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Phi

Phi (/f/;[1] uppercase Φ, lowercase φ or ϕ; Ancient Greek: ϕεῖ pheî [pʰéî̯]; Modern Greek: φι fi [fi]) is the 21st letter of the Greek alphabet.

Archaic form of Phi

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 (Ϙ, ϙ), 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 (encoded as the Unicode character U+03C6 φ GREEK SMALL LETTER PHI) is used as a mathematical or scientific symbol. Some uses, such as the golden ratio,[contradictory] require the old-fashioned 'closed' glyph, which is separately encoded as the Unicode character U+03D5 ϕ GREEK PHI SYMBOL.

Use as a symbol

The lowercase letter φ (or often its variant, ϕ) is often used to represent the following:

The uppercase letter Φ is used as a symbol for:

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 LaTeX Usage
U+03A6 GREEK CAPITAL LETTER PHI   Φ   Used in Greek texts
U+03C6 GREEK SMALL LETTER PHI   or   φ   or   Used in Greek texts
U+03D5 GREEK PHI SYMBOL   ϕ (ϕ)   Used in mathematical and technical contexts.[7] (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).

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 family (used widely in editions of classical Greek texts), have a "stroked" glyph in this position ( ), most other typefaces have "loopy" glyphs. This also applies to the "Didot" (or "apla") typefaces employed in most Greek book printing ( ), as well as 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.[7]

For use as a phonetic symbol in 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 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 phi-like characters:


Character information
Preview Φ φ
Unicode name GREEK CAPITAL LETTER PHI GREEK SMALL LETTER PHI COPTIC CAPITAL LETTER FI COPTIC SMALL LETTER FI
Encodings decimal hex dec hex dec hex dec hex
Unicode 934 U+03A6 966 U+03C6 11434 U+2CAA 11435 U+2CAB
UTF-8 206 166 CE A6 207 134 CF 86 226 178 170 E2 B2 AA 226 178 171 E2 B2 AB
Numeric character reference Φ Φ φ φ Ⲫ Ⲫ ⲫ ⲫ
Named character reference Φ φ
Character information
Preview ɸ
Unicode name LATIN SMALL LETTER PHI LATIN SMALL LETTER TAILLESS PHI
Encodings decimal hex dec hex
Unicode 632 U+0278 11383 U+2C77
UTF-8 201 184 C9 B8 226 177 183 E2 B1 B7
Numeric character reference ɸ ɸ ⱷ ⱷ
Character Name Appearance
U+1D60 MODIFIER LETTER SMALL GREEK PHI
U+1D69 GREEK SUBSCRIPT SMALL LETTER PHI
U+1DB2 MODIFIER LETTER SMALL PHI
U+2CAA COPTIC CAPITAL LETTER FI
U+2CAB COPTIC SMALL LETTER FI
U+2C77 LATIN SMALL LETTER TAILLESS PHI
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 SANS-SERIF BOLD CAPITAL PHI 𝝫
U+1D785 MATHEMATICAL SANS-SERIF BOLD SMALL PHI 𝞅
U+1D78D MATHEMATICAL SANS-SERIF BOLD PHI SYMBOL 𝞍
U+1D7A5 MATHEMATICAL SANS-SERIF BOLD ITALIC CAPITAL PHI 𝞥
U+1D7BF MATHEMATICAL SANS-SERIF BOLD ITALIC SMALL PHI 𝞿
U+1D7C7 MATHEMATICAL SANS-SERIF BOLD ITALIC PHI SYMBOL 𝟇

See also

References

  1. ^ "phi". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.)
  2. ^ Brixhe, C. "History of the Alphabet", in Christidēs & al.'s A History of Ancient Greek. 2007.
  3. ^ Weisstein, Eric W. "Golden Ratio". mathworld.wolfram.com. Retrieved 2020-08-10.
  4. ^ "Euler's Totient Function | Brilliant Math & Science Wiki". brilliant.org. Retrieved 2020-08-10.
  5. ^ Evans, Dylans (1996). An introductory dictionary of Lacanian psychoanalysis. Routledge. p. 145. ISBN 978-0-415-13523-8.
  6. ^ "Clock Circuits".
  7. ^ a b "Representative Glyphs for Greek Phi". UTR #25: Unicode support for mathematics (PDF).

External links

  •   The dictionary definition of Φ at Wiktionary
  •   The dictionary definition of φ at Wiktionary
  •   The dictionary definition of phi at Wiktionary
  •   Media related to Phi (letter) at Wikimedia Commons

Phi
similar, symbols, disambiguation, this, article, about, letter, greek, alphabet, irrational, number, found, nature, golden, ratio, other, uses, disambiguation, uppercase, lowercase, ancient, greek, ϕεῖ, pheî, pʰéî, modern, greek, φι, 21st, letter, greek, alpha. For similar symbols see O disambiguation 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 f aɪ 1 uppercase F lowercase f or ϕ Ancient Greek ϕeῖ phei pʰei Modern Greek fi fi fi is the 21st letter of the Greek alphabet Archaic form of Phi 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 Ϙ ϙ and initially represented the sound kʷʰ before shifting to Classical Greek pʰ 2 In traditional Greek numerals phi has a value of 500 fʹ or 500 000 f The Cyrillic letter Ef F f descends from phi As with other Greek letters lowercase phi encoded as the Unicode character U 03C6 f GREEK SMALL LETTER PHI is used as a mathematical or scientific symbol Some uses such as the golden ratio contradictory require the old fashioned closed glyph which is separately encoded as the Unicode character U 03D5 ϕ GREEK PHI SYMBOL Contents 1 Use as a symbol 2 Computing 3 See also 4 References 5 External linksUse as a symbolThis article needs additional citations for verification Please help improve this article by adding citations to reliable sources Unsourced material may be challenged and removed Find sources Phi news newspapers books scholar JSTOR August 2020 Learn how and when to remove this template message The lowercase letter f 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 Schrodinger equation and bra ket notation ϕ ps displaystyle langle phi psi rangle The golden ratio 1 5 2 displaystyle tfrac 1 sqrt 5 2 approx 1 618033 988 749 894 848 204 586 834 in mathematics 3 art and architecture Euler s totient function f n in number theory 4 also called Euler s phi function The cyclotomic polynomial functions Fn x of algebra The number of phases in a power system in electrical engineering for example 1ϕ for single phase 3ϕ for three phase In algebra group or ring homomorphisms In probability theory ϕ x 2p 1 2e x2 2 is the probability density function of the standard normal distribution In probability theory fX t E eitX is the characteristic function of a random variable X An angle typically the second angle mentioned after 8 theta It can denote 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 z axis The convention in physics is to use phi as the azimuthal angle from the x axis 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 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 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 form In philosophy f 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 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 macrophageThe uppercase letter F is used as a symbol for The golden ratio conjugate 0 618 in mathematics The cumulative distribution function cdf of standard normal distribution in statistics The magnetic flux and electric flux in physics with subscripts distinguishing the two In philosophy F is often used as shorthand for a generic act Also in lowercase citation needed A common symbol for the parametrization of a surface in vector calculus In Lacanian algebra F stands for the imaginary phallus and also represents phallic signification F stands in for castration 5 dubious discuss The diameter symbol in engineering is often erroneously referred to as phi and the diameter symbol is sometimes erroneously typeset as F This symbol is used to indicate the diameter of a circular section for example 14 means the diameter of the circle is 14 units A Clock signal in electronics is often called Phi or uses the symbol 6 ComputingIn Unicode there are multiple forms of the phi letter Character Name Correct appearance Your browser LaTeX UsageU 03A6 GREEK CAPITAL LETTER PHI F displaystyle Phi F F displaystyle Phi Used in Greek textsU 03C6 GREEK SMALL LETTER PHI or f f displaystyle varphi or ϕ displaystyle phi Used in Greek textsU 03D5 GREEK PHI SYMBOL ϕ displaystyle phi ϕ ϕ ϕ displaystyle phi Used in mathematical and technical contexts 7 Italicized U 0278 LATIN SMALL LETTER PHI ɸ displaystyle Used in IPA to symbolise a voiceless bilabial fricativeIn ordinary Greek text the character U 03C6 f is used exclusively although this character has considerable glyphic variation sometimes represented with a glyph more like the representative glyph shown for U 03C6 f 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 family used widely in editions of classical Greek texts have a stroked glyph in this position most other typefaces have loopy glyphs This also applies to the Didot or apla typefaces employed in most Greek book printing as well as 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 f displaystyle varphi at U 03D5 7 For use as a phonetic symbol in 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 serifs at the top and bottom In HTML XHTML the upper and lowercase phi character entity references are amp Phi F and amp phi f respectively In LaTeX the math symbols are Phi F displaystyle Phi phi ϕ displaystyle phi and varphi f displaystyle varphi The Unicode standard also includes the following variants of phi and phi like characters Character information Preview F f Ⲫ ⲫUnicode name GREEK CAPITAL LETTER PHI GREEK SMALL LETTER PHI COPTIC CAPITAL LETTER FI COPTIC SMALL LETTER FIEncodings decimal hex dec hex dec hex dec hexUnicode 934 U 03A6 966 U 03C6 11434 U 2CAA 11435 U 2CABUTF 8 206 166 CE A6 207 134 CF 86 226 178 170 E2 B2 AA 226 178 171 E2 B2 ABNumeric character reference amp 934 wbr amp x3A6 wbr amp 966 wbr amp x3C6 wbr amp 11434 wbr amp x2CAA wbr amp 11435 wbr amp x2CAB wbr Named character reference amp Phi amp phi Character information Preview ɸ ⱷUnicode name LATIN SMALL LETTER PHI LATIN SMALL LETTER TAILLESS PHIEncodings decimal hex dec hexUnicode 632 U 0278 11383 U 2C77UTF 8 201 184 C9 B8 226 177 183 E2 B1 B7Numeric character reference amp 632 wbr amp x278 wbr amp 11383 wbr amp x2C77 wbr Character Name AppearanceU 1D60 MODIFIER LETTER SMALL GREEK PHI ᵠU 1D69 GREEK SUBSCRIPT SMALL LETTER PHI ᵩU 1DB2 MODIFIER LETTER SMALL PHI ᶲU 2CAA COPTIC CAPITAL LETTER FI ⲪU 2CAB COPTIC SMALL LETTER FI ⲫU 2C77 LATIN SMALL LETTER TAILLESS PHI ⱷU 1D6BD MATHEMATICAL BOLD CAPITAL PHI 𝚽U 1D6D7 MATHEMATICAL BOLD SMALL PHI 𝛗U 1D6DF MATHEMATICAL BOLD PHI SYMBOL 𝛟U 1D6F7 MATHEMATICAL ITALIC CAPITAL PHI 𝛷U 1D711 MATHEMATICAL ITALIC SMALL PHI 𝜑U 1D719 MATHEMATICAL ITALIC PHI SYMBOL 𝜙U 1D731 MATHEMATICAL BOLD ITALIC CAPITAL PHI 𝜱U 1D74B MATHEMATICAL BOLD ITALIC SMALL PHI 𝝋U 1D753 MATHEMATICAL BOLD ITALIC PHI SYMBOL 𝝓U 1D76B MATHEMATICAL SANS SERIF BOLD CAPITAL PHI 𝝫U 1D785 MATHEMATICAL SANS SERIF BOLD SMALL PHI 𝞅U 1D78D MATHEMATICAL SANS SERIF BOLD PHI SYMBOL 𝞍U 1D7A5 MATHEMATICAL SANS SERIF BOLD ITALIC CAPITAL PHI 𝞥U 1D7BF MATHEMATICAL SANS SERIF BOLD ITALIC SMALL PHI 𝞿U 1D7C7 MATHEMATICAL SANS SERIF BOLD ITALIC PHI SYMBOL 𝟇See alsoF f Ef Latin F f Ef Cyrillic 中 Psi and phi type figurine Փ Armenian Deposition geology Golden ratioReferences phi Oxford English Dictionary Online ed Oxford University Press Subscription or participating institution membership required Brixhe C History of the Alphabet in Christides amp al s A History of Ancient Greek 2007 Weisstein Eric W Golden Ratio mathworld wolfram com Retrieved 2020 08 10 Euler s Totient Function Brilliant Math amp Science Wiki brilliant org Retrieved 2020 08 10 Evans Dylans 1996 An introductory dictionary of Lacanian psychoanalysis Routledge p 145 ISBN 978 0 415 13523 8 Clock Circuits a b Representative Glyphs for Greek Phi UTR 25 Unicode support for mathematics PDF External links The dictionary definition of F at Wiktionary The dictionary definition of f at Wiktionary The dictionary definition of phi at Wiktionary Media related to Phi letter at Wikimedia Commons Retrieved from https en wikipedia org w index php title Phi amp oldid 1125601791, wikipedia, wiki, book, books, library,

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