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{{Short description|Eighteenth letter of the Greek alphabet}}
{{hatnote group|{{Not to be confused with|Stigma (disambiguation){{!}}Stigma}}{{Other uses|Sigma (disambiguation)}}}}
{{pp-vandalism|small=yes}}
{{Greek alphabet|letter=sigma}}

'''Sigma ''' ({{IPAc-en|'|s|ɪ|g|m|ə}} {{respell|SIG|mə}};<ref>{{OED|sigma}}</ref> [[uppercase]] '''Σ''', [[lowercase]] '''σ''', lowercase in word-final position '''ς'''; {{lang-grc-gre|σίγμα}}) is the eighteenth letter of the [[Greek alphabet]]. In the system of [[Greek numerals]], it has a value of 200. In general mathematics, uppercase '''Σ''' is used as an [[operator (mathematics)|operator]] for [[summation]]. When used at the end of a [[Letter case|letter-case]] word (one that does not use [[all caps]]), the [[final form]] (ς) is used. In ''{{Langx|grc|Ὀδυσσεύς|label=none}}'' (Odysseus), for example, the two lowercase sigmas (σ) in the center of the name are distinct from the word-final sigma (ς) at the end. The [[Latin alphabet|Latin]] letter [[S]] derives from sigma while the [[Cyrillic script|Cyrillic]] letter [[Es (Cyrillic)|Es]] derives from a [[#Lunate sigma|lunate form]] of this letter.

==History==
The shape (Σς) and alphabetic position of sigma is derived from the [[Phoenician alphabet|Phoenician]] letter <span style="background-color: white;">{{Phoenician letter|sin}}</span> ([[Shin (letter)|''shin'']]).

Sigma's original name may have been ''san'', but due to the complicated early history of the Greek [[Archaic Greek alphabets|epichoric alphabets]], ''san'' came to be identified as a separate letter in the Greek alphabet, represented as [[San (letter)|'''Ϻ''']].<ref name="woodard">{{cite encyclopedia|encyclopedia=Encyclopedia of Ancient Greece|editor-first=Nigel Guy|editor-last=Wilson|title=Alphabet|first=Roger D.|last=Woodard|publisher=Routledge|place=London|page=38|date=2006}}<br /></ref>
[[Herodotus]] reports that "san" was the name given by the [[Doric Greek|Dorians]] to the same letter called "sigma" by the [[Ionic Greek|Ionians]].<ref group="lower-roman">"the same letter, which the Dorians call "san", but the Ionians 'sigma'..." [translated from [[Ancient Greek]]: "{{Langx|grc|τὠυτὸ γράμμα, τὸ Δωριέες μὲν σὰν καλέουσι ,Ἴωνες δὲ σίγμα|italic=yes|label=none}}"] ([[Herodotus]] 1.139)</ref><ref>[[Herodotus]], ''[[Histories (Herodotus)|Histories]]'' 1.139 — [[Michael Everson|Everson, Michael]] and [[Nicholas Sims-Williams]]. 2002. "[https://archive.today/20120628161421/http://www.tlg.uci.edu/~opoudjis/unicode/nonattic.html Non-Attic letters]," transcribed by N. Nicholas. Archived from the [http://www.tlg.uci.edu/~opoudjis/unicode/nonattic.html original] 2020-06-28.</ref>

According to one hypothesis,<ref name="jeffery25">{{cite book|last=Jeffery|first=Lilian H.|title=The Local Scripts of Archaic Greece|publisher=Clarendon|year=1961|place=Oxford|pages=25–7}}</ref> the name "sigma" may continue that of Phoenician ''[[samekh]]'' (<span style="background-color: white;">{{Phoenician letter|samekh}}</span>), the letter continued through Greek ''xi'', represented as [[Xi (letter)|'''Ξ''']]. Alternatively, the name may have been a Greek innovation that simply meant 'hissing', from the root of {{lang|grc|σίζω}} ({{Langx|grc|sízō|label=none|italic=yes}}, from [[Proto-Greek language|Proto-Greek]] ''*sig-jō'' 'I hiss').<ref name="woodard" />

=== Lunate sigma ===
[[File:Madaba Jerusalem Mosaic.jpg|thumb|left|The [[Madaba Map]], a sixth-century mosaic of [[Jerusalem]] ({{lang|grc|Η [[Agia (disambiguation)|ΑΓΙΑ]] [[polis|ΠΟΛΙ{{strong|Ϲ}}]]}}) uses the lunate sigma]]
[[File:Metochion Gethsimanis, Jerusalem.jpg|thumb|right|upright|A plaque reading "[[Metochion]] of [[Gethsemane]]" ({{lang|grc|Μετόχιον Γεθσημανῆς}}) in Jerusalem, with a lunate sigma both at the end and in the middle of the word]]
In handwritten Greek during the [[Hellenistic period]] (4th–3rd century BC), the [[epigraphic]] form of Σ was simplified into a C-like shape,<ref>Thompson, Edward M. (1912). ''Introduction to Greek and Latin Paleography.'' Oxford: Clarendon. p.&nbsp;108, 144.</ref> which has also been found on coins from the 4th century BC onward.<ref>Hopkins, Edward C. D. (2004). "[http://www.parthia.com/fonts/letterforms_usage.htm Letterform Usage | Numismatica Font Projects]" ''Parthia''.</ref> This became the universal standard form of sigma during late antiquity and the Middle Ages.

Today, it is known as ''lunate sigma'' (uppercase '''Ϲ''', lowercase '''ϲ'''), because of its [[crescent]]-like shape, and is still widely used in decorative typefaces in Greece, especially in religious and church contexts, as well as in some modern print editions of classical Greek texts.

A dotted lunate sigma (''sigma periestigmenon'', '''Ͼ''') was used by [[Aristarchus of Samothrace]] (220–143 BC) as an editorial sign indicating that the line marked as such is at an incorrect position. Similarly, a reversed sigma (''[[antisigma]]'', '''Ͻ'''), may mark a line that is out of place. A dotted antisigma (''antisigma periestigmenon'', '''Ͽ''') may indicate a line after which rearrangements should be made, or to variant readings of uncertain priority.

In Greek inscriptions from the late first century BC onwards, '''Ͻ''' was an abbreviation indicating that a man's father's name is the same as his own name, thus Dionysodoros son of Dionysodoros would be written Διονυσόδωρος Ͻ (''Dionysodoros Dionysodorou'').<ref>{{cite journal |last1=de Lisle |first1=Christopher |title=Attic Inscriptions in UK Collections: Ashmolean Museum, Oxford |journal=AIUK |date=2020 |volume=11 |page=11 |url=https://www.atticinscriptions.com/papers/aiuk-11/ |access-date=2 June 2022 |issn=2054-6769}}</ref><ref>{{cite journal |last1=Follet |first1=Simone |title=Les deux archontes Pamménès du Ier siècle a.c. à Athènes |journal=Revue des Études Grecques |date=2000 |volume=113 |pages=188–192|doi=10.3406/reg.2000.4402 }}</ref>

In [[Unicode]], the above variations of lunate sigma are encoded as {{Unichar|03F9|html=}};
{{Unichar|03FD|size=100%|html=}},
{{Unichar|03FE|size=100%|html=}}, and
{{Unichar|03FF|size=100%|html=}}.

===Derived alphabets===
Sigma was adopted in the [[Old Italic alphabets]] beginning in the 8th century BC.
At that time a simplified three-stroke version, omitting the lowermost stroke, was already found in Western Greek alphabets,
and was incorporated into classical [[Etruscan alphabet|Etruscan]] and [[Old Italic scripts|Oscan]], as well as in the earliest [[History of the Latin script|Latin]] [[epigraphy]] (early Latin ''[[S]]''), such as the [[Duenos inscription]].
The alternation between three and four (and occasionally more than four) strokes was also adopted into the [[Elder Futhark|early runic alphabet]] (early form of the [[Sowilō|''s''-rune]]).
Both the [[Anglo-Saxon runes]] and the [[Younger Futhark]] consistently use the simplified three-stroke version.

The letter [[Es (Cyrillic)|С]] of [[Cyrillic script]] originates in the [[#Lunate sigma|lunate form]] of Sigma.

==Uses==

===Language and linguistics===
* In both [[Ancient Greek|Ancient]] and [[Modern Greek]], the sigma represents the [[voiceless alveolar fricative]] {{IPA-el|s|}}. In Modern Greek, this sound is [[voice (phonetics)|voiced]] to the [[voiced alveolar fricative]] {{IPA-el|z|}} when occurring before {{IPA-el|m|}}, {{IPA-el|n|}}, {{IPA-el|v|}}, {{IPA-el|ð|}} or {{IPA-el|ɣ|}}.
* The uppercase form of sigma ('''Σ''') was re-borrowed into the [[Latin alphabet]]—more precisely, the [[Africa Alphabet|International African Alphabet]]—to serve as the uppercase of modern [[esh (letter)|''esh'']] (lowercase: '''ʃ''').
* In [[phonology]], '''σ''' is used to represent [[syllable]]s.
* In [[linguistics]], '''Σ''' represents the set of symbols that form an alphabet (see also [[Computer Science|computer science]]).
* In [[historical linguistics]], '''Σ''' is used to represent a [[Common Brittonic]] consonant with a sound between {{IPAblink|s}} and {{IPAblink|h}}; perhaps an [[Aspirated consonant|aspirated]] [[Voiceless postalveolar fricative|[ʃʰ<nowiki>]</nowiki>]].<ref>{{Cite web|url=http://dlib.bc.edu/islandora/object/bc-ir:102094|title=Celtic initial consonant mutations - nghath and bhfuil?|first=Kevin M.|last=Conroy|date=21 February 2008|via=dlib.bc.edu}}</ref>

=== Science and mathematics ===

==== Mathematics ====
* In general mathematics, lowercase '''σ''' is commonly used to represent unknown angles, additionally serving as a shorthand for "[[countable set|countably]]", whereas '''Σ''' is regularly used as the [[operator (mathematics)|operator]] for [[summation]], e.g.:
<math display="block">\sum_{k=0}^5k= 0 + 1 + 2 + 3 + 4 + 5 = 15</math>
* In [[mathematical logic]], <math>\Sigma^0_n</math> is used to denote the set of formulae with bounded quantifiers beginning with existential quantifiers, alternating <math>n-1</math> times between existential and universal quantifiers. This notation reflects an indirect analogy between the relationship of summation and products on one hand, and existential and universal quantifiers on the other. See the article on the [[arithmetic hierarchy]].
* In [[statistics]], '''σ''' represents the [[standard deviation]] of population or [[probability distribution]] (where [[Mu (letter)|mu]] or '''μ''' is used for the mean).
* In [[topology]], '''σ-compact''' [[topological space]] is one that can be written as a countable [[union (set theory)|union]] of [[compact space|compact subsets]].
* In [[mathematical analysis]] and in [[probability theory]], there is a type of [[Field of sets|algebra of sets]] known as '''[[sigma-algebra|σ-algebra]]''' (aka '''σ-field'''). Sigma algebra also includes terms such as:
** '''σ(''A'')''', denoting the [[Σ-algebra#σ-algebras generated by families of sets|generated sigma-algebra]] of a set ''A''
** '''[[Σ-finite measure]]''' (see [[Measure (mathematics)|measure theory]])
* In [[number theory]], '''σ''' is included in various [[divisor function]]s, especially the '''sigma function''' or sum-of-divisors function.
* In [[applied mathematics]], '''σ(''T'')''' denotes the spectrum of a [[linear map]] ''T''.
* In [[complex analysis]], '''σ''' is used in the [[Weierstrass functions#Weierstrass sigma-function|Weierstrass sigma-function]].
* In [[probability theory]] and [[statistics]], '''Σ''' denotes the [[covariance matrix]] of a set of [[random variable]]s, sometimes in the form <math>\;|\!\!\!\Sigma</math> to distinguish it from the summation operator.
* [[Spectral theory|Theoretical spectral analysis]] uses '''σ''' as standard deviation opposed to lowercase [[Mu (letter)|mu]] as the absolute mean value.

====Biology, physiology, and medicine====
* In [[biology]], the [[sigma receptor]] ('''σ–receptors''') is a type of [[cell surface receptor]].
* In [[biochemistry]], the '''[[Sigma factor|σ factor]]''' (or specificity factor) is a protein found in [[RNA polymerase]].
* In [[Bone|bone physiology]], the [[bone remodeling period]]—i.e., the [[Sigma (bone remodeling)|life span]] of a basic multicellular unit—has historically been referred to as the '''sigma period'''
* In early 20th-century [[physiology]] literature, '''σ''' had been used to represent [[millisecond]]s<ref name="Hill1935">{{Cite journal | last1 = Hill | first1 = A. V. | title = Units and Symbols | doi = 10.1038/136222a0 | journal = Nature | volume = 136 | issue = 3432 | pages = 222 | year = 1935 | bibcode = 1935Natur.136..222H | s2cid = 4087300 | doi-access = free }}</ref>

====Business, finance, and economics====
* In [[finance]], '''σ''' is the symbol used to represent [[Volatility (finance)|volatility]] of [[stock]]s, usually measured by the [[standard deviation]] of [[logarithmic return]]s.
* In [[accounting]], '''Σ''' indicates the [[Balance (accounting)|balance]] of invoice classes and the overall amount of debts and demands.
* In [[macroeconomics]], '''σ''' is used in equations to represent the [[elasticity of substitution]] between two inputs.
* In the [[machine industry]], [[Six Sigma]] ('''6σ''') is a quality model based on the [[standard deviation]].

====Chemistry====
* [[Sigma bond]]s ('''σ bonds''') are the strongest type of [[covalent]] [[chemical bond]].
* In [[organic chemistry]], '''σ''' symbolizes the sigma constant of [[Hammett equation]].
* In [[alchemy]], '''Σ''' was sometimes used to represent [[sugar]].

====Engineering and computer science====
* In [[Computer Science|computer science]], '''Σ''' represents the set of symbols that form an alphabet (see also [[linguistics]])
* [[Relational algebra]] uses the values <math>\sigma_{a \theta b}( R )</math> and <math>\sigma_{a \theta v}( R )</math> to denote [[Selection (relational algebra)|selections]], which are a type of [[unary operation]].
* In [[machine learning]], '''σ''' is used in the formula that derives the [[Sigmoid function]].
* In [[radar jamming]] or [[electronic warfare]], [[radar cross-section]]s (RCS) are commonly represented as '''σ''' when measuring the size of a target's image on radar.
* In [[signal processing]], '''σ''' denotes the [[damping ratio]] of a system parameter.
* In [[theoretical computer science]], '''Σ''' serves as the [[busy beaver]] function.
* In [[civil engineering]], '''σ''' refers to the normal stress applied on a material or structure.

====Physics====
* In [[nuclear physics|nuclear]] and [[particle physics]], '''σ''' is used to denote [[cross section (physics)|cross sections]] in general (see also [[Radar cross-section|RCS]]), while '''Σ''' represents macroscopic [[Cross section (physics)|cross section]]s [1/length].
* The symbol is to denote the [[Stefan–Boltzmann constant]].
* In relation to fundamental properties of material, '''σ''' is often used to signify [[electrical conductivity]].
* In [[electrostatics]], '''σ''' represents [[surface charge]] density.
* In [[continuum mechanics]], '''σ''' is used to signify [[Stress (physics)|stress]].
* In [[condensed matter physics]], '''Σ''' denotes [[self-energy]].
* The symbol can be used to signify [[surface tension]] (alternatively, ''[[Gamma|γ]]'' or ''[[T]]'' are also used instead).
* In [[quantum mechanics]], '''σ''' is used to indicate [[Pauli matrices]].
* In [[astronomy]], '''σ''' represents [[velocity dispersion]].
* In astronomy, the prefix '''Σ''' is used to designate double stars of the [[Catalogus Novus Stellarum Duplicium]] by [[Friedrich Georg Wilhelm von Struve]].
* In particle physics, '''Σ''' represents a class of [[baryon]]s.

=== Organizations ===
* During the 1930s, an uppercase '''Σ''' was in use as the symbol of the [[Brazilian Integralism|Ação Integralista Brasileira]], a fascist political party in [[Brazil]].
* [[Sigma Corporation]] uses the name of the letter but not the letter itself, but in many Internet forums, photographers refer to the company or its lenses using the letter.
* [[Sigma Aldrich]] incorporate both the name and the character in their logo.

==Unicode==

* {{unichar|037B|html=}}
* {{unichar|037C|html=}}
* {{unichar|037D|html=}}
* {{unichar|03A3|html=}}
* {{unichar|03C2|html=}}
* {{unichar|03C3|html=}}
* {{unichar|03F2|html=}}
* {{unichar|03F9|html=}}
* {{unichar|03FD|html=}}
* {{unichar|03FE|html=}}
* {{unichar|03FF|html=}}
* {{unichar|2140|html=}}
* {{unichar|2211|html=}}
* {{unichar|23B2|html=}}{{efn|Combined with {{tt|U+23B3}} to make a double-high sigma}}
* {{unichar|23B3|html=}}
* {{unichar|2CA4|html=}}
* {{unichar|2CA5|html=}}
* {{unichar|2CEA|html=}}
* {{unichar|1D6BA|html=}}{{efn|The {{sc|mathematical}} characters should only be used for math. Stylized Greek text should be encoded using the normal Greek letters, with markup and formatting to indicate text style.}}
* {{unichar|1D6D3|html=}}
* {{unichar|1D6D4|html=}}
* {{unichar|1D6F4|html=}}
* {{unichar|1D70D|html=}}
* {{unichar|1D70E|html=}}
* {{unichar|1D72E|html=}}
* {{unichar|1D747|html=}}
* {{unichar|1D748|html=}}
* {{unichar|1D768|html=}}
* {{unichar|1D781|html=}}
* {{unichar|1D782|html=}}
* {{unichar|1D7A2|html=}}
* {{unichar|1D7BB|html=}}
* {{unichar|1D7BC|html=}}
{{notelist}}

==See also==
{{Commons|Σ|the letter sigma}}
{{Wiktionary|Σ|σ|ς}}
* [[Antisigma]]
* [[Greek letters used in mathematics, science, and engineering]]
* [[Sampi]]
* [[Sho (letter)]]
* [[Stigma (letter)]]
* [[Sibilant consonant]]
* [[Summation]] ('''Σ''')
* Combining form "''sigm-''" (e.g. [[sigmodon|'''''sigm'''odon'']], [[sigmurethra|'''''sigm'''urethra'']], etc.)
* Derivative "''[[Sigmoid (disambiguation)|sigmoid]]''" (e.g. [[sigmoid sinus]], [[sigmoid colon]], [[sigmoidoscopy]], etc.)

==References==

=== Notes ===
{{Reflist|group=lower-roman}}

=== Citations ===
{{Reflist}}

[[Category:Greek letters]]
[[Category:Letters with final form]]