Theta

Eighth letter of the Greek alphabet From Wikipedia, the free encyclopedia

Theta (UK: /θiːtə/ ^ⓘ, US: /θeɪtə/ ^ⓘTHAY-tə, uppercase Θ or ϴ; lowercase θ; cursive ϑ; Ancient Greek: θῆτα thē̂ta Greek pronunciation: [tʰɛ̂ːta]; Modern: θήτα thī́ta Greek pronunciation: [θita]) 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.

Greek

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 ).

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 U+03D1 ϑ GREEK THETA SYMBOL was introduced, which always displays the open-loop form. There is also U+03F4 ϴ GREEK CAPITAL THETA SYMBOL, 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

The name ARAÐÐOVNA on a Gaulish tombstone

In the 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 thigh (/θaɪ/), thick (/θɪk/), enthuse (/ˈɛnθjuːz/), thing (/θɪŋ/), think (/θɪŋk/), thin (/θɪn/), thang (/θæŋ/), myth (/mɪθ/), smith (/smɪθ/), youth (/juːθ/), bath (/bæθ/), math (/mæθ/), birth (/bɜːrθ/), fourth (/fɔːrθ/), death (/dɛθ/), or health (/hɛlθ/).^[2] 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^[3]
Angle (bold) and angular separation in physics^[4]
An unknown angular variable in trigonometry
A special function $ϑ (z; τ)$ of several complex variables θ.
The first Chebyshev function $θ (x)$ in prime number theory^[5]
The potential temperature in meteorology^[6]
Dimensionless temperature in heat transport phenomena^[7]
The score of a test taker in item response theory^[8]
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^[9]
The statistical parameter frequently used in writing the likelihood function^[10]
The Watterson estimator $θ ̂ w$ for the population mutation rate in population genetics^[11]
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.)^[12]
The reserve ratio of banks in economic models^[13]
The ordinal collapsing function developed by Solomon Feferman^[14]
The Heaviside step function^[15]
In pharmacology, the fraction of ligand bound to a macromolecule based on the Hill Equation^[16]
The angle of the incident beam during X-ray diffraction^[17]

Upper case

The uppercase letter Θ is used as a symbol for:

Theta functions^[18]
Dimension of temperature, by SI standard (in italics)^[19]
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^[20]

Symbolism

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.^[21] A quotation attributed to the ancient Roman author Ennius (though possibly spuriously) said of it: "oh, theta, a letter much unluckier than the others".^[22]

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).^[23]

Abbreviation

In classical Athens, it was used as an abbreviation for the Greek θάνατος (Thanatos, "death") and as it vaguely resembles a human skull,^[23] theta was used as a warning symbol of death, in the same way that skull and crossbones are used in modern times.^[23] It survives on potsherds used by Athenians when voting for the death penalty.^[23] 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.^[24] 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+1D6AF 𝚯 MATHEMATICAL BOLD CAPITAL THETA^[a]
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
U+1E02A ◌𞀪 COMBINING GLAGOLITIC LETTER FITA

Notes and references

Notes

[a]
The MATHEMATICAL characters are used only in math. Stylized Greek text should be encoded using normal Greek letters, with markup and formatting to indicate text style.

References

[1]
Everson, Michael; Lilley, Chris (2019-05-26). "L2/19-179: Proposal for the addition of four Latin characters for Gaulish" (PDF).
[2]
International Phonetic Association (2021). Handbook of the International Phonetic Association: a guide to the use of the international phonetic alphabet (22. printing ed.). Cambridge: Cambridge University Press. p. 177. ISBN 978-0-521-63751-0. θ Theta voiceless dental fricative
[3]
Weisstein, Eric W. "Angle". mathworld.wolfram.com. Retrieved 2025-02-01.
[4]
Elert, Glenn (2023), "Special Symbols", The Physics Hypertextbook, hypertextbook, retrieved 2025-02-01, ϴ,φ,ϴ,φ angle, angular displacement, angular separation, rotation angle
[5]
Weisstein, Eric W. "Chebyshev Functions". mathworld.wolfram.com. Retrieved 2025-01-24.
[6]
Holton, James R.; Hakim, Gregory J. (2013). An introduction to dynamic meteorology (Fifth ed.). Amsterdam: Academic Press. p. 87. ISBN 978-0-12-384866-6. Since potential temperature θ is conserved following the motion in adiabatic flow, a parcel of air which movees adiabatically will remain on the same potential temperature surface.
[7]
Ahsan, Amimul, ed. (2011). Convection and Conduction Heat Transfer. Erscheinungsort nicht ermittelbar: IntechOpen. p. 57. ISBN 978-953-307-582-2. θ dimensionless temperature,...
[8]
van der Linden, Wim J.; Hambleton, Ronald K. (2018). Handbook of Item Response Theory, Three Volume Set. Chapman & Hall / CRC Statistics in the Social and Behavioral Sciences (1st ed.). Boca Raton: CRC Press. pp. section 10.2. ISBN 978-1-4665-1439-3. θ_p is test taker p's factor score or latent trait parameter.
[9]
Bakich, Michael E. (1995). The Cambridge guide to the constellations. Cambridge: Cambridge University Press. p. 109. ISBN 978-0-521-46520-5. To designate the 24 brightest stars in the constellation, Johannes Bayer used the lowercase letters of the Greek alphabet:
[10]
Casella, George; Berger, Roger W. (2024). Statistical inference. CRC texts in statistical science series (Second ed.). Boca Raton: CRC Press. p. 8. ISBN 978-1-032-59303-6. Likelihood is the most directly accessible inferential element for estimation statements about θ conditional on f...
[11]
Hösel, Volker (2020). Mathematical population genetics and evolution of bacterial cooperation. New Jersey: World Scientific. p. 75. ISBN 978-981-12-0549-1.
[12]
Reszat, Beate (2005). European financial systems in the global economy. Chichester, West Sussex, England ; Hoboken, NJ: Wiley. pp. 223–224. ISBN 978-0-470-87056-3. As a consequence, the critical level for joining shifts from θ to θ' -...
[13]
Damill, Mario; Rapetti, Martín; Rozenwurcel, Guillermo, eds. (2016). Macroeconomics and development: Roberto Frenkel and the economics of Latin America. Initiative for policy dialogue. New York: Columbia University Press. ISBN 978-0-231-17508-1. ...θ and θ^* are reserve ratios in DC and FC...
[14]
Feferman, Solomon (2010). Schindler, Ralf (ed.). "The proof theory of classical and constructive inductive definitions. A 40 year saga, 1968–2008" (PDF). Ways of Proof Theory: 7–30. doi:10.1515/9783110324907.7.
[15]
Weisstein, Eric W. "Heaviside Step Function". mathworld.wolfram.com. Retrieved 2025-01-24.
[16]
Ali Elkordy, Amal, ed. (2013). Applications of Calorimetry in a Wide Context: Differential Scanning Calorimetry, Isothermal Titration Calorimetry and Microcalorimetry. Erscheinungsort nicht ermittelbar: IntechOpen. p. 75. ISBN 978-953-51-0947-1. Saturation fraction, θ. The fraction of the total number of sites of the macromolecule occupied by ligand molecules.
[17]
Singh, Balwant; Arbestain, Marta Camps; Lehmann, Johannes, eds. (2017). Biochar: a guide to analytical methods. Clayton South: CSIRO Publishing. p. 246. ISBN 978-1-4987-6553-4. θ is the angle between the incident x-ray beam and the diffracting plane
[18]
"DLMF: Chapter 20 Theta Functions". dlmf.nist.gov. Retrieved 2025-01-31.
[19]
Lemons, Don S. (2017). A student's guide to dimensional analysis. New York: Cambridge University Press. p. 50. ISBN 978-1-107-16115-3. The physics of heat transfer requires two dimensions, temperature Θ and heat H, beyond those that characterize non-equilibrium dynamics.
[20]
Ursone, Pierino (2015). How to Calculate Options Prices and Their Greeks: Exploring the Black Scholes Model from Delta to Vega. The Wiley Finance Series (1st ed.). Newark: John Wiley & Sons, Incorporated. p. 2. ISBN 978-1-119-01162-0. Θ (time decay): the price change of an option in relation to time;
[21]
Mednikarova, Iveta (2001). "The Use of Θ in Latin Funerary Inscriptions". Zeitschrift für Papyrologie und Epigraphik. 136: 267–276. JSTOR 20190914.
[22]
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.
[23]
Barry, Kieren (1998), The Greek Qabalah: Alphabetic Mysticism and Numerology in the Ancient World, Samuel Weiser, p. 73, ISBN 1-57863-110-6.
[24]
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.