Phi (Letter) Information

Phi (uppercase Φ, lowercase φ, or math symbol ϕ), pronounced /ˈfaɪ/ fy or sometimes /ˈfiː/ fee in English,^[1] and [ˈfi] in modern Greek, is the 21st letter of the Greek alphabet. In modern Greek, it represents [f], a voiceless labiodental fricative. In Ancient Greek it represented [pʰ], an aspirated voiceless bilabial plosive (from which English ultimately inherits the spelling "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 (Ф, ф) arose from Φ.

Use as a symbol

The lower-case letter (or often its variant, ) is often used to represent the following:

• The golden ratio in mathematics, art, and architecture.
• Euler's totient function φ(n) in number theory; also called Euler's phi function.
• In algebra, group or ring homomorphisms
• In probability theory, ϕ(x) = (2π)^−½e^−x2/2 is the probability density function of the normal distribution.
• In probability theory, φ_X(t) = E[ e^itX ] is the characteristic function of a random variable X.
• An angle, typically the second angle mentioned, after θ (theta). Especially:
  • 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.
  • Internal or effective angle of friction.
• Electric potential in physics.
• The work function in electronics.
• 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 and navigation, latitude.
• In combustion engineering, 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.
• The symbol for a voiceless bilabial fricative in the International Phonetic Alphabet (using the straight line variant character)
• In economics, this is usually an additive term.^{[clarification needed]}
• In flight dynamics, the roll angle.
• In philosophy, φ is often used as shorthand for a generic act. (Also in upper-case.)

The upper-case letter Φ is used as a symbol for:

• 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.
• In philosophy, Φ is often used as shorthand for a generic act. (Also in lower-case.)
• 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.
• In Lacanian algebra, Φ stands for the imaginary phallus and also represents phallic signification. -Φ stands in for castration.^[2]

The diameter symbol in engineering, ⌀, is often incorrectly 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.

Computing

In Unicode, there are multiple forms of the phi letter:

In some older fonts that are not yet compatible with Unicode 3.0 from 1998, the U+03D5 GREEK PHI SYMBOL might be represented by the "loopy" symbol instead.^[3] This is no longer a correct representation. The U+03C6 GREEK SMALL LETTER PHI may be presented as either the "stroked" glyph, but preferably as the "loopy" glyph.^[3]

In HTML/XHTML, the upper and lower case phi character entity references are Φ (Φ) and φ (φ) respectively.

In LaTeX, the math symbols are \Phi (), \phi (), and \varphi ().

See also

• F, f - Ef (Latin)
• Ф, ф - Ef (Cyrillic)
• Psi and phi type figurine
• Փ (Armenian letter)

References

1. ^ [faɪ]: Collins English Dictionary, 3rd ed. (1991); New Oxford American Dictionary, 2nd ed. (2005) (transcribed "[fʌɪ] "). [fiː] is used increasingly in the media, especially when representing the golden ratio: see, for example, The Da Vinci Code and the Criminal Minds episode, "Masterpiece".
2. ^ Evans, Dylans (1996). An introductory dictionary of Lacanian psychoanalysis. Routledge. pp. 145. ISBN 9780415135238. http://books.google.com/?id=qwVVhLaiULEC&pg=PA145.
3. ^ ^{a b c} "Representative Glyphs for Greek Phi" (PDF). UTR #25: Unicode support for mathematics. http://unicode.org/reports/tr25/#_Toc231.

Categories: Greek letters

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