Theta

Theta is the eighth letter of the Greek alphabet, derived from the Phoenician letter Teth ?. In the system of Greek numerals, it has a value of 9.

Greek

In Ancient Greek, θ represented the aspirated voiceless dental plosive, but in Modern Greek it represents the voiceless dental fricative.

Forms

In its archaic form, θ was written as a cross within a circle, and later, as a line or point in circle.
In modern print, lowercase theta may appear in two different forms: one resembling the uppercase form, like a zero with a horizontal bar through it, and one more cursive style featuring an open loop, resembling a nine. For use as a mathematical symbol, where the difference between the two forms may be significant, a dedicated Unicode symbol was introduced, which always displays the open-loop form. There is also, which differs from most realizations of the normal textual uppercase letter by having a cross-bar that goes all the way across, touching the outer O-shape on both sides.

Latin

In the Latin script used for the Gaulish language, theta inspired the tau gallicum. The phonetic value of the tau gallicum is thought to have been. In addition, multiple modern languages written in Latin script use Latin theta.

Cyrillic

The early Cyrillic letter fita developed from θ. This letter existed in the Russian alphabet until the 1918 Russian orthography reform.

International Phonetic Alphabet

In the International Phonetic Alphabet, represents the voiceless dental fricative, as in thigh, thick, enthuse, thing, think, thin, thang, myth, smith, youth, bath, math, birth, fourth, death, or health. It does not represent the consonant in the, which is the voiced dental fricative counterpart, represented in the IPA by . A similar-looking lower-case symbol,, which is called barred o, is the IPA symbol for the close-mid central rounded vowel.

Mathematics and science

Lower case

The lowercase letter θ is used as a symbol for:

A plane angle in geometry
Angle and angular separation in physics
An unknown angular variable in trigonometry
A special function of several complex variables θ.
The first Chebyshev function in prime number theory
The potential temperature in meteorology
Dimensionless temperature in heat transport phenomena
The score of a test taker in item response theory
Theta Type Replication: a type of bacterial DNA replication specific to circular chromosomes
Threshold value of an artificial neuron
A Bayer designation letter applied to a star in a constellation; usually the eighth star so labelled but not necessarily the eighth-brightest as viewed from Earth
The statistical parameter frequently used in writing the likelihood function
The Watterson estimator for the population mutation rate in population genetics
Indicates a minimum optimum integration level determined by the intersection of GG and LL schedules
The reserve ratio of banks in economic models
The ordinal collapsing function developed by Solomon Feferman
The Heaviside step function
In pharmacology, the fraction of ligand bound to a macromolecule based on the Hill Equation
The angle of the incident beam during X-ray diffraction
Upper case

The uppercase letter Θ is used as a symbol for:

Theta functions
Dimension of temperature, by SI standard
An asymptotically tight bound in the analysis of algorithms
A certain ordinal number in set theory
Pentaquarks, exotic baryons in particle physics
A brain signal frequency ranging from 4–8 Hz
One of the variables known as "Greeks" in finance, representing time decay of options or the change in the intrinsic value of an option divided by the number of days until the option expires
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. A quotation attributed to the ancient Roman author Ennius said of it: "oh, theta, a letter much unluckier than the others".
According to Porphyry of Tyros, the Egyptians used an X within a circle as a symbol of the soul; having a value of nine, it was used as a symbol for Ennead. Johannes Lydus says that the Egyptians used a symbol for Kosmos in the form of theta, with a fiery circle representing the world, and a snake spanning the middle representing Agathos Daimon.

Abbreviation

In classical Athens, it was used as an abbreviation for the Greek θάνατος and as it vaguely resembles a human skull, theta was used as a warning symbol of death, in the same way that skull and crossbones are used in modern times. It survives on potsherds used by Athenians when voting for the death penalty. Petrus de Dacia in a document from 1291 relates the idea that theta was used to brand criminals as empty ciphers, and the branding rod was affixed to the crossbar spanning the circle. For this reason, the use of the number 9 was sometimes avoided where the connotation was felt to be unlucky—the mint marks of some Late Imperial Roman coins famously have the sum ΔΕ or ΕΔ substituted as a euphemism where a Θ would otherwise be expected.