Tau

Tau (uppercase Τ, lowercase τ; ταυ ) is the 19th letter of the Greek alphabet. In the system of Greek numerals it has a value of 300.

The name in English is pronounced, but in modern Greek it is. This is because the pronunciation of the combination of Greek letters αυ has changed from ancient to modern times from one of to either  or, depending on what follows (see Greek orthography).

Tau was derived from the Phoenician letter taw. 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 ( and  ), decimal references (  and  ), or hexadecimal references (  and  ).

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

Mathematics

 * The golden ratio (1.618...), although φ (phi) is more common.
 * Tau-function in number theory.
 * The divisor function in number theory, also denoted d or σ0.
 * Torsion of a curve in differential geometry.
 * Kendall tau rank correlation coefficient in statistics.
 * Tau (2π) (6.283...), a proposed alternative to $\pi$ as the circle constant.

Physics

 * In the physical sciences, tau is sometimes used as time variable, to avoid confusing t as temperature.
 * The tau lepton, an elementary particle in particle physics.
 * Tau in astronomy is a measure of opacity, or how much sunlight cannot penetrate the atmosphere.
 * Proper time in relativity.
 * The symbol for tortuosity in hydrogeology.
 * Torque, the rotational force in mechanics.
 * The lifetime of a spontaneous emission process.
 * The prefix of many stars, via the Bayer stellar designation system. (Tau Ceti is such a star.)
 * Shear stress in continuum mechanics.
 * The time constant of any device, such as an RC circuit.

Biology

 * The dose interval in pharmacokinetics.
 * Tau (protein) in biochemistry is a protein associated with microtubules and is implicated in neurodegenerative diseases such as Alzheimer's Disease and FTLD.
 * 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 core variable in General Tau Theory.

Other

 * The specific tax amount.

Symbolism

 * In ancient times, Tau was used as a symbol for life and/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 v. Tau in the Court of 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; usually members of the Secular Franciscan Order wear a wooden τ in a string with three knots around the neck.