# Sigma

Sigma (/ˈsɪɡmə/;[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

The Madaba Map, a sixth-century mosaic of Jerusalem () uses the lunate sigma
A plaque reading "Metochion of Gethsemane" (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,[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 ; 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 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

### 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.:
• ${\displaystyle \sum _{k=0}^{5}k=0+1+2+3+4+5=15}$
• In mathematical logic, ${\displaystyle \Sigma _{n}^{0}}$ is used to denote the set of formulae with bounded quantifiers beginning with existential quantifiers, alternating ${\displaystyle n-1}$ 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.

### 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 &#931; &#x3A3; &#963; &#x3C3; &#962; &#x3C2; &#1017; &#x3F9; &#1010; &#x3F2;
Named character reference &Sigma; &sigma; &sigmaf;, &sigmav;, &varsigma;
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

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 &#1021; &#x3FD; &#891; &#x37B; &#1022; &#x3FE; &#892; &#x37C; &#1023; &#x3FF; &#893; &#x37D;

### Coptic sima

Character information
Preview
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 &#11428; &#x2CA4; &#11429; &#x2CA5;

### 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 &#8721; &#x2211; &#120506; &#x1D6BA; &#120532; &#x1D6D4; &#120531; &#x1D6D3; &#120564; &#x1D6F4; &#120590; &#x1D70E;
Named character reference &Sum;, &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 &#120589; &#x1D70D; &#120622; &#x1D72E; &#120648; &#x1D748; &#120647; &#x1D747; &#120680; &#x1D768;

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 &#120706; &#x1D782; &#120705; &#x1D781; &#120738; &#x1D7A2; &#120764; &#x1D7BC; &#120763; &#x1D7BB;

## References

### Notes

1. ^ "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

1. ^ "sigma". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.)
2. ^ a b Woodard, Roger D. (2006). "Alphabet". In Wilson, Nigel Guy (ed.). Encyclopedia of Ancient Greece. London: Routledge. p. 38.
3. ^ 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.
4. ^ Jeffery, Lilian H. (1961). The Local Scripts of Archaic Greece. Oxford: Clarendon. pp. 25–7.
5. ^ Thompson, Edward M. (1912). Introduction to Greek and Latin Paleography. Oxford: Clarendon. p. 108, 144.
6. ^ Hopkins, Edward C. D. (2004). "Letterform Usage | Numismatica Font Projects" Parthia.
7. ^ de Lisle, Christopher (2020). "Attic Inscriptions in UK Collections: Ashmolean Museum, Oxford". AIUK. 11: 11. ISSN 2054-6769. Retrieved 2 June 2022.
8. ^ 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.
9. ^ Conroy, Kevin M. (21 February 2008). "Celtic initial consonant mutations - nghath and bhfuil?" – via dlib.bc.edu.
10. ^ Hill, A. V. (1935). "Units and Symbols". Nature. 136 (3432): 222. Bibcode:1935Natur.136..222H. doi:10.1038/136222a0. S2CID 4087300.
11. ^ Unicode Code Charts: Greek and Coptic (Range: 0370-03FF)