Phi
Greek alphabet  



History  
Use in other languages  
Related topics  


Phi (uppercase Φ, lowercase or ; Ancient Greek: ϕεῖ, pheî, ; modern Greek: φι, fi, ; English: ^{[1]}) is the 21st letter of the Greek alphabet. In Ancient Greek, it represented an aspirated voiceless bilabial plosive ([pʰ]), which was the origin of its usual romanization as "ph". In modern Greek, it represents a voiceless labiodental fricative ([f]) and is correspondingly romanized as "f". Its origin is uncertain but it may be that phi originated as the letter qoppa and initially represented the sound /kʷʰ/ before shifting to Classical Greek .^{[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.
Phi is also used as a symbol for the golden ratio and on other occasions in math and science. This use is separately encoded as the Unicode glyph ϕ. The modern Greek pronunciation of the letter is sometimes encountered in English (as ) when the letter is being used in this sense.^{[3]}
Contents
 Use as a symbol 1
 Computing 2
 See also 3
 References 4
Use as a symbol
The lowercase letter φ (or often its variant, ϕ) is often used to represent the following:
 The golden ratio \tfrac{1 + \sqrt{5}}{2} \approx 033988749894848204586834... in mathematics, art, and architecture. 1.618
 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)^{−½}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 organic chemistry.
 The fugacity coefficient in thermodynamics.
 The ratio of free energy destabilizations of protein mutants in phi value analysis.
 In cartography and navigation, latitude.
 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 economics, this is usually an additive term.
 In flight dynamics, the roll angle.
 In philosophy, φ is often used as shorthand for a generic act. (Also in uppercase.)
 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.
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.)
 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.^{[4]}
The diameter symbol in engineering, ⌀, is often incorrectly referred to as "phi". 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  \Phi\,\!  Φ  used in Greek texts 
U+03C6  GREEK SMALL LETTER PHI  or  φ  used in Greek texts 
U+03D5  GREEK PHI SYMBOL  \phi\,\!  ϕ  used in mathematical and technical contexts^{[5]} 
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” form) and less often with a glyph more like the representative glyph shown for U+03D5 (ϕ, the “straight“ 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 "Porson" typefaces (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 \varphi at U+03D5.^{[5]}
For use as a phonetic symbol in IPA, Unicode has a separate codepoint 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\,\!), \phi (\phi\,\!), and \varphi (\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  횽 
U+1D6D7  MATHEMATICAL BOLD SMALL PHI  훗 
U+1D6DF  MATHEMATICAL BOLD PHI SYMBOL  훟 
U+1D6F7  MATHEMATICAL ITALIC CAPITAL PHI  훷 
U+1D711  MATHEMATICAL ITALIC SMALL PHI  휑 
U+1D719  MATHEMATICAL ITALIC PHI SYMBOL  휙 
U+1D731  MATHEMATICAL BOLD ITALIC CAPITAL PHI  휱 
U+1D74B  MATHEMATICAL BOLD ITALIC SMALL PHI  흋 
U+1D753  MATHEMATICAL BOLD ITALIC PHI SYMBOL  흓 
U+1D76B  MATHEMATICAL SANSSERIF BOLD CAPITAL PHI  흫 
U+1D785  MATHEMATICAL SANSSERIF BOLD SMALL PHI  힅 
U+1D78D  MATHEMATICAL SANSSERIF BOLD PHI SYMBOL  힍 
U+1D7A5  MATHEMATICAL SANSSERIF BOLD ITALIC CAPITAL PHI  
U+1D7BF  MATHEMATICAL SANSSERIF BOLD ITALIC SMALL PHI  ힿ 
U+1D7C7  MATHEMATICAL SANSSERIF BOLD ITALIC PHI SYMBOL  
See also
 F, f: Ef (Latin)
 Ф, ф: Ef (Cyrillic)
 Psi and phi type figurine
 Փ (Armenian letter)
 Deposition (geology)
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.
 ^ See, e.g., The Da Vinci Code and the Criminal Minds episode, "Masterpiece".
 ^ Evans, Dylans (1996). An introductory dictionary of Lacanian psychoanalysis. Routledge. p. 145.
 ^ ^{a} ^{b} "Representative Glyphs for Greek Phi". UTR #25: Unicode support for mathematics (
 WorldHeritage pages with incorrect protection templates
 Articles containing Ancient Greeklanguage text
 Articles containing Greeklanguage text
 WorldHeritage articles needing clarification from October 2009
 All articles with unsourced statements
 Articles with unsourced statements from February 2013
 Greek letters