View source for Phi

{{short description|Twenty-first letter in the Greek alphabet}}
{{about|the letter of the [[Greek alphabet]]|other uses|Phi (disambiguation)|similar symbols|Ø (disambiguation)}}
{{pp-semi-indef}}

{{Greek Alphabet|letter=phi}}
[[File:Greek Phi archaic.svg|thumb|Archaic form of Phi]]

'''Phi''' ({{IPAc-en|'|f|aɪ|,_|"|f|i:|audio=LL-Q1860 (eng)-Flame, not lame-Phi.wav}} {{respell|FY|,_|FEE}};<ref>{{OED|phi}}</ref> uppercase '''Φ''', lowercase '''φ''' or '''ϕ'''; {{langx|grc|ϕεῖ}} ''pheî'' {{IPA|el|pʰéî̯|}}; [[Modern Greek]]: {{lang|el|φι}} ''fi'' {{IPA|el|fi|}}) is the twenty-first letter of the [[Greek alphabet]].

In [[Archaic Greek|Archaic]] and [[Classical Greek]] ({{circa}} 9th to 4th century BC), it represented an [[Aspiration (phonetics)|aspirated]] [[voiceless bilabial plosive]] ({{IPA|[pʰ]}}), which was the origin of its [[Romanization of Greek#Ancient Greek|usual romanization]] as {{angle bracket|ph}}. During the later part of [[Classical Antiquity]], in [[Koine Greek]] (c. 4th century BC to 4th century AD), its pronunciation shifted to a [[voiceless bilabial fricative]] ({{IPA|[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]] ({{IPA|[f]}}).
The [[Romanization of Greek#Modern Greek|romanization]] of the Modern Greek phoneme is therefore usually {{angle bracket|f}}.

It may be that phi originated as the letter [[qoppa]] (Ϙ, ϙ), and initially represented the sound {{IPA|/kʷʰ/}} before shifting to Classical Greek {{IPA|el|pʰ|}}.<ref>Brixhe, C. "History of the Alphabet", in Christidēs &&nbsp;al.'s ''A History of Ancient Greek''. 2007.</ref> In traditional [[Greek numerals]], phi has a value of 500 ({{lang|grc|φʹ}}) or 500,000 ({{lang|grc|͵φ}}). The Cyrillic letter [[Ef (Cyrillic)|Ef]] (Ф, ф) descends from phi.

Like other Greek letters, lowercase phi (encoded as the [[Unicode]] character {{unichar|03C6|Greek small letter phi}}) is used as a mathematical or scientific symbol. Some uses{{example needed|date=July 2023}} require the old-fashioned 'closed' glyph, which is separately encoded as the Unicode character {{unichar|03d5|Greek phi symbol}}.

== <span id="Functions">Use as a symbol</span> ==
{{More citations needed|date=August 2020}}

=== In lowercase ===
The lowercase letter φ (or its variant, ϕ or ɸ) is often used to represent the following:
* The letter phi is commonly used in [[physics]] to represent [[wave function]]s in [[quantum mechanics]], such as in the [[Schrödinger equation]] and [[bra–ket notation]]: <math>\langle\phi|\psi\rangle</math>.
* The [[golden ratio]] <math>\tfrac{1 + \sqrt{5}}{2} \approx</math>  {{val|1.618033988749894848204586834}}... in mathematics,<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Golden Ratio|url=https://mathworld.wolfram.com/GoldenRatio.html|access-date=2020-08-10|website=mathworld.wolfram.com|language=en}}</ref> art, and architecture. (Its reciprocal, 1/φ, is <math>\tfrac{ \sqrt{5} - 1}{2}</math> and is equal to φ − 1.)
* [[Euler's totient function]] ''φ''(''n'') in number theory;<ref>{{Cite web|title=Euler's Totient Function {{!}} Brilliant Math & Science Wiki|url=https://brilliant.org/wiki/eulers-totient-function/|access-date=2020-08-10|website=brilliant.org|language=en-us}}</ref> also called ''Euler's phi function''.
* The [[cyclotomic polynomial]] functions Φ<sub>''n''</sub>(''x'') of algebra.
* The number of [[electrical phase]]s in a power system in [[electrical engineering]], for example 1ϕ for [[single-phase electric power|single phase]], 3ϕ for [[three-phase electric power|three phase]].
* In algebra, [[group (mathematics)|group]] or [[ring (mathematics)|ring]] [[homomorphism]]s
* In [[probability theory]],  <math display="inline">\varphi(x)=(2\pi)^{-\frac{1}{2}}e^{-\frac{x^2}{2}}</math> is the [[probability density function]] of the [[standard normal distribution]].
* In [[probability theory]], {{nowrap|1=''φ''<sub>''X''</sub>(''t'') = E[''e''<sup>''itX''</sup>]}} is the [[characteristic function (probability theory)|characteristic function]] of a random variable ''X''.
* An [[angle]], typically the second angle mentioned, after ''θ'' ([[theta]]).<ref>{{Citation |last=Elert |first=Glenn |title=Special Symbols |date=2023 |work=The Physics Hypertextbook|quote='''ϴ''','''φ''',ϴ,φ angle, angular displacement, angular separation, rotation angle |url=https://physics.info/symbols/ |access-date=2025-02-01 |publisher=hypertextbook |language=en}}</ref> It can denote:
** The argument of a [[complex number]].
** The [[phase (waves)|phase of a wave]] in signal processing.
** In [[spherical coordinates]], mathematicians usually refer to ϕ as the polar angle (from the ''z''-[[Coordinate axis|axis]]). The convention in physics is to use ϕ as the azimuthal angle (from the ''x''-axis).
** One of the [[dihedral angle]]s in the backbones of [[protein]]s in a [[Ramachandran plot]]
** Internal or effective [[angle of friction]].
** In [[cartography]] and [[navigation]], ɸ denotes [[latitude]]; likewise, in [[geodesy]] it denotes [[geodetic latitude]].
** In [[ship stability]], ɸ represents the [[angle of list]].
** In [[aircraft flight mechanics]] as the symbol for [[bank angle]] (sometimes represented with the letter theta, which is also used for pitch angle).
* 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 [[Thermodynamic free energy|free energy]] destabilizations of protein [[mutant]]s in [[phi value analysis]].
* In combustion engineering, [[Air–fuel ratio#Fuel–air equivalence ratio (ϕ)|fuel–air equivalence ratio]]. The ratio between the actual fuel-air ratio to the stoichiometric fuel-air ratio.
* In granulometry, [[sedimentology]], and [[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 [[first-order 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 form ɸ).
* In [[philosophy]], φ is often used as shorthand for a [[generic act]]. (Also in uppercase.){{Citation needed|date=August 2018}}
* 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 linguistics, [[Phi_features|φ-features]] denote features such as case, number and gender in which adjectives and pronouns agree with nouns.
* 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]].
* An abbreviation for the word [[bacteriophage]]
** Mϕ is used as an abbreviation for the word [[macrophage]]

=== In uppercase ===
The uppercase Φ is used as a symbol for:
* The [[cumulative distribution function]] (cdf) of [[standard normal distribution]] in statistics.
* The [[magnetic flux]] and [[electric flux]] in physics, with subscripts distinguishing the two.
* [[Quantum yield]], the number of times a specific [[Event (particle physics)|event]] occurs per [[photon]] absorbed by a [[Quantum system|system]].<ref>{{Cite journal |last=Braslavsky |first=S. E. |date=2007-01-01 |title=Glossary of terms used in photochemistry, 3rd edition (IUPAC Recommendations 2006) |journal=Pure and Applied Chemistry |volume=79 |issue=3 |page=406 |doi=10.1351/pac200779030293 |issn=1365-3075 |s2cid=96601716 |quote=quantum yield Φ, Number of defined events occurring per photon absorbed by the system |doi-access=free}}</ref>
* In [[philosophy]], Φ is often used as shorthand for a [[generic act]]. (Also in lowercase.){{citation needed|date=February 2013}}
* 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.<ref>{{cite book|last=Evans|first=Dylans|title=An introductory dictionary of Lacanian psychoanalysis|page=145|isbn=978-0-415-13523-8|url=https://books.google.com/books?id=qwVVhLaiULEC&pg=PA145|publisher=Routledge|year=1996}}</ref>{{dubious|date=August 2018}}
* 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 of measurement|units]].
* A [[clock signal]] in electronics is often called Phi or uses the symbol.<ref>{{cite web | url=https://learnabout-electronics.org/Digital/dig51.php | title=Clock Circuits }}</ref>
* In emulsion and suspension science, Φ often denotes the volume fraction of the dispersed phase.<ref>{{Cite journal |last1=Rosti |first1=Marco E. |last2=Brandt |first2=Luca |last3=Mitra |first3=Dhrubaditya |date=2018-01-22 |title=Rheology of suspensions of viscoelastic spheres: Deformability as an effective volume fraction |url=https://link.aps.org/doi/10.1103/PhysRevFluids.3.012301 |journal=Physical Review Fluids |volume=3 |issue=1 |article-number=012301 |doi=10.1103/PhysRevFluids.3.012301|arxiv=1709.04210 |bibcode=2018PhRvF...3a2301R }}</ref>
* Archaically used in chemistry for [[phenyl group]].
* [[Integrated information theory]] (IIT), a mathematical model for consciousness

== Unicode ==
In [[Unicode]], there are multiple forms of the phi letter:
{| class="wikitable"
! Character !! Name !! Correct appearance !! Your browser !! [[LaTeX]] !! Usage
|-
| <code>{{U+|03A6}}</code> || {{nowrap|GREEK CAPITAL LETTER PHI}} || style="text-align: center;" | <math>\Phi\,\!</math> || style="font-size: larger; text-align: center;" | Φ || <math>\Phi</math> || Used in Greek texts
|-
| <code>U+03C6</code>|| GREEK SMALL LETTER PHI || style="text-align: center;" | {{nowrap|{{GrGl|phi Didot}} or {{GrGl|phi Porson}}}} || style="font-size: larger; text-align: center;" | φ || {{nowrap|<math>\varphi</math> or <math>\phi</math>}} || Used in Greek texts
|-
| <code>U+03D5</code>|| GREEK PHI SYMBOL || style="text-align: center;" | ϕ || style="font-size: larger; text-align: center;" | ϕ (''ϕ'') || <math>\phi</math> || Intended for use in modern (monotonic) Greek texts. Used italicized in mathematical and technical contexts when the "straight-line" variant glyph is preferred.<ref name="unicode-math">{{cite book | url = http://unicode.org/reports/tr25/#_Toc231 | title = UTR #25: Unicode support for mathematics | chapter = Representative Glyphs for Greek Phi | format = [[PDF]]}}</ref>
|-
| <code>U+0278</code>|| LATIN SMALL LETTER PHI || style="text-align: center;" | [[File:Xsampa-pslash.png|15px]] || style="font-size: larger; text-align: center;" | ɸ || <math></math> || Used in [[International Phonetic Alphabet|IPA]] to denote a [[voiceless bilabial fricative]]
|}

In ordinary Greek text, the character U+03C6 φ is used exclusively, though 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 (typeface)|Porson]] family (used widely in editions of classical Greek texts), have a "stroked" glyph in this position ({{GrGl|phi Porson}}), most other typefaces have "loopy" glyphs. This also applies to the "Didot" (or "''apla''") typefaces employed in most Greek book printing ({{GrGl|phi Didot}}), as well as the "Neohellenic" typeface often used for ancient texts ({{GrGl|phi Neohellenic}}).

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 <math>\varphi</math> at U+03D5.<ref name="unicode-math"/>

For use as a phonetic symbol in [[International Phonetic Alphabet|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 [[serif]]s at the top and bottom.

In [[LaTeX]], the math symbols are <code>\Phi</code> (<math>\Phi\,\!</math>), <code>\phi</code> (<math>\phi\,\!</math>), and <code>\varphi</code> (<math>\varphi\,\!</math>).

The Unicode standard includes the following variants of phi and phi-like characters:

* {{unichar|0278|html=}}
* {{unichar|03A6|html=}}
* {{unichar|03C6|html=}}
* {{unichar|03D5|html=}}
* {{unichar|1D60|html=}}
* {{unichar|1D69|html=}}
* {{unichar|1DB2|html=}}
* {{unichar|2C77|html=}}
* {{unichar|2CAA|html=}}
* {{unichar|2CAB|html=}}
* {{unichar|1D6BD|html=}}
* {{unichar|1D6D7|html=}}
* {{unichar|1D6DF|html=}}
* {{unichar|1D6F7|html=}}
* {{unichar|1D711|html=}}
* {{unichar|1D719|html=}}
* {{unichar|1D731|html=}}
* {{unichar|1D74B|html=}}
* {{unichar|1D753|html=}}
* {{unichar|1D76B|html=}}
* {{unichar|1D785|html=}}
* {{unichar|1D78D|html=}}
* {{unichar|1D7A5|html=}}
* {{unichar|1D7BF|html=}}
* {{unichar|1D7C7|html=}}

== <!-- Someone Please Add A "Charmap" Template With All The Mathematical Symbols -->See also ==
* [[F|F, f: Ef (Latin)]]
* [[Ef (Cyrillic)|Ф, ф: Ef (Cyrillic)]]
* [[Psi and phi type figurine]]
* [[Piwr (letter)|Փ (Armenian)]]

== References ==
{{reflist}}

== External links ==
* {{Wiktionary inline|Φ}}
* {{Wiktionary inline|φ}}
* {{Wiktionary inline|phi}}
* {{Commons category-inline}}

{{DEFAULTSORT:Phi (Letter)}}
[[Category:Greek letters]]
[[Category:Phonetic transcription symbols]]