Phi
Greek alphabet  



History  
Use in other languages  
Related topics  
Phi (/faɪ/;^{[1]} uppercase Φ, lowercase φ or ϕ; Ancient Greek: ϕεῖ pheî [pʰéî̯]; Modern Greek: φι fi [fi]) is the 21st letter of the Greek alphabet.
In Archaic and Classical Greek (c. 9th century BC to 4th century BC), it represented an aspirated voiceless bilabial plosive ([pʰ]), which was the origin of its usual romanization as ⟨ph⟩. During the later part of Classical Antiquity, in Koine Greek (c. 4th century BC to 4th century AD), its pronunciation shifted to that of a voiceless bilabial fricative ([ɸ]), and by the Byzantine Greek period (c. 4th century AD to 15th century AD) it developed its modern pronunciation as a voiceless labiodental fricative ([f]). The romanization of the Modern Greek phoneme is therefore usually ⟨f⟩.
It may be that phi originated as the letter qoppa (Ϙ, ϙ), and initially represented the sound /kʷʰ/ before shifting to Classical Greek [pʰ].^{[2]} In traditional Greek numerals, phi has a value of 500 (φʹ) or 500,000 (͵φ). The Cyrillic letter Ef (Ф, ф) descends from phi.
As with other Greek letters, lowercase phi (encoded as the Unicode character U+03C6 φ GREEK SMALL LETTER PHI) is used as a mathematical or scientific symbol. Some uses, such as the golden ratio,^{[contradictory]} require the oldfashioned 'closed' glyph, which is separately encoded as the Unicode character U+03D5 ϕ GREEK PHI SYMBOL.
Use as a symbol
This article needs additional citations for verification. (August 2020) 
The lowercase letter φ (or often its variant, ϕ) is often used to represent the following:
 Magnetic flux in physics
 The letter phi is commonly used in physics to represent wave functions in quantum mechanics, such as in the Schrödinger equation and bra–ket notation: .
 The golden ratio 1.618033988749894848204586834... in mathematics,^{[3]} art, and architecture.
 Euler's totient function φ(n) in number theory;^{[4]} also called Euler's phi function.
 The cyclotomic polynomial functions Φ_{n}(x) of algebra.
 The number of phases in a power system in electrical engineering, for example 1ϕ for single phase, 3ϕ for three phase.
 In algebra, group or ring homomorphisms
 In probability theory, ϕ(x) = (2π)^{−.mwparseroutput .frac{whitespace:nowrap}.mwparseroutput .frac .num,.mwparseroutput .frac .den{fontsize:80%;lineheight:0;verticalalign:super}.mwparseroutput .frac .den{verticalalign:sub}.mwparseroutput .sronly{border:0;clip:rect(0,0,0,0);height:1px;margin:1px;overflow:hidden;padding:0;position:absolute;width:1px}1⁄2}e^{−x2/2} is the probability density function of the standard normal distribution.
 In probability theory, φ_{X}(t) = E[e^{itX}] is the characteristic function of a random variable X.
 An angle, typically the second angle mentioned, after θ (theta). It can denote:
 The argument of a complex number.
 The phase of a wave in signal processing.
 In spherical coordinates, mathematicians usually refer to phi as the polar angle (from the zaxis). The convention in physics is to use phi as the azimuthal angle (from the xaxis).
 One of the dihedral angles in the backbones of proteins in a Ramachandran plot
 Internal or effective angle of friction.
 The work function of a surface, in solidstate physics.
 A shorthand representation for an aromatic functional group in organic chemistry.
 The fugacity coefficient in thermodynamics.
 The ratio of free energy destabilizations of protein mutants in phi value analysis.
 In cartography, geodesy and navigation, latitude.
 In aircraft flight mechanics as the symbol for bank angle (sometimes represented with the letter theta, which is also used for pitch angle).
 In combustion engineering, fuel–air equivalence ratio. The ratio between the actual fuelair ratio to the stoichiometric fuelair ratio.
 In soil engineering, φ is a logarithmic unit of sediment grain size, defined such that a change of 1 φ in grain size corresponds to a factor of 2 in grain diameter.
 A sentence in firstorder logic.
 The Veblen function in set theory.
 Porosity in geology and hydrology.
 Strength (or resistance) reduction factor in structural engineering, used to account for statistical variabilities in materials and construction methods.
 The symbol for a voiceless bilabial fricative in the International Phonetic Alphabet (using the straightline variant form).
 In philosophy, φ is often used as shorthand for a generic act. (Also in uppercase.)^{[citation needed]}
 In perceptual psychology, the phi phenomenon is the apparent motion caused by the successive viewing of stationary objects, such as the frames of a motion picture.
 In lexicalfunctional grammar, the function that maps elements from the cstructure to the fstructure.
 In ecology, site survival probability, or the probability that a species will continue to occupy a site if it was there the previous year.
 The logo of La France Insoumise, a leftist French political party.
 An abbreviation for the word bacteriophage
 Mϕ is used as an abbreviation for the word macrophage
The uppercase letter Φ is used as a symbol for:
 The golden ratio conjugate −0.618... in mathematics.
 The cumulative distribution function (cdf) of standard normal distribution in statistics.
 The magnetic flux and electric flux in physics, with subscripts distinguishing the two.
 In philosophy, Φ is often used as shorthand for a generic act. (Also in lowercase.)^{[citation needed]}
 A common symbol for the parametrization of a surface in vector calculus.
 In Lacanian algebra, Φ stands for the imaginary phallus and also represents phallic signification; −Φ stands in for castration.^{[5]}^{[dubious – discuss]}
The diameter symbol in engineering, ⌀, is often erroneously referred to as "phi", and the diameter symbol is sometimes erroneously typeset as Φ. This symbol is used to indicate the diameter of a circular section; for example, "⌀14" means the diameter of the circle is 14 units.
 A clock signal in electronics is often called Phi or uses the symbol.^{[6]}
Computing
In Unicode, there are multiple forms of the phi letter:
Character  Name  Correct appearance  Your browser  LaTeX  Usage 

U+03A6  GREEK CAPITAL LETTER PHI  Φ  Used in Greek texts  
U+03C6  GREEK SMALL LETTER PHI  or  φ  or  Used in Greek texts 
U+03D5  GREEK PHI SYMBOL  ϕ (ϕ)  Used in mathematical and technical contexts.^{[7]} (Italicized.)  
U+0278  LATIN SMALL LETTER PHI  ɸ  Used in IPA to symbolise a voiceless bilabial fricative 
In ordinary Greek text, the character U+03C6 φ is used exclusively, although this character has considerable glyphic variation, sometimes represented with a glyph more like the representative glyph shown for U+03C6 (φ, the "loopy" or "open" form), and less often with a glyph more like the representative glyph shown for U+03D5 (ϕ, the "stroked" or "closed" form).
Because Unicode represents a character in an abstract way, the choice between glyphs is purely a matter of font design. While some Greek typefaces, most notably those in the Porson family (used widely in editions of classical Greek texts), have a "stroked" glyph in this position (), most other typefaces have "loopy" glyphs. This also applies to the "Didot" (or "apla") typefaces employed in most Greek book printing (), as well as the "Neohellenic" typeface often used for ancient texts ().
It is necessary to have the stroked glyph available for some mathematical uses, and U+03D5 GREEK PHI SYMBOL is designed for this function. Prior to Unicode version 3.0 (1998), the glyph assignments in the Unicode code charts were the reverse, and thus older fonts may still show a loopy form at U+03D5.^{[7]}
For use as a phonetic symbol in IPA, Unicode has a separate code point U+0278, LATIN SMALL LETTER PHI, because only the stroked glyph is considered correct in this use. It typically appears in a form adapted to a Latin typographic environment, with a more upright shape than normal Greek letters and with serifs at the top and bottom.
In HTML/XHTML, the upper and lowercase phi character entity references are Φ
(Φ) and φ
(φ), respectively.
In LaTeX, the math symbols are \Phi
(), \phi
(), and \varphi
().
The Unicode standard also includes the following variants of phi and philike characters:
Preview  Φ  φ  Ⲫ  ⲫ  

Unicode name  GREEK CAPITAL LETTER PHI  GREEK SMALL LETTER PHI  COPTIC CAPITAL LETTER FI  COPTIC SMALL LETTER FI  
Encodings  decimal  hex  dec  hex  dec  hex  dec  hex 
Unicode  934  U+03A6  966  U+03C6  11434  U+2CAA  11435  U+2CAB 
UTF8  206 166  CE A6  207 134  CF 86  226 178 170  E2 B2 AA  226 178 171  E2 B2 AB 
Numeric character reference  Φ 
Φ 
φ 
φ 
Ⲫ 
Ⲫ 
ⲫ 
ⲫ 
Named character reference  Φ  φ 
Preview  ɸ  ⱷ  

Unicode name  LATIN SMALL LETTER PHI  LATIN SMALL LETTER TAILLESS PHI  
Encodings  decimal  hex  dec  hex 
Unicode  632  U+0278  11383  U+2C77 
UTF8  201 184  C9 B8  226 177 183  E2 B1 B7 
Numeric character reference  ɸ 
ɸ 
ⱷ 
ⱷ 
Character  Name  Appearance 

U+1D60  MODIFIER LETTER SMALL GREEK PHI  ᵠ 
U+1D69  GREEK SUBSCRIPT SMALL LETTER PHI  ᵩ 
U+1DB2  MODIFIER LETTER SMALL PHI  ᶲ 
U+2CAA  COPTIC CAPITAL LETTER FI  Ⲫ 
U+2CAB  COPTIC SMALL LETTER FI  ⲫ 
U+2C77  LATIN SMALL LETTER TAILLESS PHI  ⱷ 
U+1D6BD  MATHEMATICAL BOLD CAPITAL PHI  𝚽 
U+1D6D7  MATHEMATICAL BOLD SMALL PHI  𝛗 
U+1D6DF  MATHEMATICAL BOLD PHI SYMBOL  𝛟 
U+1D6F7  MATHEMATICAL ITALIC CAPITAL PHI  𝛷 
U+1D711  MATHEMATICAL ITALIC SMALL PHI  𝜑 
U+1D719  MATHEMATICAL ITALIC PHI SYMBOL  𝜙 
U+1D731  MATHEMATICAL BOLD ITALIC CAPITAL PHI  𝜱 
U+1D74B  MATHEMATICAL BOLD ITALIC SMALL PHI  𝝋 
U+1D753  MATHEMATICAL BOLD ITALIC PHI SYMBOL  𝝓 
U+1D76B  MATHEMATICAL SANSSERIF BOLD CAPITAL PHI  𝝫 
U+1D785  MATHEMATICAL SANSSERIF BOLD SMALL PHI  𝞅 
U+1D78D  MATHEMATICAL SANSSERIF BOLD PHI SYMBOL  𝞍 
U+1D7A5  MATHEMATICAL SANSSERIF BOLD ITALIC CAPITAL PHI  𝞥 
U+1D7BF  MATHEMATICAL SANSSERIF BOLD ITALIC SMALL PHI  𝞿 
U+1D7C7  MATHEMATICAL SANSSERIF BOLD ITALIC PHI SYMBOL  𝟇 
See also
 F, f: Ef (Latin)
 Ф, ф: Ef (Cyrillic)
 中
 Psi and phi type figurine
 Փ (Armenian)
 Deposition (geology)
 Golden ratio
References
 ^ "phi". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.)
 ^ Brixhe, C. "History of the Alphabet", in Christidēs & al.'s A History of Ancient Greek. 2007.
 ^ Weisstein, Eric W. "Golden Ratio". mathworld.wolfram.com. Retrieved 20200810.
 ^ "Euler's Totient Function  Brilliant Math & Science Wiki". brilliant.org. Retrieved 20200810.
 ^ Evans, Dylans (1996). An introductory dictionary of Lacanian psychoanalysis. Routledge. p. 145. ISBN 9780415135238.
 ^ "Clock Circuits".
 ^ ^{a} ^{b} "Representative Glyphs for Greek Phi". UTR #25: Unicode support for mathematics (PDF).
External links
 The dictionary definition of Φ at Wiktionary
 The dictionary definition of φ at Wiktionary
 The dictionary definition of phi at Wiktionary
 Media related to Phi (letter) at Wikimedia Commons