Tau (; uppercase Τ, lowercase τ or

\boldsymbol\tau

; ) is the nineteenth letter of the

Greek alphabet The Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BC. It was derived from the earlier Phoenician alphabet, and is the earliest known alphabetic script to systematically write vowels as wel ...

, representing the voiceless dental or alveolar plosive . In the system of

Greek numerals Greek numerals, also known as Ionic, Ionian, Milesian, or Alexandrian numerals, is a numeral system, system of writing numbers using the letters of the Greek alphabet. In modern Greece, they are still used for ordinal number (linguistics), ordi ...

, it has a value of 300. The name in English is pronounced or , but in

Greek Greek may refer to: Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group *Greek language, a branch of the Indo-European language family **Proto-Greek language, the assumed last common ancestor of all kno ...

it is . This is because the pronunciation of the combination of Greek letters αυ can have the pronunciation of either , or , depending on what follows and if a diaeresis is present on the second vowel (see

Greek orthography The orthography of the Modern Greek, modern Greek language was standardised in 1976 and simplified the diacritics in 1982. There are relatively few differences between the orthography of Ancient Greek and Modern Greek. Some time prior to that, on ...

). Tau was derived from the Phoenician letter taw (𐤕). Letters that arose from tau include Roman T and Cyrillic Te (Т, т).

Modern usage

The lower-case letter τ is used as a symbol for: *

Specific tax A per unit tax, or specific tax, is a tax A tax is a mandatory financial charge or levy imposed on an individual or legal entity by a governmental organization to support government spending and public expenditures collectively or to regulat ...

amount

Biology

* The expressed period of the

freerunning Freerunning is an athletic and acrobatic discipline incorporating an aesthetic element, and can be considered either a sport or a performance art, or both. Freerunning is similar to parkour, from which it is derived, but emphasizes artistry over ...

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 Pharmacokinetics (from Ancient Greek ''pharmakon'' "drug" and ''kinetikos'' "moving, putting in motion"; see chemical kinetics), sometimes abbreviated as PK, is a branch of pharmacology dedicated to describing how the body affects a specific su ...

* The core variable in general tau theory *

Tau Tau (; uppercase Τ, lowercase τ or \boldsymbol\tau; ) is the nineteenth letter of the Greek alphabet, representing the voiceless alveolar plosive, voiceless dental or alveolar plosive . In the system of Greek numerals, it has a value of 300 ...

in biochemistry, a protein associated with

microtubule Microtubules are polymers of tubulin that form part of the cytoskeleton and provide structure and shape to eukaryotic cells. Microtubules can be as long as 50 micrometres, as wide as 23 to 27 nanometer, nm and have an inner diameter bet ...

s and implicated in neurodegenerative diseases such as

Alzheimer's disease Alzheimer's disease (AD) is a neurodegenerative disease and the cause of 60–70% of cases of dementia. The most common early symptom is difficulty in remembering recent events. As the disease advances, symptoms can include problems wit ...

, some forms of

frontotemporal lobar degeneration Frontotemporal lobar degeneration (FTLD) is a pathological process that occurs in frontotemporal dementia. It is characterized by atrophy in the frontal lobe and temporal lobe of the brain, with sparing of the parietal and occipital lobes. Com ...

, and

chronic traumatic encephalopathy Chronic traumatic encephalopathy (CTE) is a neurodegenerative disease linked to repeated trauma to the head. The encephalopathy symptoms can include behavioral problems, mood problems, and problems with thinking. The disease often gets wor ...

Mathematics

Divisor function In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as ''the'' divisor function, it counts the ''number of divisors of an integer'' (includi ...

in number theory, also denoted ''d'' or σ₀The 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. *

Ramanujan tau function The Ramanujan tau function, studied by , is the function \tau : \mathbb\to\mathbb defined by the following identity: :\sum_\tau(n)q^n=q\prod_\left(1-q^n\right)^ = q\phi(q)^ = \eta(z)^=\Delta(z), where q=\exp(2\pi iz) with \mathrm(z)>0, \phi is t ...

Golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \fr ...

(1.618...), although φ (

phi Phi ( ; uppercase Φ, lowercase φ or ϕ; ''pheî'' ; Modern Greek: ''fi'' ) is the twenty-first letter of the Greek alphabet. In Archaic and Classical Greek (c. 9th to 4th century BC), it represented an aspirated voiceless bilabial plos ...

) is more common *

Kendall tau rank correlation coefficient In statistics, the Kendall rank correlation coefficient, commonly referred to as Kendall's τ coefficient (after the Greek letter τ, tau), is a statistic used to measure the ordinal association between two measured quantities. A τ test is a ...

in statistics *

Stopping time In probability theory, in particular in the study of stochastic processes, a stopping time (also Markov time, Markov moment, optional stopping time or optional time ) is a specific type of "random time": a random variable whose value is interpre ...

in stochastic processes.The archived version of this source may take a few minutes to render the

TeX Tex, TeX, TEX, may refer to: People and fictional characters * Tex (nickname), a list of people and fictional characters with the nickname * Tex Earnhardt (1930–2020), U.S. businessman * Joe Tex (1933–1982), stage name of American soul singer ...

math codes properly. *

, the ratio of the circumference to the radius of a circle, which is equal to 2 (6.28318...) Tau is also used to calculate how many rads are in a circle. * Tau functions, several *

Torsion of a curve In the differential geometry of curves in three dimensions, the torsion of a curve measures how sharply it is twisting out of the osculating plane. Taken together, the curvature and the torsion of a space curve are analogous to the curvature o ...

in differential geometry *

Translation Translation is the communication of the semantics, meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The English la ...

in Euclidean geometry (although the Latin letter T is used more often) * The

Prouhet–Thue–Morse constant In mathematics, the Prouhet–Thue–Morse constant, named for , Axel Thue, and Marston Morse, is the number—denoted by —whose binary expansion 0.01101001100101101001011001101001... is given by the Prouhet–Thue–Morse sequence. That is, : ...

Physics

Proper time In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. The proper time interval between two events on a world line is the change in proper time ...

in relativity *

Shear stress Shear stress (often denoted by , Greek alphabet, Greek: tau) is the component of stress (physics), stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross secti ...

in continuum mechanics * The lifetime of a

spontaneous emission Spontaneous emission is the process in which a Quantum mechanics, quantum mechanical system (such as a molecule, an atom or a subatomic particle) transits from an excited state, excited energy state to a lower energy state (e.g., its ground state ...

process *

, an elementary particle in particle physics * Tau in

astronomy Astronomy is a natural science that studies celestial objects and the phenomena that occur in the cosmos. It uses mathematics, physics, and chemistry in order to explain their origin and their overall evolution. Objects of interest includ ...

is a measure of

optical depth In physics, optical depth or optical thickness is the natural logarithm of the ratio of incident to ''transmitted'' radiant power through a material. Thus, the larger the optical depth, the smaller the amount of transmitted radiant power throu ...

, or how much sunlight cannot penetrate the atmosphere * In the physical sciences, tau is sometimes used as

time Time is the continuous progression of existence that occurs in an apparently irreversible process, irreversible succession from the past, through the present, and into the future. It is a component quantity of various measurements used to sequ ...

variable, to avoid confusing t as temperature *

Time constant In physics and engineering, the time constant, usually denoted by the Greek language, Greek letter (tau), is the parameter characterizing the response to a step input of a first-order, LTI system theory, linear time-invariant (LTI) system.Concre ...

(also

relaxation time Relaxation stands quite generally for a release of tension, a return to equilibrium. In the sciences, the term is used in the following ways: * Relaxation (physics), and more in particular: ** Relaxation (NMR), processes by which nuclear magneti ...

) of any system, such as an

RC circuit A resistor–capacitor circuit (RC circuit), or RC filter or RC network, is an electric circuit composed of resistors and capacitors. It may be driven by a voltage source, voltage or current source and these will produce different responses. A fi ...

Torque In physics and mechanics, torque is the rotational analogue of linear force. It is also referred to as the moment of force (also abbreviated to moment). The symbol for torque is typically \boldsymbol\tau, the lowercase Greek letter ''tau''. Wh ...

, the rotational force in mechanics * The symbol for

tortuosity Tortuosity is widely used as a critical parameter to predict transport properties of porous media, such as rocks and soils. But unlike other standard microstructural properties, the concept of tortuosity is vague with multiple definitions and vari ...

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 Life, also known as biota, refers to matter that has biological processes, such as Cell signaling, signaling and self-sustaining processes. It is defined descriptively by the capacity for homeostasis, Structure#Biological, organisation, met ...

resurrection Resurrection or anastasis is the concept of coming back to life after death. Reincarnation is a similar process hypothesized by other religions involving the same person or deity returning to another body. The disappearance of a body is anothe ...

, whereas the eighth letter of the Greek alphabet,

theta Theta (, ) uppercase Θ or ; lowercase θ or ; ''thē̂ta'' ; Modern: ''thī́ta'' ) is the eighth letter of the Greek alphabet, derived from the Phoenician letter Teth 𐤈. In the system of Greek numerals, it has a value of 9. Gree ...

, was considered the symbol of

death Death is the end of life; the irreversible cessation of all biological functions that sustain a living organism. Death eventually and inevitably occurs in all organisms. The remains of a former organism normally begin to decompose sh ...

. * 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 The voiceless alveolar, dental and postalveolar plosives (or stops) are types of consonantal sounds used in almost all spoken languages. The symbol in the International Phonetic Alphabet that represents voiceless dental, alveolar, and postal ...

, IPA /t/ * The symbolism of the

cross A cross is a religious symbol consisting of two Intersection (set theory), intersecting Line (geometry), lines, usually perpendicular to each other. The lines usually run vertically and horizontally. A cross of oblique lines, in the shape of t ...

was connected not only to the letter

chi __NOTOC__ Chi may refer to: __NOTOC__ Greek *Chi (letter) (Χ or χ), the twenty-second letter of the Greek alphabet Chinese * ''Chi'' (length) (尺), a traditional unit of length, about ⅓ meter *Chi (mythology) (螭), a dragon * Chi (surname) ...

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 Lucian of Samosata (Λουκιανὸς ὁ Σαμοσατεύς, 125 – after 180) was a Hellenized Syrian satirist, rhetorician and pamphleteer who is best known for his characteristic tongue-in-cheek style, with which he frequently ridi ...

, 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 Sigma ( ; uppercase Σ, lowercase σ, lowercase in word-final position ς; ) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. In general mathematics, uppercase Σ is used as an operator ...

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 Gibbeting is the use of a gallows-type structure from which the dead or dying bodies of criminals were hanged on public display to deter other existing or potential criminals. Occasionally, the gibbet () was also used as a method of public ex ...

named ''stauros'' after him by men'' *

is usually considered as the symbol of

Franciscan The Franciscans are a group of related organizations in the Catholic Church, founded or inspired by the Italian saint Francis of Assisi. They include three independent Religious institute, religious orders for men (the Order of Friars Minor bei ...

orders due to St. Francis' love for it, symbol of the redemption and of the

Cross A cross is a religious symbol consisting of two Intersection (set theory), intersecting Line (geometry), lines, usually perpendicular to each other. The lines usually run vertically and horizontally. A cross of oblique lines, in the shape of t ...

. Almost all Franciscan churches have painted a tau with two crossing arms, both with

stigmata Stigmata (, plural of , 'mark, spot, brand'), in Roman Catholicism, Catholicism, are bodily wounds, scars and pain which appear in locations corresponding to the Crucifixion of Jesus, crucifixion Five Holy Wounds, wounds of Jesus in Christian ...

, the one of Jesus and the other of Francis; members of the

Secular Franciscan Order The Secular Franciscan Order (; abbreviated OFS) is part of the Third Order of Saint Francis, third branch of the Franciscans, Franciscan family formed by Catholic Church, Catholic men and women who seek to observe the Gospel of Jesus in Christia ...

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:Unicode:
Greek and Coptic (Range: 0370-03FF)
. * * * * * * * * * * * * * *

Notes

References

{{Authority control Greek letters Cross symbols