Phi
Greek alphabet  



History  
Use in other languages  
Related topics  
Phi (/faɪ/;^{[1]} uppercase Φ, lowercase φ or ϕ; Ancient Greek: ϕεῖ pheî [pʰé͜e]; 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 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 is used as a mathematical or scientific symbol. Some uses, such as the golden ratio, require the oldfashioned 'closed' glyph, which is separately encoded as the Unicode character U+03D5 ϕ GREEK PHI SYMBOL.
Use as a symbol
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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, art, and architecture.
 Euler's totient function φ(n) in number theory; also called Euler's phi function.
 The cyclotomic polynomial functions Φ_{n}(x) of algebra.
 In algebra, group or ring homomorphisms
 In probability theory, ϕ(x) = (2π)^{−1⁄2}e^{−x²/2} is the probability density function of the 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). Especially:
 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 fuel air ratio to the stoichiometric fuel air ratio.
 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 straight line variant character)
 In flight dynamics, the roll angle.
 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 magnetic flux and electric flux in physics, with subscripts distinguishing the two.
 The cumulative distribution function of the normal distribution in mathematics and statistics.
 In philosophy, Φ is often used as shorthand for a generic act. (Also in lowercase.)^{[citation needed]}
 The number of phases in a power system in electrical engineering, for example 1Φ for single phase, 3Φ for three phase.
 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.^{[3]}^{[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.
Computing
In Unicode, there are multiple forms of the phi letter:
Character  Name  Correct appearance  Your browser  Usage 

U+03A6  GREEK CAPITAL LETTER PHI  Φ  Used in Greek texts  
U+03C6  GREEK SMALL LETTER PHI  or  φ  Used in Greek texts 
U+03D5  GREEK PHI SYMBOL  ϕ (ϕ)  Used in mathematical and technical contexts.^{[4]} (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). Unicode makes an effort to distinguish the two by generally calling the loopy form "small letter phi" or "small phi", and by calling the stroked form "phi symbol", but this isn't exclusively true on all variants.
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 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 goes for the "Didot" (or "apla") typefaces employed in most Greek book printing (), as well as for 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.^{[4]}
For use as a phonetic symbol in IPA, Unicode has a separate code point U+0278, LATIN SMALL LETTER PHI, because in this use only the stroked glyph is considered correct. 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:
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
References
 ^ Oxford English Dictionary, 3rd ed. "phi, n." Oxford University Press (Oxford), 2005.
 ^ Brixhe, C. "History of the Alphabet", in Christidēs & al.'s A History of Ancient Greek. 2007.
 ^ Evans, Dylans (1996). An introductory dictionary of Lacanian psychoanalysis. Routledge. p. 145. ISBN 9780415135238.
 ^ ^{a} ^{b} "Representative Glyphs for Greek Phi". UTR #25: Unicode support for mathematics (PDF).
External links
 The dictionary definition of φ at Wiktionary