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, 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:

  • Magnetic flux in physics
  • The letter phi is commonly used in physics to represent wave functions in quantum mechanics, such as in the Schrödinger equation and bra–ket notation: .
  • The golden ratio 1.618033988749894848204586834... in mathematics,[3] art, and architecture.
  • Euler's totient function φ(n) in number theory;[4] also called Euler's phi function.
  • The cyclotomic polynomial functions Φn(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) = (2π)12ex2/2 is the probability density function of the standard normal distribution.
  • In probability theory, φX(t) = E[eitX] is the characteristic function of a random variable X.
  • An angle, typically the second angle mentioned, after θ (theta). It can denote:
  • 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, 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.
  • 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

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:

CharacterNameCorrect appearanceYour browserLaTeXUsage
U+03A6GREEK CAPITAL LETTER PHIΦUsed in Greek texts
U+03C6GREEK SMALL LETTER PHI or φ or Used in Greek texts
U+03D5GREEK PHI SYMBOLϕ (ϕ)Used in mathematical and technical contexts.[6] (Italicized.)
U+0278LATIN SMALL LETTER PHIɸUsed 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.[6]

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
PreviewΦφ
Unicode name GREEK CAPITAL LETTER PHI GREEK SMALL LETTER PHI COPTIC CAPITAL LETTER FI COPTIC SMALL LETTER FI
Encodingsdecimalhexdechexdechexdechex
Unicode934U+03A6966U+03C611434U+2CAA11435U+2CAB
UTF-8206 166CE A6207 134CF 86226 178 170E2 B2 AA226 178 171E2 B2 AB
Numeric character referenceΦΦφφⲪⲪⲫⲫ
Named character referenceΦφ
Character information
Previewɸ
Unicode name LATIN SMALL LETTER PHI LATIN SMALL LETTER TAILLESS PHI
Encodingsdecimalhexdechex
Unicode632U+027811383U+2C77
UTF-8201 184C9 B8226 177 183E2 B1 B7
Numeric character referenceɸɸⱷⱷ
CharacterNameAppearance
U+1D60MODIFIER LETTER SMALL GREEK PHI
U+1D69GREEK SUBSCRIPT SMALL LETTER PHI
U+1DB2MODIFIER LETTER SMALL PHI
U+2CAACOPTIC CAPITAL LETTER FI
U+2CABCOPTIC SMALL LETTER FI
U+2C77LATIN SMALL LETTER TAILLESS PHI
U+1D6BDMATHEMATICAL BOLD CAPITAL PHI𝚽
U+1D6D7MATHEMATICAL BOLD SMALL PHI𝛗
U+1D6DFMATHEMATICAL BOLD PHI SYMBOL𝛟
U+1D6F7MATHEMATICAL ITALIC CAPITAL PHI𝛷
U+1D711MATHEMATICAL ITALIC SMALL PHI𝜑
U+1D719MATHEMATICAL ITALIC PHI SYMBOL𝜙
U+1D731MATHEMATICAL BOLD ITALIC CAPITAL PHI𝜱
U+1D74BMATHEMATICAL BOLD ITALIC SMALL PHI𝝋
U+1D753MATHEMATICAL BOLD ITALIC PHI SYMBOL𝝓
U+1D76BMATHEMATICAL SANS-SERIF BOLD CAPITAL PHI𝝫
U+1D785MATHEMATICAL SANS-SERIF BOLD SMALL PHI𝞅
U+1D78DMATHEMATICAL SANS-SERIF BOLD PHI SYMBOL𝞍
U+1D7A5MATHEMATICAL SANS-SERIF BOLD ITALIC CAPITAL PHI𝞥
U+1D7BFMATHEMATICAL SANS-SERIF BOLD ITALIC SMALL PHI𝞿
U+1D7C7MATHEMATICAL SANS-SERIF BOLD ITALIC PHI SYMBOL𝟇

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. "Representative Glyphs for Greek Phi". UTR #25: Unicode support for mathematics (PDF).
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