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| {{other uses}}
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| {{Wiktionary|Σ|σ|ς}}
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| {{Greek Alphabet|letter=sigma}}
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| '''Sigma''' (upper-case '''Σ''', lower-case '''σ''', lower-case in word-final position '''ς'''; [[Ancient Greek|Greek]] σίγμα) is the eighteenth letter of the [[Greek alphabet]], and carries the 's' sound. In the system of [[Greek numerals]], it has a value of 200. When used at the end of a word, and the word is not all [[upper case|upper-case]], the final form (ς) is used, e.g. {{unicode|Ὀδυσσεύς}} (Odysseus). Note the two sigmas in the center of the name, and the word-final sigma at the end.
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| ==History==
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| The shape and alphabetic position of Sigma is derived from [[Phoenician alphabet|Phoenician]] [[Shin (letter)|shin]] {{Unicode|𐤔}} [[Image:Phoenician sin.svg|20px]].
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| ===Etymology===
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| The name of sigma, according to one hypothesis,<ref name="jeffery25">{{cite book|last=Jeffery|first=Lilian H.|title=The local scripts of archaic Greece|place=Oxford|publisher=Clarendon|year=1961|pages=25–27}}</ref> may continue that of Phoenician [[Samekh]]. According to a different theory,<ref name="woodard">{{cite encyclopedia|encyclopedia=Encyclopedia of ancient Greece|editor-first=Nigel Guy|editor-last= Wilson|title=Alphabet|first=Roger D.| last=Woodard|year=2006| publisher=Routldedge| place=London| page=38}}</ref> its original name may have been "''[[San (letter)|San]]'' " (the name today associated with another, obsolete letter), while "sigma" was a Greek innovation that simply meant "hissing", based on a nominalization of a verb {{lang|grc|σίζω}} (''sízō'', from earlier *''sig-jō'', meaning 'I hiss').
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| ===Uppercase of esh===
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| The uppercase form of sigma was re-borrowed into the [[Latin alphabet]] to serve as the uppercase of modern [[esh (letter)|esh]] (lowercase: {{unicode|ʃ}}).
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| === Lunate sigma ===
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| [[File:Madaba Jerusalem Mosaic.jpg|thumb|left|The [[Madaba Map]], a sixth-century mosaic map of [[Jerusalem]] ("Η ΑΓΙΑ ПОΛΙΣ") uses the lunate sigma.]]
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| [[File:Metochion Gethsimanis, Jerusalem.jpg|thumb|right|upright|A plaque reading "Μετόχιον Γεθσημανῆς", i.e. [[Metochion]] of [[Gethsemane]], in Jerusalem with a lunate sigma both at the end and in the middle of the word]]
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| In handwritten Greek during the Hellenistic period (4th and 3rd centuries BC), the epigraphic form of Σ was simplified into a C-like shape.<ref>Edward M. Thompson (1912), ''Introduction to Greek and Latin paleography'', Oxford: Clarendon. p. 108, 144</ref> It is also found on coins from the fourth century BC onward.<ref>[http://www.parthia.com/fonts/letterforms_usage.htm Parthia.com]: ''Numismatica Font Projects''.</ref> This became the universal standard form of Sigma during late antiquity and the Middle Ages. It is today known as ''lunate sigma'' (upper-case {{Unicode|Ϲ}}, lower-case {{Unicode|ϲ}}), because of its [[crescent]]-like shape.
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| It 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. The forms of the [[Cyrillic script|Cyrillic letter]] [[Es (Cyrillic)|С]] (representing {{IPA|/s/}}) and [[Coptic alphabet|Coptic letter]] {{coptic|Ⲥ}} ''sima'' are derived from lunate sigma.
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| A dotted lunate sigma (''sigma periestigmenon'', encoded at U+03FE {{unicode|Ͼ}}) was used by [[Aristarchus of Samothrace]] as an editorial sign indicating that the line so marked is at an incorrect position. Similarly, an [[antisigma]] or reversed sigma ({{unicode|Ͻ}}) may mark a line that is out of place. A dotted [[antisigma]] or dotted reversed sigma (''antisigma periestigmenon'': {{unicode|Ͽ}}) may indicate a line after which rearrangements should be made, or to variant readings of uncertain priority.
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| ==Uses==
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| ===Greek===
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| In both [[Ancient Greek|Ancient]] and [[Modern Greek]], the sigma represents the [[voiceless alveolar fricative]] {{IPA|/s/}}. Both in Ancient and Modern Greek, this sound is [[voice (phonetics)|voiced]] to {{IPA|/z/}} before {{IPA|/m/}} or {{IPA|/n/}}.
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| ===Berber===
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| Upper-case Σ may be used in the [[Berber Latin alphabet]] for <nowiki>[</nowiki>[[ʕ]]<nowiki>]</nowiki>, though the [[INALCO]] standard uses [[Ɛ]] instead.
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| ===Science and mathematics===
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| ====Upper-case====
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| Upper-case Σ is used as a symbol for:
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| * the [[summation]] operator
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| * a class of [[baryon]]s in [[particle physics]]
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| * macroscopic [[Cross section (physics)|cross section]]s in [[nuclear physics|nuclear]] and particle physics
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| * [[self-energy]] in [[condensed matter physics]]
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| * the [[Balance (accounting)|balance]] of the invoice classes and the overall amount of the debts and demands in [[economics]]
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| * the set of symbols that form an alphabet in [[linguistics]] and [[Computer Science|computer science]]
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| * the [[covariance matrix]] of a set of random variables in [[probability theory]] and [[statistics]], sometimes in the form <math>|\!\!\!\Sigma</math> to distinguish it from the summation operator.
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| ====Lower-case====
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| Lower-case σ is used for:
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| * [[sigma bond]]s in [[chemistry]]
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| * to represent an unknown angle in mathematics
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| * [[Velocity dispersion]] in [[astronomy]]
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| * [[Sigma constant]] in science
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| * the [[sigma receptor]] in [[biology]]
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| * the [[standard deviation]] of a [[Standard deviation#Estimating_population_standard_deviation|population]] or [[probability distribution]] in [[statistics]]
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| * a quality model for business, [[Six Sigma]], based on the [[standard deviation]], often referred to as "'''6σ'''"
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| * [[sigma-algebra]]s, sigma-fields, and [[Sigma-finite measure|sigma-finiteness]] in [[Measure (mathematics)|measure theory]]; in general terms, the symbol ''σ'' serves as a shorthand for "[[countable set|countably]]", e.g. a ''σ''-compact [[topological space]] is one that can be written as a countable [[union (set theory)|union]] of [[compact space|compact subsets]].
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| * the [[Sigma-algebra#Generated_.CF.83-algebra|generated sigma-algebra]] of a set <math>A</math> is denoted <math>\sigma(A)</math>
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| * the [[divisor function|sum-of-divisors function]] in [[number theory]]
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| * the [[Stefan–Boltzmann constant]]
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| * the "[[sigma factor]]" of [[RNA polymerase]]
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| * a measure of [[electrical conductivity]]
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| * the [[Surface charge]] density in electrostatics
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| * [[Normal stress|Normal]] [[Stress (physics)|stress]] in [[continuum mechanics]]
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| * volatility of a [[stock]] generally needed for [[options pricing]]
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| * a [[syllable]] in [[phonology]]
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| * the [[spectrum of a matrix]] <math>A</math>, denoted as <math>\sigma(A)</math>, in [[applied mathematics]]
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| * [[surface tension]]
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| * the [[unary operation]] of [[Selection (relational algebra)|selection]] on a database relation in [[relational algebra]]
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| * the [[Pauli matrices]] in [[quantum mechanics]]
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| * a target's [[radar cross-section]] (RCS) in [[radar jamming]] or [[Electronic warfare|Electronic Warfare]] (EW)
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| * the [[Sigma (bone remodeling)|life span]] of a basic multicellular unit (BMU) in [[bone remodeling]]
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| * the Damping Parameter in Signal processing
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| * a millisecond in early 20th-century physiology literature.<ref name=Hill1935>{{cite doi|10.1038/136222a0}}</ref>
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| ===Politics===
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| During the 1930s, an upper-case Σ was in use as the symbol of the [[Brazilian Integralism|Ação Integralista Brasileira]], a political party in [[Brazil]].
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| ==Companies==
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| [[Sigma Corporation]] uses the name of the letter but not the letter itself, however in many Internet forums photographers refer to the company or its lenses using the sigma letter.
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| ==Character encodings==
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| * Greek Sigma
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| {{charmap
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| |03A3|name1=Greek Capital Letter Sigma
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| |03C3|name2=Greek Small Letter Sigma
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| |03C2|name3=Greek Small Letter Final Sigma
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| |03F9|name4=Greek Capital Lunate Sigma Symbol
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| |03F2|name5=Greek Lunate Sigma Symbol
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| |map2=DOS Greek|map2char1=91|map2char2=A9|map2char3=AA
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| |map3=DOS Greek-2|map3char1=CF|map3char2=EC|map3char3=ED
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| |map4=Windows 1253|map4char1=D3|map4char2=F3|map4char3=F2
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| |namedref1=[[TeX]]|ref1char1=\Sigma|ref1char2=\sigma|ref1char3=\varsigma
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| }}
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| <ref>Unicode Code Charts: [http://www.unicode.org/charts/PDF/U0370.pdf Greek and Coptic (Range: 0370-03FF)]</ref>
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| {{charmap
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| |03FD|name1=Greek Capital Reversed Lunate Sigma Symbol
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| |037B|name2=Greek Small Reversed Lunate Sigma Symbol
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| |03FE|name3=Greek Capital Dotted Lunate Sigma Symbol
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| |037C|name4=Greek Small Dotted Lunate Sigma Symbol
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| |03FF|name5=Greek Capital Reversed Dotted Lunate Sigma Symbol
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| |037D|name6=Greek Small Reversed Dotted Lunate Sigma Symbol
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| }}
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| * Coptic Sima
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| {{charmap
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| |2CA4|name1=Coptic Capital Letter Sima
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| |2CA5|name2=Coptic Small Letter Sima
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| }}
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| * Mathematical Sigma
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| {{charmap
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| |2211|name1=N-ary Summation
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| |1D6BA|name2=Mathematical Bold<br />Capital Sigma
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| |1D6D4|name3=Mathematical Bold<br />Small Sigma
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| |1D6D3|name4=Mathematical Bold<br />Small Final Sigma
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| |1D6F4|name5=Mathematical Italic<br />Capital Sigma
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| |1D70E|name6=Mathematical Italic<br />Small Sigma
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| }}
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| {{charmap
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| |1D70D|name1=Mathematical Italic<br />Small Final Sigma
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| |1D72E|name2=Mathematical Bold Italic<br />Capital Sigma
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| |1D748|name3=Mathematical Bold Italic<br />Small Sigma
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| |1D747|name4=Mathematical Bold Italic<br />Small Final Sigma
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| |1D768|name5=Mathematical Sans-Serif<br />Bold Capital Sigma
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| }}
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| {{charmap
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| |1D782|name1=Mathematical Sans-Serif<br />Bold Small Sigma
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| |1D781|name2=Mathematical Sans-Serif<br />Bold Small Final Sigma
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| |1D7A2|name3=Mathematical Sans-Serif<br />Bold Italic Capital Sigma
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| |1D7BC|name4=Mathematical Sans-Serif<br />Bold Italic Small Sigma
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| |1D7BB|name5=Mathematical Sans-Serif<br />Bold Italic Small Final Sigma
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| }}
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| These characters are used only as mathematical symbols. Stylized Greek text should be encoded using the normal Greek letters, with markup and formatting to indicate text style.
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| ==See also==
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| *[[Antisigma]]
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| *[[Greek letters used in mathematics, science, and engineering]]
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| *[[Sampi]]
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| *[[Sibilant consonant]]
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| *[[∑]], or [[summation]]
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| *"Sigm-" as a combining form, as in [[sigmodon|''sigm''odon]], [[sigmurethra|''sigm''urethra]], or in the derivative "sigmoid", as in [[sigmoid sinus]], [[sigmoid colon]], [[sigmoidoscopy]], etc.
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| ==References==
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| {{Reflist}}
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| [[Category:Greek letters]]
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