View source for Theta

{{Short description|Eighth letter of the Greek alphabet}}
{{Hatnote group|
{{Redirect|θ|the letter used in Latin scripts|Latin theta|the letter used in IPA|Voiceless dental fricative||Theta (disambiguation)}}
{{Distinguish|text=the Cyrillic letters [[Fita]] or [[Oe (Cyrillic)|Oe]]}}
}}
{{See also-text|[[Ø (disambiguation)|Ø]] and similar symbols}}
{{More citations needed|date=April 2009}}

{{Greek Alphabet|letter=theta}}

'''Theta''' ({{IPAc-en|UK|ˈ|θ|iː|t|ə|audio=LL-Q1860 (eng)-Flame, not lame-Theta.wav}}, {{IPAc-en|US|ˈ|θ|eɪ|t|ə|audio=LL-Q1860 (eng)-Vivian S Zitek-theta.wav}}) uppercase '''Θ'''; lowercase '''θ'''; {{langx|grc|{{linktext|θῆτα}}}} ''thē̂ta'' {{IPA|el|tʰɛ̂ːta|}}; [[Modern Greek|Modern]]: {{lang|el|{{linktext|θήτα}}|}} ''thī́ta'' {{IPA|el|ˈθita|}}) is the eighth letter of the [[Greek alphabet]], derived from the [[Phoenician alphabet|Phoenician letter]] [[Teth]] 𐤈. In the system of [[Greek numerals]], it has a value of 9. 

==Greek==
In [[Ancient Greek]], θ represented the [[aspiration (phonetics)|aspirated]] [[voiceless dental plosive]] {{IPA|el|t̪ʰ|IPA}}, but in [[Modern Greek]] it represents the [[voiceless dental fricative]] {{IPA|el|θ|IPA}}.

{{anchor|cursive theta|Cursive theta|{{not a typo|ϑ}}}}<!--linked from [[Romanization of Greek]], [[Cursive theta]], {{not a typo|[[ϑ]]}}, etc.-->

==Forms==
[[File:Theta uc lc with variant.svg|thumb|right|Modern print theta, in a typical uppercase and two different lowercase glyph forms.]]
In its [[Archaic Greek alphabets|archaic]] form, θ was written as a cross within a circle (as in the Etruscan [[Image:Earth symbol.svg|16px|A symbol of a cross within a circle]] or [[Image:Cross within circle 2.svg|16px|Another symbol of a cross within a circle]]), and later, as a line or point in circle ([[File:Circled dash.svg|16px|The symbol of a line within a circle]] or [[Image:Sun symbol.svg|16px|The symbol of a point within a 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 {{unichar|03D1|GREEK THETA SYMBOL}} was introduced, which always displays the open-loop form. There is also {{unichar|03F4|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==
{{Details|Latin theta}}
[[File:Tau gallic CourDOr Metz 3147.jpg|thumb|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 {{IPA|[t͡s]}}.<ref>{{Cite web|url=https://www.unicode.org/L2/L2019/19179-n5044-tau-gallicum.pdf|title=L2/19-179: Proposal for the addition of four Latin characters for Gaulish|date=2019-05-26|first1=Michael|last1=Everson|first2=Chris|last2=Lilley}}</ref> 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 [[Reforms of Russian orthography#The post-revolution reform|1918 Russian orthography reform]].

==International Phonetic Alphabet==
In the [[International Phonetic Alphabet]] (IPA), {{IPA|[θ]}} represents the [[voiceless dental fricative]], as in '''''th'''ick'' or '''''th'''in''.<ref>{{Cite book |last=International Phonetic Association |title=Handbook of the International Phonetic Association: a guide to the use of the international phonetic alphabet |date=2021 |publisher=Cambridge University Press |isbn=978-0-521-63751-0 |edition=22. printing |location=Cambridge |pages=177 |quote=θ Theta voiceless dental fricative}}</ref> It does not represent the consonant in '''''th'''e'', which is the [[voiced dental fricative]] counterpart, represented in the IPA by {{IPA|[ð]}} (cf. [[Eth]]). A similar-looking lower-case symbol, {{IPA|[ɵ]}}, which is called ''barred o'', is the IPA symbol for the [[close-mid central rounded vowel]].

==Mathematics and science==
{{^|This is a disambiguation section, so references are in the linked target articles per [[MOS:DABNOLINK]].  Definitions WITHOUT a link, however, DO require a citation!}}

===Lower case===
The lowercase letter θ is used as a symbol for:
*A [[angle|plane angle]] in geometry<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Angle |url=https://mathworld.wolfram.com/Angle.html |access-date=2025-02-01 |website=mathworld.wolfram.com |language=en}}</ref>
*[[Angle]] (bold) and angular separation in physics<ref>{{Citation |last=Elert |first=Glenn |title=Special Symbols |date=2023 |work=The Physics Hypertextbook|quote='''ϴ''','''φ''',ϴ,φ angle, angular displacement, angular separation, rotation angle |url=https://physics.info/symbols/ |access-date=2025-02-01 |publisher=hypertextbook |language=en}}</ref>
*An unknown angular variable in [[trigonometry]]
*A [[theta function|special function]] {{math|''ϑ''(''z''; ''τ'')}} of several complex variables θ.
*The first [[Chebyshev function]] {{math|''θ''(''x'')}} in prime number theory<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Chebyshev Functions |url=https://mathworld.wolfram.com/ChebyshevFunctions.html |access-date=2025-01-24 |website=mathworld.wolfram.com |language=en}}</ref>
*The [[potential temperature]] in [[meteorology]]<ref>{{Cite book |last1=Holton |first1=James R. |title=An introduction to dynamic meteorology |last2=Hakim |first2=Gregory J. |date=2013 |publisher=Academic Press |isbn=978-0-12-384866-6 |edition=Fifth |location=Amsterdam |pages=87 |quote=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.}}</ref>
*Dimensionless temperature in heat [[transport phenomena]]<ref>{{Cite book |title=Convection and Conduction Heat Transfer |date=2011 |publisher=IntechOpen |isbn=978-953-307-582-2 |editor-last=Ahsan |editor-first=Amimul |location=Erscheinungsort nicht ermittelbar |pages=57 |quote=θ   dimensionless temperature,...}}</ref>
*The score of a test taker in [[item response theory]]<ref>{{Cite book |last1=van der Linden |first1=Wim J. |title=Handbook of Item Response Theory, Three Volume Set |last2=Hambleton |first2=Ronald K. |date=2018 |publisher=CRC Press |isbn=978-1-4665-1439-3 |edition=1st |series=Chapman & Hall / CRC Statistics in the Social and Behavioral Sciences |location=Boca Raton |pages=section 10.2 |quote=''θ''{{sub|p}} is test taker p's factor score or latent trait parameter.}}</ref>
*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<ref>{{Cite book |last=Bakich |first=Michael E. |title=The Cambridge guide to the constellations |date=1995 |publisher=Cambridge University Press |isbn=978-0-521-46520-5 |location=Cambridge |pages=109 |quote=To designate the 24 brightest stars in the constellation, Johannes Bayer used the lowercase letters of the Greek alphabet:}}</ref>
*The [[statistical parameter]] frequently used in writing the [[likelihood function]]<ref>{{Cite book |last1=Casella |first1=George |title=Statistical inference |last2=Berger |first2=Roger W. |date=2024 |publisher=CRC Press |isbn=978-1-032-59303-6 |edition=Second |series=CRC texts in statistical science series |location=Boca Raton |pages=8 |quote=Likelihood is the most directly accessible inferential element for estimation statements about θ conditional on f...}}</ref>
*The [[Watterson estimator]] {{mvar|θ&#770;<sub>w</sub>}} for the [[population mutation rate]] in [[population genetics]]<ref>{{Cite book |last=Hösel |first=Volker |title=Mathematical population genetics and evolution of bacterial cooperation |date=2020 |publisher=World Scientific |isbn=978-981-12-0549-1 |location=New Jersey |pages=75}}</ref>
*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.)<ref>{{Cite book |last=Reszat |first=Beate |title=European financial systems in the global economy |date=2005 |publisher=Wiley |isbn=978-0-470-87056-3 |location=Chichester, West Sussex, England ; Hoboken, NJ |pages=223–224 |quote=As a consequence, the critical level for joining shifts from θ to θ' -...}}</ref>
*The reserve ratio of banks in economic models<ref>{{Cite book |title=Macroeconomics and development: Roberto Frenkel and the economics of Latin America |date=2016 |publisher=Columbia University Press |isbn=978-0-231-17508-1 |editor-last=Damill |editor-first=Mario |series=Initiative for policy dialogue |location=New York |quote=...θ and θ{{sup|*}} are reserve ratios in DC and FC... |editor-last2=Rapetti |editor-first2=Martín |editor-last3=Rozenwurcel |editor-first3=Guillermo}}</ref>
*The [[ordinal collapsing function]] developed by [[Solomon Feferman]]<ref>{{Cite journal |last=Feferman |first=Solomon |author-link=Solomon Feferman |date=2010 |editor-last=Schindler |editor-first=Ralf |title=The proof theory of classical and constructive inductive definitions. A 40 year saga, 1968–2008. |url=https://math.stanford.edu/~feferman/papers/id-saga.pdf |journal=Ways of Proof Theory |pages=7–30|doi=10.1515/9783110324907.7 }}</ref>
*The [[Heaviside step function]]<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Heaviside Step Function |url=https://mathworld.wolfram.com/HeavisideStepFunction.html |access-date=2025-01-24 |website=mathworld.wolfram.com |language=en}}</ref>
*In [[pharmacology]], the fraction of ligand bound to a macromolecule based on the [[Hill equation (biochemistry)|Hill Equation]]<ref>{{Cite book |title=Applications of Calorimetry in a Wide Context: Differential Scanning Calorimetry, Isothermal Titration Calorimetry and Microcalorimetry |date=2013 |publisher=IntechOpen |isbn=978-953-51-0947-1 |editor-last=Ali Elkordy |editor-first=Amal |location=Erscheinungsort nicht ermittelbar |pages=75 |quote='''Saturation fraction, θ.''' The fraction of the total number of sites of the macromolecule occupied by ligand molecules.}}</ref>
*The angle of the incident beam during [[X-ray diffraction]]<ref>{{Cite book |title=Biochar: a guide to analytical methods |date=2017 |publisher=CSIRO Publishing |isbn=978-1-4987-6553-4 |editor-last=Singh |editor-first=Balwant |location=Clayton South |pages=246 |quote=θ is the angle between the incident x-ray beam and the diffracting plane |editor-last2=Arbestain |editor-first2=Marta Camps |editor-last3=Lehmann |editor-first3=Johannes}}</ref>

===Upper case===
The uppercase letter Θ is used as a symbol for:
*[[Theta function|Theta functions]]<ref>{{Cite web |title=DLMF: Chapter 20 Theta Functions |url=https://dlmf.nist.gov/20 |access-date=2025-01-31 |website=dlmf.nist.gov}}</ref>
*[[Dimensional analysis|Dimension]] of [[temperature]], by [[SI unit|SI standard]] (in italics)<ref>{{Cite book |last=Lemons |first=Don S. |title=A student's guide to dimensional analysis |date=2017 |publisher=Cambridge University Press |isbn=978-1-107-16115-3 |location=New York |pages=50 |quote=The physics of heat transfer requires two dimensions, temperature Θ and heat H, beyond those that characterize non-equilibrium dynamics.}}</ref>
*An asymptotically tight bound in the [[analysis of algorithms]] ([[big O notation]])
*A certain [[ordinal number]] in [[Θ (set theory)|set theory]]
*[[Pentaquark]]s, exotic [[baryon]]s in [[particle physics]]
*A [[Electroencephalography|brain signal frequency]] (beta, alpha, theta, delta) ranging from 4–8&nbsp;Hz
*One of the variables known as [[Greeks (finance)|"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<ref>{{Cite book |last=Ursone |first=Pierino |title=How to Calculate Options Prices and Their Greeks: Exploring the Black Scholes Model from Delta to Vega |date=2015 |publisher=John Wiley & Sons, Incorporated |isbn=978-1-119-01162-0 |edition=1st |series=The Wiley Finance Series |location=Newark |pages=2 |quote=Θ (time decay): the price change of an option in relation to time;}}</ref>

==Symbolism==
[[File:Retiarius vs secutor from Borghese mosaic.jpg|thumb|300px|upright|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]].<ref name="Mednik">{{cite journal |last=Mednikarova  |first=Iveta |title=The Use of Θ in Latin Funerary Inscriptions |journal=[[Zeitschrift für Papyrologie und Epigraphik]] |volume=136 |date=2001 |pages=267–276 |jstor=20190914 }}</ref> A quotation attributed to the ancient Roman author [[Ennius]] (though possibly spuriously) said of it: "oh, theta, a letter much unluckier than the others".<ref>{{cite book |last1=Ennius |first1=Quintus |author1-link=Ennius |editor1-last=Goldberg |editor1-first=Sander M. |editor2-last=Manuwald |editor2-first=Gesine |editor2-link=Gesine Manuwald |title=Fragmentary Republican Latin, Volume I: Ennius, Testimonia. Epic Fragments. |date=2018 |publisher=Harvard University Press |location=Cambridge, MA |pages=456–457}}</ref>

According to [[Porphyry of Tyros]], the [[Ancient Egypt|Egyptian]]s 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 [[cosmos|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'').<ref name="Barry_1">{{Citation| last = Barry
| first = Kieren
| title = The Greek Qabalah: Alphabetic Mysticism and Numerology in the Ancient World
| publisher = [[Samuel Weiser]]
| page = 73
| year = 1998
| isbn=1-57863-110-6
}}.</ref>

===Abbreviation===
{{main|Theta nigrum}}
In classical Athens, it was used as an abbreviation for the [[Greek language|Greek]] θάνατος (''Thanatos'', "death") and as it vaguely resembles a human skull,<ref name="Barry_1"/> theta was used as a warning symbol of [[Thanatos|death]], in the same way that [[skull and crossbones (poison)|skull and crossbones]] are used in modern times.<ref name="Barry_1"/> It survives on [[potsherd]]s used by Athenians when voting for the [[death penalty]].<ref name="Barry_1"/> [[Petrus de Dacia (mathematician)|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.<ref>{{Citation
| last = Kaplan
| first = Robert
| title = The Nothing that Is: A Natural History of Zero
| publisher = [[Oxford University Press]]
| year = 2000
| pages = 66
| isbn = 0-19-514237-3
| bibcode = 2000tnti.book.....K
| location = Oxford; New York
}}.</ref> 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==

* {{unichar|0398|html=}}
* {{unichar|03B8|html=}}
* {{unichar|03D1|html=}}
* {{unichar|03F4|html=}}
* {{unichar|0472|html=}}
* {{unichar|0473|html=}}
* {{unichar|1DBF|html=}}
* {{unichar|2C2A|html=}}
* {{unichar|2C5A|html=}}
* {{unichar|2C90|html=}}
* {{unichar|2C91|html=}}
* {{unichar|2DF4|cwith=◌}}
* {{unichar|1CF83|html=}}
* {{unichar|1D6AF|html=}}{{efn|The {{sc|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.}}
* {{unichar|1D6B9|html=}}
* {{unichar|1D6C9|html=}}
* {{unichar|1D6DD|html=}}
* {{unichar|1D6E9|html=}}
* {{unichar|1D6F3|html=}}
* {{unichar|1D703|html=}}
* {{unichar|1D717|html=}}
* {{unichar|1D723|html=}}
* {{unichar|1D72D|html=}}
* {{unichar|1D73D|html=}}
* {{unichar|1D751|html=}}
* {{unichar|1D75D|html=}}
* {{unichar|1D767|html=}}
* {{unichar|1D777|html=}}
* {{unichar|1D78B|html=}}
* {{unichar|1D797|html=}}
* {{unichar|1D7A1|html=}}
* {{unichar|1D7B1|html=}}
* {{unichar|1D7C5|html=}}
* {{unichar|1E02A|cwith=◌}}

==See also==
{{Wiktionary|Θ|θ}}
* {{annotated link|ʘ}}

==Notes and references==

=== Notes ===
{{notelist}}
{{NoteFoot}}

=== References ===
{{Reflist}}

[[Category:Greek letters]]
[[Category:Phonetic transcription symbols]]