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| {{for|similar symbols|Ø (disambiguation)}}
| | Not much to tell about me I think.<br>Yes! Im a part of this community.<br>I just wish Im useful at all<br><br>Feel free to surf to my page; [http://hemorrhoidtreatmentfix.com/prolapsed-hemorrhoid-treatment prolapsed hemorrhoids treatment] |
| {{pp-semi-indef}}
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| {{about|the letter of the Greek alphabet}}
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| {{Greek Alphabet|letter=phi}}
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| '''Phi''' (uppercase Φ, lowercase φ, or math symbol {{Unicode|ϕ}}, also '''fi'''), pronounced {{IPAc-en|'|f|aɪ}} {{respell|FY|'}} or sometimes {{IPAc-en|'|f|iː}} {{respell|FEE|'}} in English,<ref>{{IPA|[faɪ]}}: ''Collins English Dictionary'', 3rd ed. (1991); ''[[New Oxford American Dictionary]]'', 2nd ed. (2005) (transcribed "{{IPA|[fʌɪ]}} "). {{IPA|[fiː]}} is used increasingly in the media, especially when representing the [[golden ratio]]: see, for example, ''[[The Da Vinci Code (film)|The Da Vinci Code]]'' and the ''[[Criminal Minds]]'' episode, "Masterpiece".</ref> and {{IPA-el|ˈfi|}} in [[Modern Greek language|modern Greek]], is the 21st letter of the [[Greek alphabet]]. In modern Greek, it represents {{IPA|[f]}}, a [[voiceless labiodental fricative]]. In Ancient Greek, it represented {{IPA|[pʰ]}}, an [[Aspiration (phonetics)|aspirated]] [[voiceless bilabial plosive]] (from which English ultimately inherits the spelling "[[ph (digraph)|ph]]" in words derived from Greek). In the system of [[Greek numerals]], it has a value of 500 (φʹ) or 500,000 (͵φ). The Cyrillic letter [[Ef (Cyrillic)|Ef]] (Ф, ф) arose from Φ.
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| == Use as a symbol ==
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| The lower-case letter φ (or often its variant, {{Unicode|ϕ}}) is often used to represent the following:
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| *The [[golden ratio]] <math>\tfrac{1 + \sqrt{5}}{2} \approx</math> 1.618033988749894848204586834... in mathematics, art, and architecture.
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| *[[Euler's totient function]] ''φ''(''n'') in number theory; also called ''Euler's phi function''.
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| *In algebra, [[Group (mathematics)|group]] or [[Ring (mathematics)|ring]] [[homomorphism]]s
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| * In [[probability theory]], {{nowrap|1=''ϕ''(''x'') = (2π)<sup>−½</sup>''e''<sup>−''x''<sup>2</sup>/2</sup>}} is the [[probability density function]] of the [[normal distribution]].
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| * 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''.
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| *An [[angle]], typically the second angle mentioned, after ''θ'' ([[theta]]). Especially:
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| **The argument of a [[complex number]].
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| **The [[phase (waves)|phase of a wave]] in signal processing.
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| **In [[spherical coordinates]], mathematicians usually refer to phi as the polar angle (from the ''z''-[[Coordinate axis|axis]]). The convention in physics is to use phi as the azimuthal angle (from the ''x''-axis).
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| **One of the [[dihedral angle]]s in the backbones of [[protein]]s in a [[Ramachandran plot]]
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| **Internal or effective [[angle of friction]].
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| *The [[work function]] of a surface, in [[solid-state physics]].
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| *A [[shorthand]] representation for an [[aromatic]] [[functional group]] in [[organic chemistry]].
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| *The [[fugacity]] coefficient in [[thermodynamics]].
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| *The ratio of [[Thermodynamic free energy|free energy]] destabilizations of protein [[mutant]]s in [[phi value analysis]].
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| *In [[cartography]] and [[navigation]], [[latitude]].
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| *In combustion engineering, equivalence ratio. The ratio between the actual fuel air ratio to the stoichiometric fuel air ratio.
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| *A sentence in [[first-order logic]].
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| *[[Porosity]] in geology and [[hydrology]].
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| *Strength (or resistance) reduction factor in [[structural engineering]], used to account for statistical variabilities in materials and construction methods.
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| *The symbol for a [[voiceless bilabial fricative]] in the [[International Phonetic Alphabet]] (using the straight line variant character)
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| *In economics, this is usually an additive term.{{Clarify|date=October 2009}}
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| *In [[flight dynamics]], the roll angle.
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| *In [[philosophy]], φ is often used as shorthand for a generic act. (Also in upper-case.)
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| The upper-case letter Φ is used as a symbol for:
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| *The [[Golden_ratio#Golden_ratio_conjugate|golden ratio conjugate]] −0.618... in mathematics.
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| *The [[magnetic flux]] and [[electric flux]] in physics, with subscripts distinguishing the two.
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| *The [[cumulative distribution function]] of the [[normal distribution]] in [[mathematics]] and [[statistics]].
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| *In [[philosophy]], Φ is often used as shorthand for a generic act. (Also in lower-case.){{fact|date=February 2013}}
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| *The number of phases in a power system in [[electrical engineering]], for example 1Φ for [[single phase]], 3Φ for [[three phase]].
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| *A common symbol for the parametrization of a surface in vector [[calculus]].
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| *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|pages=145|isbn=978-0-415-13523-8|url=http://books.google.com/?id=qwVVhLaiULEC&pg=PA145|publisher=Routledge|year=1996}}</ref>
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| The [[diameter symbol]] in [[engineering]], [[⌀]], is often {{Citation needed span|text=incorrectly|date=February 2012}} 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 of measurement|units]].
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| == Computing ==
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| In [[Unicode]], there are multiple forms of the phi letter:
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| {| class="wikitable"
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| ! Character !! Name !! Correct appearance !! Your browser !! Usage
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| | {{U+|03A6}} || GREEK CAPITAL LETTER PHI || style="text-align: center;" | <math>\Phi\,\!</math> || style="font-size: larger; text-align: center;" | {{Unicode|Φ}} || used in Greek texts
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| | U+03C6 || GREEK SMALL LETTER PHI || style="text-align: center;" | {{GrGl|phi Didot}} or {{GrGl|phi Porson}} || style="font-size: larger; text-align: center;" | {{Unicode|φ}} || used in Greek texts
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| | U+03D5 || GREEK PHI SYMBOL || style="text-align: center;" | <math>\phi\,\!</math> || style="font-size: larger; text-align: center;" | {{Unicode|ϕ}} || used in mathematical and technical contexts<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>
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| | U+0278 || LATIN SMALL LETTER PHI || style="text-align: center;" | [[File:Xsampa-pslash.png|15px]] || style="font-size: larger; text-align: center;" | {{Unicode|ɸ}} || used in [[IPA]] to symbolise a [[voiceless bilabial fricative]]
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| |}
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| 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).
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| 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 ({{GrGl|phi Porson}}), most other typefaces have "loopy" glyphs. This goes for the "Didot" (or "''apla''") typefaces employed in most Greek book printing ({{GrGl|phi Didot}}), as well as for the "Neohellenic" typeface often used for ancient texts ({{GrGl|phi Neohellenic}}).
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| 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"/>
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| 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 [[serif]]s at the top and bottom.
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| In HTML/[[XHTML]], the upper- and lower-case phi [[character entity reference]]s are <tt>&Phi;</tt> (Φ) and <tt>&phi;</tt> (φ), respectively.
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| In [[LaTeX]], the math symbols are <tt>\Phi</tt> (<math>\Phi\,\!</math><!-- doesn't display correctly? -->), <tt>\phi</tt> (<math>\phi\,\!</math>), and <tt>\varphi</tt> (<math>\varphi\,\!</math>).
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| The Unicode standard also includes the following variants of phi and phi-like characters:
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| {| class="wikitable"
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| ! Character !! Name !! Appearance
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| | U+1D60 || MODIFIER LETTER SMALL GREEK PHI || style="text-align: center" | {{Unicode|ᵠ}}
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| | U+1D69 || GREEK SUBSCRIPT SMALL LETTER PHI || style="text-align: center" | {{Unicode|ᵩ}}
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| | U+1DB2 || MODIFIER LETTER SMALL PHI || style="text-align: center" | {{Unicode|ᶲ}}
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| | U+2CAA || COPTIC CAPITAL LETTER FI || style="text-align: center" | {{Unicode|Ⲫ}}
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| | U+2CAB || COPTIC SMALL LETTER FI || style="text-align: center" | {{Unicode|ⲫ}}
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| | U+2C77 || LATIN SMALL LETTER TAILLESS PHI || style="text-align: center" | {{Unicode|ⱷ}}
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| | U+1D6BD || MATHEMATICAL BOLD CAPITAL PHI || style="text-align: center" | {{Unicode|𝚽}}
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| | U+1D6D7 || MATHEMATICAL BOLD SMALL PHI || style="text-align: center" | {{Unicode|𝛗}}
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| | U+1D6DF || MATHEMATICAL BOLD PHI SYMBOL || style="text-align: center" | {{Unicode|𝛟}}
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| | U+1D6F7 || MATHEMATICAL ITALIC CAPITAL PHI || style="text-align: center" | {{Unicode|𝛷}}
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| | U+1D711 || MATHEMATICAL ITALIC SMALL PHI || style="text-align: center" | {{Unicode|𝜑}}
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| | U+1D719 || MATHEMATICAL ITALIC PHI SYMBOL || style="text-align: center" | {{Unicode|𝜙}}
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| | U+1D731 || MATHEMATICAL BOLD ITALIC CAPITAL PHI || style="text-align: center" | {{Unicode|𝜱}}
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| | U+1D74B || MATHEMATICAL BOLD ITALIC SMALL PHI || style="text-align: center" | {{Unicode|𝝋}}
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| | U+1D753 || MATHEMATICAL BOLD ITALIC PHI SYMBOL || style="text-align: center" | {{Unicode|𝝓}}
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| | U+1D76B || MATHEMATICAL SANS-SERIF BOLD CAPITAL PHI || style="text-align: center" | {{Unicode|𝝫}}
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| | U+1D785 || MATHEMATICAL SANS-SERIF BOLD SMALL PHI || style="text-align: center" | {{Unicode|𝞅}}
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| | U+1D78D || MATHEMATICAL SANS-SERIF BOLD PHI SYMBOL || style="text-align: center" | {{Unicode|𝞍}}
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| | U+1D7A5 || MATHEMATICAL SANS-SERIF BOLD ITALIC CAPITAL PHI || style="text-align: center" | {{Unicode|𝞥}}
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| | U+1D7BF || MATHEMATICAL SANS-SERIF BOLD ITALIC SMALL PHI || style="text-align: center" | {{Unicode|𝞿}}
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| | U+1D7C7 || MATHEMATICAL SANS-SERIF BOLD ITALIC PHI SYMBOL || style="text-align: center" | {{Unicode|𝟇}}
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| |}
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| == See also ==
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| {{Wiktionary|Φ|φ|phi}}
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| *[[F|F, f: Ef (Latin)]]
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| *[[Ef (Cyrillic)|Ф, ф: Ef (Cyrillic)]]
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| *[[Psi and phi type figurine]]
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| * [[Փ]] ([[Armenian alphabet|Armenian letter]])
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| *[[Deposition (geology)]]
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| == References ==
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| {{reflist}}
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| {{DEFAULTSORT:Phi (Letter)}}
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| [[Category:Greek letters]]
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