Sigma

Sigma (/ˈsɪɡmə/ SIG-mə;^[1] uppercase Σしぐま, lowercase σしぐま, lowercase in word-final position ς; ‹See Tfd›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しぐま 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しぐま) 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くしー. 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しぐま 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しぐま) 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しぐま is used to represent syllables.
In linguistics, Σしぐま represents the set of symbols that form an alphabet (see also computer science).
In historical linguistics, Σしぐま 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しぐま is commonly used to represent unknown angles, additionally serving as a shorthand for "countably", whereas Σしぐま 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しぐま represents the standard deviation of population or probability distribution (where mu or μみゅー is used for the mean).
In topology, σしぐま-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しぐま-algebra (aka σしぐま-field). Sigma algebra also includes terms such as:
- σしぐま(A), denoting the generated sigma-algebra of a set A
- Σしぐま-finite measure (see measure theory)
In number theory, σしぐま is included in various divisor functions, especially the sigma function or sum-of-divisors function.
In applied mathematics, σしぐま(T) denotes the spectrum of a linear map T.
In complex analysis, σしぐま is used in the Weierstrass sigma-function.
In probability theory and statistics, Σしぐま 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しぐま as standard deviation opposed to lowercase mu as the absolute mean value.

Biology, physiology, and medicine

In biology, the sigma receptor (σしぐま–receptors) is a type of cell surface receptor.
In biochemistry, the σしぐま 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しぐま had been used to represent milliseconds^[10]

Business, finance, and economics

In finance, σしぐま is the symbol used to represent volatility of stocks, usually measured by the standard deviation of logarithmic returns.
In accounting, Σしぐま indicates the balance of invoice classes and the overall amount of debts and demands.
In macroeconomics, σしぐま is used in equations to represent the elasticity of substitution between two inputs.
In the machine industry, Six Sigma (6σしぐま) is a quality model based on the standard deviation.

Chemistry

Sigma bonds (σしぐま bonds) are the strongest type of covalent chemical bond.
In organic chemistry, σしぐま symbolizes the sigma constant of Hammett equation.
In alchemy, Σしぐま was sometimes used to represent sugar.

Engineering and computer science

In computer science, Σしぐま 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しぐま 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しぐま when measuring the size of a target's image on radar.
In signal processing, σしぐま denotes the damping ratio of a system parameter.
In theoretical computer science, Σしぐま serves as the busy beaver function.
In civil engineering, σしぐま refers to the normal stress applied on a material or structure.

Physics

In nuclear and particle physics, σしぐま is used to denote cross sections in general (see also RCS), while Σしぐま represents macroscopic cross sections [1/length].
The symbol is to denote the Stefan–Boltzmann constant.
In relation to fundamental properties of material, σしぐま is often used to signify electrical conductivity.
In electrostatics, σしぐま represents surface charge density.
In continuum mechanics, σしぐま is used to signify stress.
In condensed matter physics, Σしぐま denotes self-energy.
The symbol can be used to signify surface tension (alternatively, γがんま or T are also used instead).
In quantum mechanics, σしぐま is used to indicate Pauli matrices.
In astronomy, σしぐま represents velocity dispersion.
In astronomy, the prefix Σしぐま is used to designate double stars of the Catalogus Novus Stellarum Duplicium by Friedrich Georg Wilhelm von Struve.
In particle physics, Σしぐま represents a class of baryons.

Organizations

During the 1930s, an uppercase Σしぐま 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.

Unicode

U+037B ͻ GREEK SMALL REVERSED LUNATE SIGMA SYMBOL
U+037C ͼ GREEK SMALL DOTTED LUNATE SIGMA SYMBOL
U+037D ͽ GREEK SMALL REVERSED DOTTED LUNATE SIGMA SYMBOL
U+03A3 Σしぐま GREEK CAPITAL LETTER SIGMA (Σ)
U+03C2 ς GREEK SMALL LETTER FINAL SIGMA (&sigmaf;, ς, ς)
U+03C3 σしぐま GREEK SMALL LETTER SIGMA (σ)
U+03F2 ϲ GREEK LUNATE SIGMA SYMBOL
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
U+03FF Ͽ GREEK CAPITAL REVERSED DOTTED LUNATE SIGMA SYMBOL
U+2140 ⅀ DOUBLE-STRUCK N-ARY SUMMATION
U+2211 ∑ N-ARY SUMMATION (&Sum;, ∑)
U+23B2 ⎲ SUMMATION TOP^[a]
U+23B3 ⎳ SUMMATION BOTTOM
U+2CA4 Ⲥ COPTIC CAPITAL LETTER SIMA
U+2CA5 ⲥ COPTIC SMALL LETTER SIMA
U+2CEA ⳪ COPTIC SYMBOL SHIMA SIMA
U+1D6BA 𝚺 MATHEMATICAL BOLD CAPITAL SIGMA^[b]
U+1D6D3 𝛓 MATHEMATICAL BOLD SMALL FINAL SIGMA
U+1D6D4 𝛔 MATHEMATICAL BOLD SMALL SIGMA
U+1D6F4 𝛴 MATHEMATICAL ITALIC CAPITAL SIGMA
U+1D70D 𝜍 MATHEMATICAL ITALIC SMALL FINAL SIGMA
U+1D70E 𝜎 MATHEMATICAL ITALIC SMALL SIGMA
U+1D72E 𝜮 MATHEMATICAL BOLD ITALIC CAPITAL SIGMA
U+1D747 𝝇 MATHEMATICAL BOLD ITALIC SMALL FINAL SIGMA
U+1D748 𝝈 MATHEMATICAL BOLD ITALIC SMALL SIGMA
U+1D768 𝝨 MATHEMATICAL SANS-SERIF BOLD CAPITAL SIGMA
U+1D781 𝞁 MATHEMATICAL SANS-SERIF BOLD SMALL FINAL SIGMA
U+1D782 𝞂 MATHEMATICAL SANS-SERIF BOLD SMALL SIGMA
U+1D7A2 𝞢 MATHEMATICAL SANS-SERIF BOLD ITALIC CAPITAL SIGMA
U+1D7BB 𝞻 MATHEMATICAL SANS-SERIF BOLD ITALIC SMALL FINAL SIGMA
U+1D7BC 𝞼 MATHEMATICAL SANS-SERIF BOLD ITALIC SMALL SIGMA

^ Combined with U+23B3 to make a double-high sigma
^ The 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.

References

Notes

^ "the same letter, which the Dorians call "san", but the Ionians 'sigma'..." [translated from Ancient Greek: "τたうὠυうぷしろんτたうὸ γράμμα, τたうὸ Δωριέες μみゅーὲνにゅー σしぐまὰνにゅー καλέουσι ,Ἴωνες δでるたὲ 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.

[12] Combined with U+23B3 to make a double-high sigma

[13] The 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.

[3] "the same letter, which the Dorians call "san", but the Ionians 'sigma'..." [translated from Ancient Greek: "τたうὠυうぷしろんτたうὸ γράμμα, τたうὸ Δωριέες μみゅーὲνにゅー σしぐまὰνにゅー καλέουσι ,Ἴωνες δでるたὲ 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.

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