Phi, pronounced FY or sometimes FEE in English, 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. In the
system of Greek numerals, it has a value of 500 or 500,000. It may be that phi originated
as the letter qoppa, and shifted as Ancient Greek became Classical Greek . The Cyrillic
letter Ef descends from Φ.
Use as a symbol The lower-case letter φ is often used to
represent the following: The golden ratio 1.618033988749894848204586834...
in mathematics, art, and architecture. Euler's totient function φ(n) in number theory;
also called Euler's phi function. The cyclotomic polynomial functions Φn(x)
of algebra. In algebra, group or ring homomorphisms
In probability theory, ϕ(x) =−½e−x2/2 is the probability density function of the
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. 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. The convention
in physics is to use phi as the azimuthal angle.
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 and navigation, latitude.
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.
Porosity in geology and hydrology. Strength 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
In economics, this is usually an additive term.
In flight dynamics, the roll angle. In philosophy, φ is often used as shorthand
for a generic act. 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. 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. 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 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 and less often with
a glyph more like the representative glyph shown for U+03D5.
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 "Porson" typefaces,
have a "stroked" glyph in this position, most other typefaces have "loopy" glyphs. This
goes for the "Didot" typefaces employed in most Greek book printing, as well as for 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, the
glyph assignments in the Unicode code charts were the reverse, and thus older fonts may
still show a loopy form at U+03D5. For use as a phonetic symbol in IPA, Unicode
has a separate codepoint U+0278, LATIN SMALL LETTER PHI, because in this use only the stroked
glyph is considered correct. 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 lower-case phi
character entity references are Φ and 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:
See also F, f: Ef
Ф, ф: Ef Psi and phi type figurine
Փ Deposition
References