Sigma

Sigma (/ˈsɪɡmə/ SIG-mə;^[1] uppercase Σ, lowercase σ, lowercase in word-final 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 letter-case 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 word-final 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 Proto-Greek *sig-jō '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 C-like 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 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).^[7]^[8]

In Unicode, the above variations of lunate sigma are encoded as U+03F9 Ϲ GREEK CAPITAL LUNATE SIGMA SYMBOL; U+03FD Ͻ GREEK CAPITAL REVERSED LUNATE SIGMA SYMBOL, U+03FE Ͼ GREEK CAPITAL DOTTED LUNATE SIGMA SYMBOL, and U+03FF Ͽ GREEK CAPITAL REVERSED DOTTED LUNATE SIGMA SYMBOL.

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 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 s-rune). Both the Anglo-Saxon runes and the Younger Futhark consistently use the simplified three-stroke 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 [s]. In Modern Greek, this sound is voiced to the voiced alveolar fricative [z] when occurring before [m], [n], [v], [ð] or [ɣ].
The uppercase form of sigma (Σ) was re-borrowed 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.:

$\sum _{k=0}^{5}k=0+1+2+3+4+5=15$

In mathematical logic, $\Sigma _{n}^{0}$ is used to denote the set of formulae with bounded quantifiers beginning with existential quantifiers, alternating $n-1$ 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, σ-compact 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 σ-algebra (aka σ-field). Sigma algebra also includes terms such as:
- σ(A), denoting the generated sigma-algebra of a set A
- Σ-finite measure (see measure theory)
In number theory, σ is included in various divisor functions, 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 sigma-function.
In probability theory and statistics, Σ denotes the covariance matrix of a set of random variables, sometimes in the form $\;|\!\!\!\Sigma$ 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 20th-century 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 $\sigma _{a\theta b}(R)$ and $\sigma _{a\theta v}(R)$ 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 cross-sections (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 self-energy.
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

Character information
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
UTF-8	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		&sigmaf;, ς, ς
DOS Greek	145	91	169	A9	170	AA
DOS Greek-2	207	CF	236	EC	237	ED
Windows 1253	211	D3	243	F3	242	F2
TeX	\Sigma		\sigma		\varsigma

^[11]

Character information
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
UTF-8	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

Character information
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
UTF-8	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.

Character information
Preview	∑		𝚺		𝛔		𝛓		𝛴		𝜎
Unicode name	N-ARY 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
UTF-8	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
UTF-16	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	&Sum;, ∑

Character information
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 SANS-SERIF 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
UTF-8	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
UTF-16	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

Character information
Preview	𝞂		𝞁		𝞢		𝞼		𝞻
Unicode name	MATHEMATICAL SANS-SERIF BOLD SMALL SIGMA		MATHEMATICAL SANS-SERIF BOLD SMALL FINAL SIGMA		MATHEMATICAL SANS-SERIF BOLD ITALIC CAPITAL SIGMA		MATHEMATICAL SANS-SERIF BOLD ITALIC SMALL SIGMA		MATHEMATICAL SANS-SERIF 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
UTF-8	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
UTF-16	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

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 Sims-Williams. 2002. "Non-Attic letters," transcribed by N. Nicholas. Archived from the original 2020-06-28.
^ 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 2054-6769. 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: 0370-03FF)

[3] "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)

[1] "sigma". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.)

[woodard-2] Woodard, Roger D. (2006). "Alphabet". In Wilson, Nigel Guy (ed.). Encyclopedia of Ancient Greece. London: Routledge. p. 38.

[4] Herodotus, Histories 1.139 — Everson, Michael and Nicholas Sims-Williams. 2002. "Non-Attic letters," transcribed by N. Nicholas. Archived from the original 2020-06-28.

[jeffery25-5] Jeffery, Lilian H. (1961). The Local Scripts of Archaic Greece. Oxford: Clarendon. pp. 25–7.

[6] Thompson, Edward M. (1912). Introduction to Greek and Latin Paleography. Oxford: Clarendon. p. 108, 144.

[7] Hopkins, Edward C. D. (2004). "Letterform Usage | Numismatica Font Projects" Parthia.

[8] Lisle, Christopher (2020). "Attic Inscriptions in UK Collections: Ashmolean Museum, Oxford". AIUK. 11: 11. ISSN 2054-6769. Retrieved 2 June 2022.

[9] 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.

[10] Conroy, Kevin M. (21 February 2008). "Celtic initial consonant mutations - nghath and bhfuil?" – via dlib.bc.edu.

[Hill1935-11] Hill, A. V. (1935). "Units and Symbols". Nature. 136 (3432): 222. Bibcode:1935Natur.136..222H. doi:10.1038/136222a0. S2CID 4087300.

[12] Unicode Code Charts: Greek and Coptic (Range: 0370-03FF)

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