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Phi (Φ, φ) Definition

Phi (Φ, φ) is the 21st letter of the Greek alphabet. In the system of traditional Greek numerals, phi has a value of 500 (φʹ) or 500,000 (͵φ).

In Archaic and Classical Greek (c. 9th century BC to 4th century BC), it represented an aspirated voiceless bilabial plosive ([ph]), which was the origin of its usual romanization as ⟨ph⟩. During the later part of Classical Antiquity, in Koine Greek (c. 4th century BC to 4th century AD), its pronunciation shifted to that of a voiceless bilabial fricative ([ɸ]), and by the Byzantine Greek period (c. 4th century AD to 15th century AD) it developed its modern pronunciation as a voiceless labiodental fricative ([f]). The romanization of the Modern Greek phoneme is therefore usually ⟨f⟩. It may be that phi originated as the letter qoppa and initially represented the sound /kwh/ before shifting to Classical Greek [ph]. The Cyrillic letter Ef (Ф, ф) descends from phi. As with other Greek letters, lowercase phi is used as a mathematical or scientific symbol. Some uses, such as the golden ratio, require the old-fashioned 'closed' glyph.

Usage in Mathematics & Science

The uppercase Phi (Φ) is used 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 lowercase.)

  • 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 lowercase Phi (φ) is used for:

  • Magnetic flux in physics.

  • The letter phi is commonly used in physics to represent wave functions in quantum mechanics, such as in the Schrödinger equation and bra–ket notation.

  • The golden ratio 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) = (2π)−​1⁄2e−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 (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 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, geodesy and navigation, latitude.

  • In aircraft flight mechanics as the symbol for bank angle (sometimes represented with the letter theta, which is also used for pitch angle).

  • 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.

  • The Veblen function in set theory.

  • 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 flight dynamics, the roll angle.

  • In philosophy, φ is often used as shorthand for a generic act. (Also in uppercase.)

  • In perceptual psychology, the phi phenomenon is the apparent motion caused by the successive viewing of stationary objects, such as the frames of a motion picture.

  • In lexical-functional grammar, the function that maps elements from the c-structure to the f-structure.

  • In ecology, site survival probability, or the probability that a species will continue to occupy a site if it was there the previous year.

  • The logo of La France Insoumise, a leftist French political party.

  • An abbreviation for the word bacteriophage.

    • Mφ is used as an abbreviation for the word macrophage.

Greek Alphabet

The letters of the Ancient Greek Alphabet, which are frequently utilized in math and science:

Greek Alphabet

Symbol

Letter

Symbol

Letter

Uppercase

Lowercase

Uppercase

Lowercase

Α

α

Alpha

Ν

ν

Nu

Β

β

Beta

Ξ

ξ

Xi

Γ

γ

Gamma

Ο

ο

Omicron

Δ

δ

Delta

Π

π

Pi

Ε

ε

Epsilon

Ρ

ρ

Rho

Ζ

ζ

Zeta

Σ

σ

Sigma

Η

η

Eta

Τ

τ

Tau

Θ

θ

Theta

Υ

υ

Upsilon

Ι

ι

Iota

Φ

φ

Phi

Κ

κ

Kappa

Χ

χ

Chi

Λ

λ

Lambda

Ψ

ψ

Psi

Μ

μ

Mu

Ω

ω

Omega

Related Definitions

Sources

“Phi.” Wikipedia, Wikimedia Foundation, 6 Apr. 2020, en.wikipedia.org/wiki/Phi.

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