Phi (letter)

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Phi (uppercase Φ, lowercase φ or ϕ), pronounced [fi] in modern Greek and as [faɪ] in English, 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 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
• Any function in mathematics.
• 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
• A sentence in first-order logic.

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

• 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:
• GREEK SMALL LETTER PHI (φ): letter phi, used in Greek texts.
• U+03D5 GREEK PHI SYMBOL ($phi,!$): phi symbol, for mathematical and technical contexts.

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 this samples to check your browser:

Phi letter: right $varphi,!$; possible wrong φ

Phi symbol: right $phi,!$; possible wrong ϕ

References

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Last updated on Monday March 10, 2008 at 05:01:08 PDT (GMT -0700)
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