Theta

Theta (UK: /ˈθiːtə/, US: /ˈθeɪtə/) uppercase Θ or ϴ; lowercase θ^{[note 1]} or ϑ; Ancient Greek: θῆτα thē̂ta [tʰɛ̂ːta]; Modern: θήτα thī́ta [ˈθita]) 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

The alphabet on a black figure vessel, with a point-and-circle theta

In Ancient Greek, θ represented the aspirated voiceless dental plosive IPA: [t̪ʰ], but in Modern Greek it represents the voiceless dental fricative IPA: [θ].

Forms

In its archaic form, θ was written as a cross within a circle (as in the Etruscan or ), and later, as a line or point in circle ( or ).

Greek theta variant in cursive form.

The cursive form ϑ was retained by Unicode as U+03D1 ϑ GREEK THETA SYMBOL, separate from U+03B8 θ GREEK SMALL LETTER THETA. (There is also U+03F4 ϴ GREEK CAPITAL THETA SYMBOL.) For the purpose of writing Greek text, the two can be font variants of a single character, but θ and ϑ are also used as distinct symbols in technical and mathematical contexts. Extensive lists of examples follow below at Mathematics and Science. U+03D1 ϑ GREEK THETA SYMBOL (script theta) is also common in biblical and theological usage e.g. πρόϑεσις (prothesis) instead of πρόθεσις (means placing in public or laying out a corpse).

Latin

The name ARAÐÐOVNA on a Gaulish tombstone

In Latin script used for the Gaulish language, theta inspired the tau gallicum (Ꟈ ꟈ). The phonetic value of the tau gallicum is thought to have been [t͡s].^[1] In addition, multiple modern languages written in Latin script use Latin theta.

Cyrillic

The early Cyrillic letter fita (Ѳ, ѳ) developed from θ. This letter existed in the Russian alphabet until the 1918 Russian orthography reform.

International Phonetic Alphabet

In the International Phonetic Alphabet (IPA), [θ] represents the voiceless dental fricative, as in thick or thin. It does not represent the consonant in the, which is the voiced dental fricative counterpart, represented in the IPA by [ð] (cf. Eth). A similar-looking lower-case symbol, [ɵ], 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
An unknown variable in trigonometry
The voiceless dental fricative, spelled θ
A special function of several complex variables
One of the Chebyshev functions in prime number theory
The potential temperature in meteorology
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 GG-LL schedules are tools used in analyzing the potential benefits of a country pegging their domestic currency to a foreign currency.)
The reserve ratio of banks in economic models
The ordinal collapsing function developed by Solomon Feferman^[2]
Heaviside step function
In pharmacology, the fraction of ligand bound to a macromolecule based on the Hill Equation

Upper case

The uppercase letter Θ is used as a symbol for:

Dimension of temperature, by SI standard (in italics)
Dimensionless temperature in transport phenomena
An asymptotically tight bound in the analysis of algorithms (big O notation)
A certain ordinal number in set theory
Pentaquarks, exotic baryons in particle physics
A brain signal frequency (beta, alpha, theta, delta) 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
A variable indicating temperature difference in heat transfer
Measuring the angle of incident X-ray beam during XRD

Symbolism

θ (θάνατος, death) in a mosaic

In ancient times, Tau was used as a symbol for life or resurrection, whereas the eighth letter of the Greek alphabet (ninth in the archaic form anciently used for numeration), theta, was considered the symbol of death.^[3] A quotation attributed to the ancient Roman author Ennius (though possibly spuriously) said of it: "oh, theta, a letter much unluckier than the others".^[4]

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 (literally: good spirit).^[5]

Abbreviation

In classical Athens, it was used as an abbreviation for the Greek θάνατος (Thanatos, "death") and as it vaguely resembles a human skull,^[5] theta was used as a warning symbol of death, in the same way that skull and crossbones are used in modern times.^[5] It survives on potsherds used by Athenians when voting for the death penalty.^[5] 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.^[6] 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 (delta and epsilon, that is 4 and 5) substituted as a euphemism where a Θ (9) would otherwise be expected.

Unicode

U+0398 Θ GREEK CAPITAL LETTER THETA (Θ)
U+03B8 θ GREEK SMALL LETTER THETA (θ)
U+03D1 ϑ GREEK THETA SYMBOL (&thetasym;, ϑ, ϑ)
U+03F4 ϴ GREEK CAPITAL THETA SYMBOL
U+0472 Ѳ CYRILLIC CAPITAL LETTER FITA
U+0473 ѳ CYRILLIC SMALL LETTER FITA
U+1DBF ᶿ MODIFIER LETTER SMALL THETA
U+2C2A Ⱚ GLAGOLITIC CAPITAL LETTER FITA
U+2C5A ⱚ GLAGOLITIC SMALL LETTER FITA
U+2C90 Ⲑ COPTIC CAPITAL LETTER THETHE
U+2C91 ⲑ COPTIC SMALL LETTER THETHE
U+2DF4 ◌ⷴ COMBINING CYRILLIC LETTER FITA
U+1CF83 𜾃 ZNAMENNY NEUME FITA
U+1E02A ◌𞀪 COMBINING GLAGOLITIC LETTER FITA

These characters are used only as mathematical symbols. Stylized Greek text should be encoded using normal Greek letters, with markup and formatting to indicate text style:

U+1D6AF 𝚯 MATHEMATICAL BOLD CAPITAL THETA
U+1D6B9 𝚹 MATHEMATICAL BOLD CAPITAL THETA SYMBOL
U+1D6C9 𝛉 MATHEMATICAL BOLD SMALL THETA
U+1D6DD 𝛝 MATHEMATICAL BOLD THETA SYMBOL
U+1D6E9 𝛩 MATHEMATICAL ITALIC CAPITAL THETA
U+1D6F3 𝛳 MATHEMATICAL ITALIC CAPITAL THETA SYMBOL
U+1D703 𝜃 MATHEMATICAL ITALIC SMALL THETA
U+1D717 𝜗 MATHEMATICAL ITALIC THETA SYMBOL
U+1D723 𝜣 MATHEMATICAL BOLD ITALIC CAPITAL THETA
U+1D72D 𝜭 MATHEMATICAL BOLD ITALIC CAPITAL THETA SYMBOL
U+1D73D 𝜽 MATHEMATICAL BOLD ITALIC SMALL THETA
U+1D751 𝝑 MATHEMATICAL BOLD ITALIC THETA SYMBOL
U+1D75D 𝝝 MATHEMATICAL SANS-SERIF BOLD CAPITAL THETA
U+1D767 𝝧 MATHEMATICAL SANS-SERIF BOLD CAPITAL THETA SYMBOL
U+1D777 𝝷 MATHEMATICAL SANS-SERIF BOLD SMALL THETA
U+1D78B 𝞋 MATHEMATICAL SANS-SERIF BOLD THETA SYMBOL
U+1D797 𝞗 MATHEMATICAL SANS-SERIF BOLD ITALIC CAPITAL THETA
U+1D7A1 𝞡 MATHEMATICAL SANS-SERIF BOLD ITALIC CAPITAL THETA SYMBOL
U+1D7B1 𝞱 MATHEMATICAL SANS-SERIF BOLD ITALIC SMALL THETA
U+1D7C5 𝟅 MATHEMATICAL SANS-SERIF BOLD ITALIC THETA SYMBOL

Notes and references

Notes

^ which resembles digit 0 with a horizontal line.

References

^ Everson, Michael; Lilley, Chris (2019-05-26). "L2/19-179: Proposal for the addition of four Latin characters for Gaulish" (PDF).
^ Feferman, Solomon (2010). Schindler, Ralf (ed.). "The Proof Theory of Classical and Constructive Inductive Definitions. A Forty Year Saga, 1968 – 2008" (PDF). Ways of Proof Theory: 7–30.
^ Mednikarova, Iveta (2001). "The Use of Θ in Latin Funerary Inscriptions". Zeitschrift für Papyrologie und Epigraphik. 136: 267–276. JSTOR 20190914.
^ Ennius, Quintus (2018). Goldberg, Sander M.; Manuwald, Gesine (eds.). Fragmentary Republican Latin, Volume I: Ennius, Testimonia. Epic Fragments. Cambridge, MA: Harvard University Press. pp. 456–457.
^ ^a ^b ^c ^d Barry, Kieren (1998), The Greek Qabalah: Alphabetic Mysticism and Numerology in the Ancient World, Samuel Weiser, p. 73, ISBN 1-57863-110-6.
^ Kaplan, Robert (2000), The Nothing that Is: A Natural History of Zero, Oxford; New York: Oxford University Press, p. 66, Bibcode:2000tnti.book.....K, ISBN 0-19-514237-3.

[1] which resembles digit 0 with a horizontal line.

[2] Everson, Michael; Lilley, Chris (2019-05-26). "L2/19-179: Proposal for the addition of four Latin characters for Gaulish" (PDF).

[3] Feferman, Solomon (2010). Schindler, Ralf (ed.). "The Proof Theory of Classical and Constructive Inductive Definitions. A Forty Year Saga, 1968 – 2008" (PDF). Ways of Proof Theory: 7–30.

[Mednik-4] Mednikarova, Iveta (2001). "The Use of Θ in Latin Funerary Inscriptions". Zeitschrift für Papyrologie und Epigraphik. 136: 267–276. JSTOR 20190914.

[5] Ennius, Quintus (2018). Goldberg, Sander M.; Manuwald, Gesine (eds.). Fragmentary Republican Latin, Volume I: Ennius, Testimonia. Epic Fragments. Cambridge, MA: Harvard University Press. pp. 456–457.

[Barry_1-6] Barry, Kieren (1998), The Greek Qabalah: Alphabetic Mysticism and Numerology in the Ancient World, Samuel Weiser, p. 73, ISBN 1-57863-110-6.

[7] Kaplan, Robert (2000), The Nothing that Is: A Natural History of Zero, Oxford; New York: Oxford University Press, p. 66, Bibcode:2000tnti.book.....K, ISBN 0-19-514237-3.

[note 1]

[1]

[2]

[3]

[4]

[5]

[6]