Sigma

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]

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]

InUnicode, 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.

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 СofCyrillic script originates in the lunate form of Sigma.

• 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).
• Inphonology, σ is used to represent syllables.
• Inlinguistics, Σ represents the set of symbols that form an alphabet (see also computer science).
• Inhistorical linguistics, Σ is used to represent a Common Brittonic consonant with a sound between [s] and [h]; perhaps an aspirated [ʃʰ].^[9]

• 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.:

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

• Inmathematical 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.
• Instatistics, σ represents the standard deviation of population or probability distribution (where muorμ is used for the mean).
• Intopology, σ-compact topological space is one that can be written as a countable unionofcompact subsets.
• Inmathematical 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)
• Innumber theory, σ is included in various divisor functions, especially the sigma function or sum-of-divisors function.
• Inapplied mathematics, σ(T) denotes the spectrum of a linear map T.
• Incomplex analysis, σ is used in the Weierstrass sigma-function.
• Inprobability theory and statistics, Σ denotes the covariance matrix of a set of random variables, sometimes in the form ${\displaystyle \;|\!\!\!\Sigma }$  to distinguish it from the summation operator.
• Theoretical spectral analysis uses σ as standard deviation opposed to lowercase mu as the absolute mean value.

• Inbiology, the sigma receptor (σ–receptors) is a type of cell surface receptor.
• Inbiochemistry, the σ factor (or specificity factor) is a protein found in RNA polymerase.
• Inbone 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]

• Infinance, σ is the symbol used to represent volatilityofstocks, usually measured by the standard deviationoflogarithmic returns.
• Inaccounting, Σ indicates the balance of invoice classes and the overall amount of debts and demands.
• Inmacroeconomics, 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.

• Sigma bonds (σ bonds) are the strongest type of covalent chemical bond.
• Inorganic chemistry, σ symbolizes the sigma constant of Hammett equation.
• Inalchemy, Σ was sometimes used to represent sugar.

• Incomputer science, Σ represents the set of symbols that form an alphabet (see also linguistics)
• Relational algebra uses the values ${\displaystyle \sigma _{a\theta b}($R)}   and ${\displaystyle \sigma _{a\theta v}($R)}   to denote selections, which are a type of unary operation.
• Inmachine learning, σ is used in the formula that derives the Sigmoid function.
• Inradar jammingorelectronic warfare, radar cross-sections (RCS) are commonly represented as σ when measuring the size of a target's image on radar.
• Insignal processing, σ denotes the damping ratio of a system parameter.
• Intheoretical computer science, Σ serves as the busy beaver function.
• Incivil engineering, σ refers to the normal stress applied on a material or structure.

• Innuclear 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.
• Inelectrostatics, σ represents surface charge density.
• Incontinuum mechanics, σ is used to signify stress.
• Incondensed matter physics, Σ denotes self-energy.
• The symbol can be used to signify surface tension (alternatively, γorT are also used instead).
• Inquantum mechanics, σ is used to indicate Pauli matrices.
• Inastronomy, σ represents velocity dispersion.
• In astronomy, the prefix Σ is used to designate double stars of the Catalogus Novus Stellarum DupliciumbyFriedrich Georg Wilhelm von Struve.
• In particle physics, Σ represents a class of baryons.

• 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.

σ

ς

Ϲ

ϲ

GREEK CAPITAL LETTER SIGMA

GREEK SMALL LETTER SIGMA

GREEK SMALL LETTER FINAL SIGMA

GREEK CAPITAL LUNATE SIGMA SYMBOL

GREEK LUNATE SIGMA SYMBOL

decimal

hex

dec

hex

dec

hex

dec

hex

dec

hex

931

U+03A3

963

U+03C3

962

U+03C2

1017

U+03F9

1010

U+03F2

206 163

CE A3

207 131

CF 83

207 130

CF 82

207 185

CF B9

207 178

CF B2

Σ

Σ

σ

σ

ς

ς

Ϲ

Ϲ

ϲ

ϲ

Σ

σ

ς, ς, ς

145

91

169

A9

170

AA

207

CF

236

EC

237

ED

211

D3

243

F3

242

F2

\Sigma

\sigma

\varsigma

^[11]

ͻ

Ͼ

ͼ

Ͽ

ͽ

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

decimal

hex

dec

hex

dec

hex

dec

hex

dec

hex

dec

hex

1021

U+03FD

891

U+037B

1022

U+03FE

892

U+037C

1023

U+03FF

893

U+037D

207 189

CF BD

205 187

CD BB

207 190

CF BE

205 188

CD BC

207 191

CF BF

205 189

CD BD

Ͻ

Ͻ

ͻ

ͻ

Ͼ

Ͼ

ͼ

ͼ

Ͽ

Ͽ

ͽ

ͽ

ⲥ

COPTIC CAPITAL LETTER SIMA

COPTIC SMALL LETTER SIMA

decimal

hex

dec

hex

11428

U+2CA4

11429

U+2CA5

226 178 164

E2 B2 A4

226 178 165

E2 B2 A5

Ⲥ

Ⲥ

ⲥ

ⲥ

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.

𝚺

𝛔

𝛓

𝛴

𝜎

N-ARY SUMMATION

MATHEMATICAL BOLD
CAPITAL SIGMA

MATHEMATICAL BOLD
SMALL SIGMA

MATHEMATICAL BOLD
SMALL FINAL SIGMA

MATHEMATICAL ITALIC
CAPITAL SIGMA

MATHEMATICAL ITALIC
SMALL SIGMA

decimal

hex

dec

hex

dec

hex

dec

hex

dec

hex

dec

hex

8721

U+2211

120506

U+1D6BA

120532

U+1D6D4

120531

U+1D6D3

120564

U+1D6F4

120590

U+1D70E

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

8721

2211

55349 57018

D835 DEBA

55349 57044

D835 DED4

55349 57043

D835 DED3

55349 57076

D835 DEF4

55349 57102

D835 DF0E

∑

∑

𝚺

𝚺

𝛔

𝛔

𝛓

𝛓

𝛴

𝛴

𝜎

𝜎

∑, ∑

𝜮

𝝈

𝝇

𝝨

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

decimal

hex

dec

hex

dec

hex

dec

hex

dec

hex

120589

U+1D70D

120622

U+1D72E

120648

U+1D748

120647

U+1D747

120680

U+1D768

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

55349 57101

D835 DF0D

55349 57134

D835 DF2E

55349 57160

D835 DF48

55349 57159

D835 DF47

55349 57192

D835 DF68

𝜍

𝜍

𝜮

𝜮

𝝈

𝝈

𝝇

𝝇

𝝨

𝝨

𝞁

𝞢

𝞼

𝞻

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

decimal

hex

dec

hex

dec

hex

dec

hex

dec

hex

120706

U+1D782

120705

U+1D781

120738

U+1D7A2

120764

U+1D7BC

120763

U+1D7BB

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

55349 57218

D835 DF82

55349 57217

D835 DF81

55349 57250

D835 DFA2

55349 57276

D835 DFBC

55349 57275

D835 DFBB

𝞂

𝞂

𝞁

𝞁

𝞢

𝞢

𝞼

𝞼

𝞻

𝞻

• 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.)