Phi

For similar symbols, see Ø (disambiguation).

This article is about the letter of the Greek alphabet. For other uses, see Phi (disambiguation).

Greek alphabet
Αα	Alpha	Νν	Nu
Ββ	Beta	Ξξ	Xi
Γγ	Gamma	Οο	Omicron
Δδ	Delta	Ππ	Pi
Εε	Epsilon	Ρρ	Rho
Ζζ	Zeta	Σσς	Sigma
Ηη	Eta	Ττ	Tau
Θθ	Theta	Υυ	Upsilon
Ιι	Iota	Φφ	Phi
Κκ	Kappa	Χχ	Chi
Λλ	Lambda	Ψψ	Psi
Μμ	Mu	Ωω	Omega
History
Archaic local variants
Ligatures (ϛ, ȣ, ϗ) Diacritics
Numerals (6) (90) (900)
In other languages
Bactrian Coptic Albanian
Scientific symbols
Book Category Commons

Phi (uppercase Φ, lowercase φ, or math symbol ϕ, also fi), pronounced /ˈfaɪ/ FY or sometimes /ˈfiː/ 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 (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 φ (or often its variant, ϕ) is often used to represent the following:

• The golden ratio $\tfrac{1 + \sqrt{5}}{2} \approx 1.618033988749894848204586834\ldots$ 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.
• 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.
• 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.)^{[citation needed]}
• 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^{[citation needed]} 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:

Character	Name	Correct appearance	Your browser	Usage
U+03A6	GREEK CAPITAL LETTER PHI	$\Phi\,\!$	Φ	used in Greek texts
U+03C6	GREEK SMALL LETTER PHI	or	φ	used in Greek texts
U+03D5	GREEK PHI SYMBOL	$\phi\,\!$	ϕ	used in mathematical and technical contexts
U+0278	LATIN SMALL LETTER PHI		ɸ	used 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” form) and less often with a glyph more like the representative glyph shown for U+03D5 (ϕ, the “straight“ 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 "Porson" typefaces (used widely in editions of classical Greek texts), have a "stroked" glyph in this position (), most other typefaces have "loopy" glyphs. This goes for the "Didot" (or "apla") 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 (1998), the glyph assignments in the Unicode code charts were the reverse, and thus older fonts may still show a loopy form $\varphi$ 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 (Φ) and φ (φ) respectively.

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

The Unicode standard also includes the following variants of phi and phi-like characters:

Character	Name	Appearance
U+1D60	MODIFIER LETTER SMALL GREEK PHI	ᵠ
U+1D69	GREEK SUBSCRIPT SMALL LETTER PHI	ᵩ
U+1DB2	MODIFIER LETTER SMALL PHI	ᶲ
U+2CAA	COPTIC CAPITAL LETTER FI	Ⲫ
U+2CAB	COPTIC SMALL LETTER FI	ⲫ
U+2C77	LATIN SMALL LETTER TAILLESS PHI	ⱷ
U+1D6BD	MATHEMATICAL BOLD CAPITAL PHI	𝚽
U+1D6D7	MATHEMATICAL BOLD SMALL PHI	𝛗
U+1D6DF	MATHEMATICAL BOLD PHI SYMBOL	𝛟
U+1D6F7	MATHEMATICAL ITALIC CAPITAL PHI	𝛷
U+1D711	MATHEMATICAL ITALIC SMALL PHI	𝜑
U+1D719	MATHEMATICAL ITALIC PHI SYMBOL	𝜙
U+1D731	MATHEMATICAL BOLD ITALIC CAPITAL PHI	𝜱
U+1D74B	MATHEMATICAL BOLD ITALIC SMALL PHI	𝝋
U+1D753	MATHEMATICAL BOLD ITALIC PHI SYMBOL	𝝓
U+1D76B	MATHEMATICAL SANS-SERIF BOLD CAPITAL PHI	𝝫
U+1D785	MATHEMATICAL SANS-SERIF BOLD SMALL PHI	𝞅
U+1D78D	MATHEMATICAL SANS-SERIF BOLD PHI SYMBOL	𝞍
U+1D7A5	MATHEMATICAL SANS-SERIF BOLD ITALIC CAPITAL PHI	𝞥
U+1D7BF	MATHEMATICAL SANS-SERIF BOLD ITALIC SMALL PHI	𝞿
U+1D7C7	MATHEMATICAL SANS-SERIF BOLD ITALIC PHI SYMBOL	𝟇

See also

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