This article is about the letter of the Greek alphabet. For the irrational number found in nature, see Golden ratio. For other uses, see Phi (disambiguation).
Phi (/faɪ/; 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, 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 (final centuries BC), its pronunciation shifted to that of 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]. 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 old-fashioned 'closed' glyph, which is separately encoded as the Unicode character U+03D5 ϕ GREEK PHI SYMBOL.
7000161803398874989♠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 Φn(x) of algebra.
In probability theory, ϕ(x) = (2π)−½e−x²/2 is the probability density function of the normal distribution.
In probability theory, φX(t) = E[eitX] is the characteristic function of a random variable X.
The phase of a wave in signal processing.
In spherical coordinates, mathematicians usually refer to phi as the polar angle (from the z-axis). The convention in physics is to use phi as the azimuthal angle (from the x-axis).
Internal or effective angle of friction.
The work function of a surface, in solid-state 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 first-order logic.
Porosity in geology and hydrology.
Strength (or resistance) reduction factor in structural engineering, used to account for statistical variabilities in materials and construction methods.
In flight dynamics, the roll angle.
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 lexical-functional grammar, the function that maps elements from the c-structure to the f-structure.
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.
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.
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.
The diameter symbol in engineering, ⌀, is often erroneously 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.
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.
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 &Phi; (Φ) and &phi; (φ), respectively.
Look up Φ, φ, or phi in Wiktionary, the free dictionary.
^ Oxford English Dictionary, 3rd ed. "phi, n." Oxford University Press (Oxford), 2005.
^ Evans, Dylans (1996). An introductory dictionary of Lacanian psychoanalysis. Routledge. p. 145. ISBN 978-0-415-13523-8.
^ a b "Representative Glyphs for Greek Phi". UTR #25: Unicode support for mathematics (PDF).