Sigma

Sigma (upper-case Σ, lower-case σ, lower-case 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. When used at the end of a word (when the word is not all caps), the final form (ς) is used, e.g. Ὀδυσσεύς (Odysseus); note the two sigmas in the center of the name, and the word-final sigma at the end.

History

The shape and alphabetic position of sigma is derived from Phoenician shin 𐤔 .

Etymology

The name of sigma, according to one hypothesis,^[1] may continue that of Phoenician Samekh. According to a different theory,^[2] its original name may have been 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 σίζω (sízō, from earlier *sig-jō, meaning 'I hiss').

Lunate sigma

The Madaba Map, a sixth-century mosaic of Jerusalem (Η ΑΓΙΑ ΠΟΛΙϹ) 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 and 3rd centuries BC), the epigraphic form of Σ was simplified into a C-like shape.^[3] It is also found on coins from the fourth century BC onward.^[4] This became the universal standard form of sigma during late antiquity and the Middle Ages. It is today known as lunate sigma (upper-case Ϲ, lower-case ϲ), because of its crescent-like shape.

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 letter С (representing /s/) and Coptic letter Ⲥ sima are derived from lunate sigma.

A dotted lunate sigma (sigma periestigmenon, encoded at U+03FE Ͼ) was used by Aristarchus of Samothrace (220–143 BC) as an editorial sign indicating that the line so marked is at an incorrect position. Similarly, an antisigma, or reversed sigma (Ͻ), may mark a line that is out of place. A dotted antisigma or dotted reversed sigma (antisigma periestigmenon: Ͽ) may indicate a line after which rearrangements should be made, or to variant readings of uncertain priority.

Uses

Greek

In both Ancient and Modern Greek, the sigma represents the voiceless alveolar fricative /s/. Both in Ancient and Modern Greek, this sound is voiced to /z/ before /m/ or /n/.

African Latin alphabets

Berber

Upper-case Σ may be used in the Berber Latin alphabet for [ʕ], though the INALCO standard uses Ɛ instead.

Uppercase of esh

The uppercase form of sigma was re-borrowed into the Latin alphabet – more precisely, the International African Alphabet – to serve as the uppercase of modern esh (lowercase: ʃ).

Science and mathematics

Upper-case

Upper-case Σ is used as a symbol for:

• the summation operator

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

• Sigma is the addition version of factorial.
• a class of baryons in particle physics
• macroscopic cross sections in nuclear and particle physics
• self-energy in condensed matter physics
• the balance of the invoice classes and the overall amount of the debts and demands in economics
• the set of symbols that form an alphabet in linguistics and computer science
• the covariance matrix of a set of random variables in probability theory and statistics, sometimes in the form ${\displaystyle \;|\!\!\!\Sigma }$ to distinguish it from the summation operator.
• The busy beaver function

Lower-case

Lower-case σ is used for:

• the standard deviation of a population or probability distribution in statistics
• sigma bonds in chemistry
• to represent an unknown angle in mathematics
• cross section (physics)
• velocity dispersion in astronomy
• sigma constant in organic chemistry
• the sigma receptor in biology
• a quality model for business, Six Sigma (6σ), based on the standard deviation
• sigma-algebras, sigma-fields, and sigma-finiteness in measure theory; in general terms, the symbol σ serves as a shorthand for "countably", e.g. a σ-compact topological space is one that can be written as a countable union of compact subsets.
• the generated sigma-algebra of a set $A$ is denoted $\sigma (A)$
• the sum-of-divisors function in number theory
• the Stefan–Boltzmann constant
• the "sigma factor" of RNA polymerase
• a measure of electrical conductivity
• the surface charge density in electrostatics
• normal stress in continuum mechanics
• volatility of a stock generally needed for options pricing
• a syllable in phonology
• the spectrum of a matrix $A$, denoted as $\sigma (A)$, in applied mathematics
• surface tension
• the unary operation of selection on a database relation in relational algebra
• the Pauli matrices in quantum mechanics
• a target's radar cross-section (RCS) in radar jamming or electronic warfare
• the life span of a basic multicellular unit (BMU) in bone remodeling
• the damping parameter in signal processing
• a millisecond in early 20th-century physiology literature^[5]
• the Weierstrass sigma-function

Groups

Politics

During the 1930s, an upper-case Σ was in use as the symbol of the Ação Integralista Brasileira, a fascist political party in Brazil.

Companies

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 encodings

• Greek Sigma

Character	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	decimal	hex	decimal	hex	decimal	hex	decimal	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		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

^[6]

Character	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	decimal	hex	decimal	hex	decimal	hex	decimal	hex	decimal	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	Ⲥ		ⲥ
Unicode name	COPTIC CAPITAL LETTER SIMA		COPTIC SMALL LETTER SIMA
Encodings	decimal	hex	decimal	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

Character	∑		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	decimal	hex	decimal	hex	decimal	hex	decimal	hex	decimal	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	∑

Character	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	decimal	hex	decimal	hex	decimal	hex	decimal	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	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	decimal	hex	decimal	hex	decimal	hex	decimal	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

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.

See also

• Antisigma
• Greek letters used in mathematics, science, and engineering
• Sampi
• Stigma (letter)
• Sibilant consonant
• ∑, or summation
• "Sigm-" as a combining form, as in sigmodon, sigmurethra, or in the derivative "sigmoid", as in sigmoid sinus, sigmoid colon, sigmoidoscopy, etc.

References

1. Jeffery, Lilian H. (1961). The local scripts of archaic Greece. Oxford: Clarendon. pp. 25–27.
2. Woodard, Roger D. (2006). "Alphabet". In Wilson, Nigel Guy. Encyclopedia of ancient Greece. London: Routldedge. p. 38.
3. Edward M. Thompson (1912), Introduction to Greek and Latin paleography, Oxford: Clarendon. p. 108, 144
4. Parthia.com: Numismatica Font Projects.
5. Hill, A. V. (1935). "Units and Symbols". Nature. 136 (3432): 222. doi:10.1038/136222a0.
6. Unicode Code Charts: Greek and Coptic (Range: 0370-03FF)