Sigma
Greek alphabet  



History  
Use in other languages  
Related topics  
Sigma (/ˈsɪɡmə/;^{[1]} uppercase Σ, lowercase σ, lowercase in wordfinal position ς; Greek: 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 for summation. When used at the end of a lettercase word (one that does not use all caps), the final form (ς) is used. In Ὀδυσσεύς (Odysseus), for example, the two lowercase sigmas (σ) in the center of the name are distinct from the wordfinal sigma (ς) at the end. The Latin letter S derives from sigma while the Cyrillic letter Es derives from a lunate form of this letter.
History
The shape (Σς) and alphabetic position of sigma is derived from the Phoenician letter (shin).
Sigma's original name may have been san, but due to the complicated early history of the Greek epichoric alphabets, san came to be identified as a separate letter in the Greek alphabet, represented as Ϻ.^{[2]} Herodotus reports that "san" was the name given by the Dorians to the same letter called "sigma" by the Ionians.^{[i]}^{[3]}
According to one hypothesis,^{[4]} the name "sigma" may continue that of Phoenician samekh (), the letter continued through Greek xi, represented as Ξ. Alternatively, the name may have been a Greek innovation that simply meant 'hissing', from the root of σίζω (sízō, from ProtoGreek *sigjō 'I hiss').^{[2]}
Lunate sigma
In handwritten Greek during the Hellenistic period (4th–3rd century BC), the epigraphic form of Σ was simplified into a Clike shape,^{[5]} which has also been found on coins from the 4th century BC onward.^{[6]} 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 crescentlike 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).^{[7]}^{[8]}
In Unicode, the above variations of lunate sigma are encoded as U+03F9 Ϲ ; U+03FD Ͻ , U+03FE Ͼ , and U+03FF Ͽ .
Derived alphabets
Sigma was adopted in the Old Italic alphabets beginning in the 8th century BC. At that time a simplified threestroke version, omitting the lowermost stroke, was already found in Western Greek alphabets, and was incorporated into classical Etruscan and Oscan, as well as in the earliest 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 early runic alphabet (early form of the srune). Both the AngloSaxon runes and the Younger Futhark consistently use the simplified threestroke version.
The letter С of Cyrillic script originates in the lunate form of Sigma.
Uses
Language and linguistics
 In both Ancient and Modern Greek, the sigma represents the voiceless alveolar fricative IPA: [s]. In Modern Greek, this sound is voiced to the voiced alveolar fricative IPA: [z] when occurring before IPA: [m], IPA: [n], IPA: [v], IPA: [ð] or IPA: [ɣ].
 The uppercase form of sigma (Σ) was reborrowed into the Latin alphabet—more precisely, the International African Alphabet—to serve as the uppercase of modern esh (lowercase: ʃ).
 In phonology, σ is used to represent syllables.
 In linguistics, Σ represents the set of symbols that form an alphabet (see also computer science).
 In historical linguistics, Σ is used to represent a Common Brittonic consonant with a sound between [s] and [h]; perhaps an aspirated [ʃʰ].^{[9]}
Science and mathematics
Mathematics
 In general mathematics, lowercase σ is commonly used to represent unknown angles, as well as serving as a shorthand for "countably", whereas Σ is regularly used as the operator for summation, e.g.:
 In mathematical logic, is used to denote the set of formulae with bounded quantifiers beginning with existential quantifiers, alternating 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 or μ is used for the mean).
 In topology, σcompact topological space is one that can be written as a countable union of compact subsets.
 In mathematical analysis and in probability theory, there is a type of algebra of sets known as σalgebra (aka σfield). Sigma algebra also includes terms such as:
 σ(A), denoting the generated sigmaalgebra of a set A
 σfinite measure (see measure theory)
 In number theory, σ is included in various divisor functions, especially the sigma function or sumofdivisors function.
 In applied mathematics, σ(T) denotes the spectrum of a linear map T.
 In complex analysis, σ is used in the Weierstrass sigmafunction.
 In probability theory and statistics, Σ denotes the covariance matrix of a set of random variables, sometimes in the form to distinguish it from the summation operator.
 Theoretical spectral analysis uses σ as standard deviation opposed to lowercase 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 σ factor (or specificity factor) is a protein found in RNA polymerase.
 In bone physiology, the bone remodeling period—i.e., the life span of a basic multicellular unit—has historically been referred to as the sigma period
 In early 20thcentury physiology literature, σ had been used to represent milliseconds^{[10]}
Business, finance, and economics
 In finance, σ is the symbol used to represent volatility of stocks, usually measured by the standard deviation of logarithmic returns.
 In accounting, Σ indicates the 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 bonds (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, Σ represents the set of symbols that form an alphabet (see also linguistics)
 Relational algebra uses the values and to denote 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 crosssections (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 and particle physics, σ is used to denote cross sections in general (see also RCS), while Σ represents macroscopic cross sections [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.
 In condensed matter physics, Σ denotes selfenergy.
 The symbol can be used to signify surface tension (alternatively, 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 baryons.
Organizations
 During the 1930s, an uppercase Σ was in use as the symbol of the 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.
Character encoding
Greek sigma
Preview  Σ  σ  ς  Ϲ  ϲ  

Unicode name  GREEK CAPITAL LETTER SIGMA  GREEK SMALL LETTER SIGMA  GREEK SMALL LETTER FINAL SIGMA  GREEK CAPITAL LUNATE SIGMA SYMBOL  GREEK LUNATE SIGMA SYMBOL  
Encodings  decimal  hex  dec  hex  dec  hex  dec  hex  dec  hex 
Unicode  931  U+03A3  963  U+03C3  962  U+03C2  1017  U+03F9  1010  U+03F2 
UTF8  206 163  CE A3  207 131  CF 83  207 130  CF 82  207 185  CF B9  207 178  CF B2 
Numeric character reference  Σ 
Σ 
σ 
σ 
ς 
ς 
Ϲ 
Ϲ 
ϲ 
ϲ 
Named character reference  Σ  σ  ς, ς, ς  
DOS Greek  145  91  169  A9  170  AA  
DOS Greek2  207  CF  236  EC  237  ED  
Windows 1253  211  D3  243  F3  242  F2  
TeX  \Sigma  \sigma  \varsigma 
^{[11]}
Preview  Ͻ  ͻ  Ͼ  ͼ  Ͽ  ͽ  

Unicode name  GREEK CAPITAL REVERSED LUNATE SIGMA SYMBOL  GREEK SMALL REVERSED LUNATE SIGMA SYMBOL  GREEK CAPITAL DOTTED LUNATE SIGMA SYMBOL  GREEK SMALL DOTTED LUNATE SIGMA SYMBOL  GREEK CAPITAL REVERSED DOTTED LUNATE SIGMA SYMBOL  GREEK SMALL REVERSED DOTTED LUNATE SIGMA SYMBOL  
Encodings  decimal  hex  dec  hex  dec  hex  dec  hex  dec  hex  dec  hex 
Unicode  1021  U+03FD  891  U+037B  1022  U+03FE  892  U+037C  1023  U+03FF  893  U+037D 
UTF8  207 189  CF BD  205 187  CD BB  207 190  CF BE  205 188  CD BC  207 191  CF BF  205 189  CD BD 
Numeric character reference  Ͻ 
Ͻ 
ͻ 
ͻ 
Ͼ 
Ͼ 
ͼ 
ͼ 
Ͽ 
Ͽ 
ͽ 
ͽ 
Coptic sima
Preview  Ⲥ  ⲥ  

Unicode name  COPTIC CAPITAL LETTER SIMA  COPTIC SMALL LETTER SIMA  
Encodings  decimal  hex  dec  hex 
Unicode  11428  U+2CA4  11429  U+2CA5 
UTF8  226 178 164  E2 B2 A4  226 178 165  E2 B2 A5 
Numeric character reference  Ⲥ 
Ⲥ 
ⲥ 
ⲥ 
Mathematical sigma
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.
Preview  ∑  𝚺  𝛔  𝛓  𝛴  𝜎  

Unicode name  NARY SUMMATION  MATHEMATICAL BOLD CAPITAL SIGMA 
MATHEMATICAL BOLD SMALL SIGMA 
MATHEMATICAL BOLD SMALL FINAL SIGMA 
MATHEMATICAL ITALIC CAPITAL SIGMA 
MATHEMATICAL ITALIC SMALL SIGMA  
Encodings  decimal  hex  dec  hex  dec  hex  dec  hex  dec  hex  dec  hex 
Unicode  8721  U+2211  120506  U+1D6BA  120532  U+1D6D4  120531  U+1D6D3  120564  U+1D6F4  120590  U+1D70E 
UTF8  226 136 145  E2 88 91  240 157 154 186  F0 9D 9A BA  240 157 155 148  F0 9D 9B 94  240 157 155 147  F0 9D 9B 93  240 157 155 180  F0 9D 9B B4  240 157 156 142  F0 9D 9C 8E 
UTF16  8721  2211  55349 57018  D835 DEBA  55349 57044  D835 DED4  55349 57043  D835 DED3  55349 57076  D835 DEF4  55349 57102  D835 DF0E 
Numeric character reference  ∑ 
∑ 
𝚺 
𝚺 
𝛔 
𝛔 
𝛓 
𝛓 
𝛴 
𝛴 
𝜎 
𝜎 
Named character reference  ∑, ∑ 
Preview  𝜍  𝜮  𝝈  𝝇  𝝨  

Unicode name  MATHEMATICAL ITALIC SMALL FINAL SIGMA 
MATHEMATICAL BOLD ITALIC CAPITAL SIGMA 
MATHEMATICAL BOLD ITALIC SMALL SIGMA 
MATHEMATICAL BOLD ITALIC SMALL FINAL SIGMA 
MATHEMATICAL SANSSERIF BOLD CAPITAL SIGMA  
Encodings  decimal  hex  dec  hex  dec  hex  dec  hex  dec  hex 
Unicode  120589  U+1D70D  120622  U+1D72E  120648  U+1D748  120647  U+1D747  120680  U+1D768 
UTF8  240 157 156 141  F0 9D 9C 8D  240 157 156 174  F0 9D 9C AE  240 157 157 136  F0 9D 9D 88  240 157 157 135  F0 9D 9D 87  240 157 157 168  F0 9D 9D A8 
UTF16  55349 57101  D835 DF0D  55349 57134  D835 DF2E  55349 57160  D835 DF48  55349 57159  D835 DF47  55349 57192  D835 DF68 
Numeric character reference  𝜍 
𝜍 
𝜮 
𝜮 
𝝈 
𝝈 
𝝇 
𝝇 
𝝨 
𝝨 
Preview  𝞂  𝞁  𝞢  𝞼  𝞻  

Unicode name  MATHEMATICAL SANSSERIF BOLD SMALL SIGMA 
MATHEMATICAL SANSSERIF BOLD SMALL FINAL SIGMA 
MATHEMATICAL SANSSERIF BOLD ITALIC CAPITAL SIGMA 
MATHEMATICAL SANSSERIF BOLD ITALIC SMALL SIGMA 
MATHEMATICAL SANSSERIF BOLD ITALIC SMALL FINAL SIGMA  
Encodings  decimal  hex  dec  hex  dec  hex  dec  hex  dec  hex 
Unicode  120706  U+1D782  120705  U+1D781  120738  U+1D7A2  120764  U+1D7BC  120763  U+1D7BB 
UTF8  240 157 158 130  F0 9D 9E 82  240 157 158 129  F0 9D 9E 81  240 157 158 162  F0 9D 9E A2  240 157 158 188  F0 9D 9E BC  240 157 158 187  F0 9D 9E BB 
UTF16  55349 57218  D835 DF82  55349 57217  D835 DF81  55349 57250  D835 DFA2  55349 57276  D835 DFBC  55349 57275  D835 DFBB 
Numeric character reference  𝞂 
𝞂 
𝞁 
𝞁 
𝞢 
𝞢 
𝞼 
𝞼 
𝞻 
𝞻 
See also
 Antisigma
 Greek letters used in mathematics, science, and engineering
 Sampi
 Sho (letter)
 Stigma (letter)
 Sibilant consonant
 Summation (Σ)
 Combining form "sigm" (e.g. sigmodon, sigmurethra, etc.)
 Derivative "sigmoid" (e.g. sigmoid sinus, sigmoid colon, sigmoidoscopy, etc.)
References
Notes
 ^ "the same letter, which the Dorians call "san", but the Ionians 'sigma'..." [translated from Ancient Greek: "τὠυτὸ γράμμα, τὸ Δωριέες μὲν σὰν καλέουσι ,Ἴωνες δὲ σίγμα"] (Herodotus 1.139)
Citations
 ^ "sigma". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.)
 ^ ^{a} ^{b} Woodard, Roger D. (2006). "Alphabet". In Wilson, Nigel Guy (ed.). Encyclopedia of Ancient Greece. London: Routledge. p. 38.
 ^ Herodotus, Histories 1.139 — Everson, Michael and Nicholas SimsWilliams. 2002. "NonAttic letters," transcribed by N. Nicholas. Archived from the original 20200628.
 ^ Jeffery, Lilian H. (1961). The Local Scripts of Archaic Greece. Oxford: Clarendon. pp. 25–7.
 ^ Thompson, Edward M. (1912). Introduction to Greek and Latin Paleography. Oxford: Clarendon. p. 108, 144.
 ^ Hopkins, Edward C. D. (2004). "Letterform Usage  Numismatica Font Projects" Parthia.
 ^ de Lisle, Christopher (2020). "Attic Inscriptions in UK Collections: Ashmolean Museum, Oxford". AIUK. 11: 11. ISSN 20546769. Retrieved 2 June 2022.
 ^ Follet, Simone (2000). "Les deux archontes Pamménès du Ier siècle a.c. à Athènes". Revue des Études Grecques. 113: 188–192. doi:10.3406/reg.2000.4402.
 ^ Conroy, Kevin M. (21 February 2008). "Celtic initial consonant mutations  nghath and bhfuil?" – via dlib.bc.edu.
 ^ Hill, A. V. (1935). "Units and Symbols". Nature. 136 (3432): 222. Bibcode:1935Natur.136..222H. doi:10.1038/136222a0. S2CID 4087300.
 ^ Unicode Code Charts: Greek and Coptic (Range: 037003FF)