Phi (letter)

Phi (uppercase Φ, lowercase φ or Unicode|ϕ), pronounced [IPA|fī] in modern Greek and as [IPA|faɪ] in English, 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 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 lower-case letter $varphi\; ,$ (or often its variant, $phi\; ,$) is used as a symbol for:
*The golden ratio 1.618... in mathematics, art, and architecture.
*Euler's totient function φ(n) in number theory; also called "Euler's phi function".
*The probability density function of the normal distribution in mathematics and statistics.
*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.
*A sentence in first-order logic.
*Porosity in geology.

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.
*Philosophy.
*Strength (or resistance) reduction factor in structural engineering, used to account for statistical variabilities in materials and construction methods.
*The number of phases in a power system in electrical engineering, for example 1Φ for single phase, 3Φ for three phase.
*The symbol of the voiceless bilabial fricative in the international phonetic alphabet.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:
* upper case:
** U+03A6 GREEK CAPITAL LETTER PHI (Φ): Greek capital letter phi
* lower case:
** U+|03C6 GREEK SMALL LETTER PHI (φ): "letter" phi, used in Greek texts.
** U+03D5 GREEK PHI SYMBOL (Unicode|ϕ): phi "symbol", for mathematical and technical contexts. [ [http://unicode.org/reports/tr25/#_Toc231 UTR #25: Unicode and Mathematics — Representative Glyphs for Greek Phi] ]

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 ($Phi,!$), phi ($phi\; ,!$), and varphi ($varphi,!$).

In some browsers (e.g. Internet Explorer 6), the shapes of the U+03C6 GREEK SMALL LETTER PHI (which should be viewed as a curl) and U+03D5 GREEK PHI SYMBOL (which should be viewed as a circle crossed by a slash) are exchanged. Compare these samples to check your browser:

:Phi letter: right $varphi,!$; possible wrong Unicode|φ
:Phi symbol: right $phi,!$; possible wrong Unicode|ϕ

ee also

*F, f - Ef (Latin)
*Ф, ф - Ef (Cyrillic)

References