Xi (letter)

Xi is the fourteenth letter of the Greek alphabet (uppercase Ξ, lowercase ξ; Greek: ξι), representing the voiceless consonant cluster [ks]. Its name is pronounced [ksi] in Modern Greek, and generally /zaɪ/ or /ksaɪ/ in English.^[1] In the system of Greek numerals, it has a value of 60. Xi was derived from the Phoenician letter samekh .

Xi is distinct from the letter chi, which gave its form to the Latin letter X.

Greek

A joined variant of Ξ (New Athena Unicode font)

Both in classical Ancient Greek and in Modern Greek, the letter Ξ represents the consonant cluster /ks/. In some archaic local variants of the Greek alphabet, this letter was missing. Instead, especially in the dialects of most of the Greek mainland and Euboea, the cluster /ks/ was represented by Χ (which in classical Greek is chi, used for /kʰ/).

Because this variant of the Greek alphabet was used in Magna Graecia (the Greek colonies in Sicily and the southern part of the Italian peninsula), the Latin alphabet borrowed Χ rather than Ξ as the Latin letter that represented the /ks/ cluster that was also present in Latin.

Cyrillic

The Xi was adopted into the early Cyrillic alphabet, as the letter ksi (Ѯ, ѯ).

Mathematics and science

Uppercase

The uppercase letter Ξ is used as a symbol in various contexts.

Pure mathematics

Harish-Chandra's Ξ function in harmonic analysis and representation theory
The Riemann Xi function in analytic number theory and complex analysis

Physics

The "cascade particles" in particle physics
The partition function under the grand canonical ensemble in statistical mechanics

Other uses

Indicating "no change of state" in Z notation in computing
Used as the Latin letter E in some logos.
Monetary units of the cryptocurrencies Ether (and less commonly ETC), equal to 10¹⁸ Wei

Lowercase

The lowercase letter ξ is used as a symbol for:

Pure mathematics

Random variables
A parameter in a generalized Pareto distribution
The symmetric function equation of the Riemann zeta function in mathematics, also known as the Riemann Xi function
A universal set in set theory
A number used in the remainder term of Taylor's theorem that falls between the limits a and b
A number used in error approximations for formulas that are applications of Taylor's theorem, such as Newton–Cotes formulas

Physics and astronomy

In fluid dynamics, the Iribarren parameter.
The initial mass function in astronomy.
The correlation function in astronomy.
Spatial frequency;^[2] also sometimes temporal frequency.
A small displacement in MHD plasma stability theory
The x-coordinate of computational space as used in computational fluid dynamics
Potential difference in physics (in volts)
The radial integral in the spin-orbit matrix operator in atomic physics.
The Killing vector in general relativity.
Average logarithmic energy decrement per collision (neutron calculations in nuclear physics)
Pippard's cohesion length in superconductors
The diameter of a crystal nucleus in nucleation theory
Microturbulence velocity in a stellar atmosphere
The dimensionless longitudinal momentum loss of a beam particle after a two-body interaction in accelerator physics.
Dimensionless distance variable used in the Lane–Emden equation

Other uses

Propositional variables in some philosophical works, first found in Wittgenstein's Tractatus
Extent of reaction, a concept in physical chemistry^[3] used often in chemical engineering kinetics and thermochemistry
Unknown stereochemistry or stereocentre configuration in a planar ring system in organic chemistry,^[4] as well as uppercase Xi for unknown R/S/E/Z configuration in general^[5]
One of the two different polypeptide chains of the human embryonic hemoglobin types Hb-Portland (ξ₂γ₂) and Hb-Gower I (ξ₂ε₂)
A parameter denoted as warped time used to derive the equations for homogeneous azeotropic distillation
State Price Density in mathematical finance
The information vector in the Information Filter, GraphSLAM, and a number of other algorithms used for robot localization and robotic mapping.
Used in Support Vector machines in cases where the data is not linearly separable.
Used in Microelectronics to represent the distance from a p-n junction to a point in the depletion region where the electric field is strongest.

Character encodings

Greek Xi / Coptic Ksi

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

Character information
Preview	Ξ		ξ		Ⲝ		ⲝ
Unicode name	GREEK CAPITAL LETTER XI		GREEK LOWERCASE LETTER XI		COPTIC CAPITAL LETTER KSI		COPTIC LOWERCASE LETTER KSI
Encodings	decimal	hex	dec	hex	dec	hex	dec	hex
Unicode	926	U+039E	958	U+03BE	11420	U+2C9C	11421	U+2C9D
UTF-8	206 158	CE 9E	206 190	CE BE	226 178 156	E2 B2 9C	226 178 157	E2 B2 9D
Numeric character reference	Ξ	Ξ	ξ	ξ	Ⲝ	Ⲝ	ⲝ	ⲝ
Named character reference	Ξ		ξ
DOS Greek	141	8D	165	A5
DOS Greek-2	189	BD	232	E8
Windows 1253	206	CE	238	EE
TeX	\Xi		\xi

Mathematical Xi

The following 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	𝚵		𝛏		𝛯		𝜉		𝜩		𝝃
Unicode name	MATHEMATICAL BOLD CAPITAL XI		MATHEMATICAL BOLD LOWERCASE XI		MATHEMATICAL ITALIC CAPITAL XI		MATHEMATICAL ITALIC LOWERCASE XI		MATHEMATICAL BOLD ITALIC CAPITAL XI		MATHEMATICAL BOLD ITALIC LOWERCASE XI
Encodings	decimal	hex	dec	hex	dec	hex	dec	hex	dec	hex	dec	hex
Unicode	120501	U+1D6B5	120527	U+1D6CF	120559	U+1D6EF	120585	U+1D709	120617	U+1D729	120643	U+1D743
UTF-8	240 157 154 181	F0 9D 9A B5	240 157 155 143	F0 9D 9B 8F	240 157 155 175	F0 9D 9B AF	240 157 156 137	F0 9D 9C 89	240 157 156 169	F0 9D 9C A9	240 157 157 131	F0 9D 9D 83
UTF-16	55349 57013	D835 DEB5	55349 57039	D835 DECF	55349 57071	D835 DEEF	55349 57097	D835 DF09	55349 57129	D835 DF29	55349 57155	D835 DF43
Numeric character reference	𝚵	𝚵	𝛏	𝛏	𝛯	𝛯	𝜉	𝜉	𝜩	𝜩	𝝃	𝝃

Character information
Preview	𝝣		𝝽		𝞝		𝞷
Unicode name	MATHEMATICAL SANS-SERIF BOLD CAPITAL XI		MATHEMATICAL SANS-SERIF BOLD LOWERCASE XI		MATHEMATICAL SANS-SERIF BOLD ITALIC CAPITAL XI		MATHEMATICAL SANS-SERIF BOLD ITALIC LOWERCASE XI
Encodings	decimal	hex	dec	hex	dec	hex	dec	hex
Unicode	120675	U+1D763	120701	U+1D77D	120733	U+1D79D	120759	U+1D7B7
UTF-8	240 157 157 163	F0 9D 9D A3	240 157 157 189	F0 9D 9D BD	240 157 158 157	F0 9D 9E 9D	240 157 158 183	F0 9D 9E B7
UTF-16	55349 57187	D835 DF63	55349 57213	D835 DF7D	55349 57245	D835 DF9D	55349 57271	D835 DFB7
Numeric character reference	𝝣	𝝣	𝝽	𝝽	𝞝	𝞝	𝞷	𝞷

Other uses

Ξ for E in a commercial logo

Media related to Xi (letter) at Wikimedia Commons

Uppercase Ξ is used as an 'E' to stylise company names/logos like Razer (styled as RΛZΞR), Tesla (styled as TΞSLA), musician Banners (styled as BANNΞRS), and in South Korean boy group ZE:A's newest logo (styled as "ZΞA") (Compare: Heavy Metal umlaut; Faux Cyrillic)

References

↑ "xi". New Oxford American Dictionary, 2nd Edition.
↑ SPIE Optipedia article: "Spatial Frequency"
↑ IUPAC Gold Book Entry: "Extent of Reaction" Archived June 29, 2012, at the Wayback Machine
↑ Buckingham, John; Cooper, Caroline M.; Purchase, Rupert (18 November 2015). Natural Products Desk Reference. CRC Press. pp. 53–. ISBN 978-1-4398-7362-5.
↑ Nomenclature of Organic Chemistry: IUPAC Recommendations and Preferred Names. 2014. pp. 1159–1160. ISBN 9780854041824. Retrieved 11 September 2017.

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[1] "xi". New Oxford American Dictionary, 2nd Edition.

[2] SPIE Optipedia article: "Spatial Frequency"

[3] IUPAC Gold Book Entry: "Extent of Reaction" Archived June 29, 2012, at the Wayback Machine

[BuckinghamCooper2015-4] Buckingham, John; Cooper, Caroline M.; Purchase, Rupert (18 November 2015). Natural Products Desk Reference. CRC Press. pp. 53–. ISBN 978-1-4398-7362-5.

[5] Nomenclature of Organic Chemistry: IUPAC Recommendations and Preferred Names. 2014. pp. 1159–1160. ISBN 9780854041824. Retrieved 11 September 2017.

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