 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