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Phi (; uppercase Φ, lowercase φ or ϕ; grc, ϕεῖ ''pheî'' ;
Modern Greek Modern Greek (, , or , ''Kiní Neoellinikí Glóssa''), generally referred to by speakers simply as Greek (, ), refers collectively to the dialects of the Greek language spoken in the modern era, including the official standardized form of the ...
: ''fi'' ) is the 21st letter of the
Greek alphabet The Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BCE. It is derived from the earlier Phoenician alphabet, and was the earliest known alphabetic script to have distinct letters for vowels as w ...
. In Archaic and
Classical Greek Ancient Greek includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic peri ...
(c. 9th century BC to 4th century BC), it represented an aspirated
voiceless bilabial plosive The voiceless bilabial plosive or stop is a type of consonantal sound used in most spoken languages. The symbol in the International Phonetic Alphabet that represents this sound is , and the equivalent X-SAMPA symbol is p. Features Features o ...
(), which was the origin of its usual romanization as . During the later part of
Classical Antiquity Classical antiquity (also the classical era, classical period or classical age) is the period of cultural history between the 8th century BC and the 5th century AD centred on the Mediterranean Sea, comprising the interlocking civilizations of ...
, in
Koine Greek Koine Greek (; Koine el, ἡ κοινὴ διάλεκτος, hē koinè diálektos, the common dialect; ), also known as Hellenistic Greek, common Attic, the Alexandrian dialect, Biblical Greek or New Testament Greek, was the common supra-reg ...
(c. 4th century BC to 4th century AD), its pronunciation shifted to that of a
voiceless bilabial fricative The voiceless bilabial fricative is a type of consonantal sound, used in some spoken languages. The symbol in the International Phonetic Alphabet that represents this sound is . Features Features of the voiceless bilabial fricative: Occ ...
(), and by the Byzantine Greek period (c. 4th century AD to 15th century AD) it developed its modern pronunciation as a
voiceless labiodental fricative The voiceless labiodental fricative is a type of consonantal sound used in a number of spoken languages. The symbol in the International Phonetic Alphabet that represents this sound is . Some scholars also posit the voiceless labiodental approx ...
(). The
romanization Romanization or romanisation, in linguistics, is the conversion of text from a different writing system to the Roman (Latin) script, or a system for doing so. Methods of romanization include transliteration, for representing written text, a ...
of the Modern Greek phoneme is therefore usually . It may be that phi originated as the letter
qoppa Koppa or qoppa (; as a modern numeral sign: ) is a letter that was used in early forms of the Greek alphabet, derived from Phoenician qoph (). It was originally used to denote the sound, but dropped out of use as an alphabetic character in fav ...
(Ϙ, ϙ), and initially represented the sound before shifting to Classical Greek . In traditional
Greek numerals Greek numerals, also known as Ionic, Ionian, Milesian, or Alexandrian numerals, are a system of writing numbers using the letters of the Greek alphabet. In modern Greece, they are still used for ordinal numbers and in contexts similar to tho ...
, phi has a value of 500 () or 500,000 (). The Cyrillic letter Ef (Ф, ф) descends from phi. As with other Greek letters, lowercase phi (encoded as the
Unicode Unicode, formally The Unicode Standard,The formal version reference is is an information technology standard for the consistent encoding, representation, and handling of text expressed in most of the world's writing systems. The standard, wh ...
character ) is used as a mathematical or scientific symbol. Some uses, such as the
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...
, require the old-fashioned 'closed' glyph, which is separately encoded as the Unicode character .


Use as a symbol

The lowercase letter φ (or often its variant, ϕ) is often used to represent the following: * Magnetic flux in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
* The letter phi is commonly used in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
to represent
wave function A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
s in
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
, such as in the
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
and bra–ket notation: \langle\phi, \psi\rangle. * The
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...
\tfrac \approx ... in mathematics, art, and architecture. * Euler's totient function ''φ''(''n'') in number theory; also called ''Euler's phi function''. * The cyclotomic polynomial functions Φ''n''(''x'') of algebra. * The number of phases in a power system in electrical engineering, for example 1ϕ for
single phase In electrical engineering, single-phase electric power (abbreviated 1φ) is the distribution of alternating current electric power using a system in which all the voltages of the supply vary in unison. Single-phase distribution is used when loads ...
, 3ϕ for
three phase Three-phase electric power (abbreviated 3φ) is a common type of alternating current used in electricity generation, transmission, and distribution. It is a type of polyphase system employing three wires (or four including an optional neutral ...
. * In algebra,
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
or
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
s * In
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
, is the
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
of the standard normal distribution. * In
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
, is the
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts: * The indicator function of a subset, that is the function ::\mathbf_A\colon X \to \, :which for a given subset ''A'' of ''X'', has value 1 at points ...
of a random variable ''X''. * An
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles a ...
, typically the second angle mentioned, after ''θ'' (
theta Theta (, ; uppercase: Θ or ; lowercase: θ or ; grc, ''thē̂ta'' ; Modern: ''thī́ta'' ) is the eighth letter of the Greek alphabet, derived from the Phoenician letter Teth . In the system of Greek numerals, it has a value of 9. Gr ...
). It can denote: ** The argument of a
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
. ** The phase of a wave in signal processing. ** In
spherical coordinates In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' meas ...
, mathematicians usually refer to phi as the polar angle (from the ''z''-
axis An axis (plural ''axes'') is an imaginary line around which an object rotates or is symmetrical. Axis may also refer to: Mathematics * Axis of rotation: see rotation around a fixed axis * Axis (mathematics), a designator for a Cartesian-coordinat ...
). The convention in physics is to use phi as the azimuthal angle (from the ''x''-axis). ** One of the
dihedral angle A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the un ...
s in the backbones of
protein Proteins are large biomolecules and macromolecules that comprise one or more long chains of amino acid residues. Proteins perform a vast array of functions within organisms, including catalysing metabolic reactions, DNA replication, res ...
s in a
Ramachandran plot In biochemistry, a Ramachandran plot (also known as a Rama plot, a Ramachandran diagram or a ,ψplot), originally developed in 1963 by G. N. Ramachandran, C. Ramakrishnan, and V. Sasisekharan, is a way to visualize energetically allowed regions ...
** Internal or effective angle of friction. * The
work function In solid-state physics, the work function (sometimes spelt workfunction) is the minimum thermodynamic work (i.e., energy) needed to remove an electron from a solid to a point in the vacuum immediately outside the solid surface. Here "immediately" ...
of a surface, in solid-state physics. * A
shorthand Shorthand is an abbreviated symbolic writing method that increases speed and brevity of writing as compared to longhand, a more common method of writing a language. The process of writing in shorthand is called stenography, from the Greek ''st ...
representation for an
aromatic In chemistry, aromaticity is a chemical property of cyclic ( ring-shaped), ''typically'' planar (flat) molecular structures with pi bonds in resonance (those containing delocalized electrons) that gives increased stability compared to satur ...
functional group In organic chemistry, a functional group is a substituent or moiety in a molecule that causes the molecule's characteristic chemical reactions. The same functional group will undergo the same or similar chemical reactions regardless of the re ...
in
organic chemistry Organic chemistry is a subdiscipline within chemistry involving the scientific study of the structure, properties, and reactions of organic compounds and organic materials, i.e., matter in its various forms that contain carbon atoms.Clayden, ...
. * The
fugacity In chemical thermodynamics, the fugacity of a real gas is an effective partial pressure which replaces the mechanical partial pressure in an accurate computation of the chemical equilibrium constant. It is equal to the pressure of an ideal gas whic ...
coefficient in
thermodynamics Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of th ...
. * The ratio of free energy destabilizations of protein
mutant In biology, and especially in genetics, a mutant is an organism or a new genetic character arising or resulting from an instance of mutation, which is generally an alteration of the DNA sequence of the genome or chromosome of an organism. It ...
s in
phi value analysis Phi value analysis, \phi analysis, or \phi-value analysis is an experimental protein engineering technique for studying the structure of the folding transition state of small protein domains that fold in a two-state manner. The structure of the f ...
. * In
cartography Cartography (; from grc, χάρτης , "papyrus, sheet of paper, map"; and , "write") is the study and practice of making and using maps. Combining science, aesthetics and technique, cartography builds on the premise that reality (or an i ...
, geodesy and
navigation Navigation is a field of study that focuses on the process of monitoring and controlling the movement of a craft or vehicle from one place to another.Bowditch, 2003:799. The field of navigation includes four general categories: land navigation, ...
,
latitude In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pol ...
. * In aircraft flight mechanics as the symbol for
bank angle A banked turn (or banking turn) is a turn or change of direction in which the vehicle banks or inclines, usually towards the inside of the turn. For a road or railroad this is usually due to the roadbed having a transverse down-slope towards the ...
(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 First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
. * The Veblen function in
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
. *
Porosity Porosity or void fraction is a measure of the void (i.e. "empty") spaces in a material, and is a fraction of the volume of voids over the total volume, between 0 and 1, or as a percentage between 0% and 100%. Strictly speaking, some tests measur ...
in geology and
hydrology Hydrology () is the scientific study of the movement, distribution, and management of water on Earth and other planets, including the water cycle, water resources, and environmental watershed sustainability. A practitioner of hydrology is call ...
. * Strength (or resistance) reduction factor in
structural engineering Structural engineering is a sub-discipline of civil engineering in which structural engineers are trained to design the 'bones and muscles' that create the form and shape of man-made structures. Structural engineers also must understand and cal ...
, used to account for statistical variabilities in materials and construction methods. * The symbol for a
voiceless bilabial fricative The voiceless bilabial fricative is a type of consonantal sound, used in some spoken languages. The symbol in the International Phonetic Alphabet that represents this sound is . Features Features of the voiceless bilabial fricative: Occ ...
in the
International Phonetic Alphabet The International Phonetic Alphabet (IPA) is an alphabetic system of phonetic notation based primarily on the Latin script. It was devised by the International Phonetic Association in the late 19th century as a standardized representation ...
(using the straight-line variant form). * In philosophy, φ is often used as shorthand for a
generic act Generic or generics may refer to: In business * Generic term, a common name used for a range or class of similar things not protected by trademark * Generic brand, a brand for a product that does not have an associated brand or trademark, other ...
. (Also in uppercase.) * In
perceptual psychology Perceptual psychology is a subfield of cognitive psychology that concerns the conscious and unconscious innate aspects of the human cognitive system: perception. A pioneer of the field was James J. Gibson. One major study was that of affordances, ...
, 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 Ecology () is the study of the relationships between living organisms, including humans, and their physical environment. Ecology considers organisms at the individual, population, community, ecosystem, and biosphere level. Ecology overl ...
, 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 La France Insoumise (FI or LFI; ; "France Unbowed") is a left-wing populist political party in France, launched in 2016 by Jean-Luc Mélenchon, then a Member of the European Parliament (MEP) and former co-president of the Left Party (PG). It aims ...
, a leftist French
political party A political party is an organization that coordinates candidates to compete in a particular country's elections. It is common for the members of a party to hold similar ideas about politics, and parties may promote specific ideological or p ...
. * An abbreviation for the word bacteriophage ** Mϕ is used as an abbreviation for the word macrophage The uppercase letter Φ is used as a symbol for: * The golden ratio conjugate −0.618... in mathematics. * The cumulative distribution function (cdf) of standard normal distribution in statistics. * The magnetic flux and
electric flux In electromagnetism, electric flux is the measure of the electric field through a given surface, although an electric field in itself cannot flow. The electric field E can exert a force on an electric charge at any point in space. The electric fi ...
in physics, with subscripts distinguishing the two. * In philosophy, Φ is often used as shorthand for a
generic act Generic or generics may refer to: In business * Generic term, a common name used for a range or class of similar things not protected by trademark * Generic brand, a brand for a product that does not have an associated brand or trademark, other ...
. (Also in lowercase.) * A common symbol for the parametrization of a surface in vector
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
. * 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 erroneously referred to as "phi", and the diameter symbol is sometimes erroneously typeset as Φ. This symbol is used to indicate the diameter of a circular section; for example, "⌀14" means the diameter of the
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
is 14
units Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * Unit (album), ...
. * A Clock signal in electronics is often called Phi or uses the symbol.


Computing

In
Unicode Unicode, formally The Unicode Standard,The formal version reference is is an information technology standard for the consistent encoding, representation, and handling of text expressed in most of the world's writing systems. The standard, wh ...
, 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 (φ, the "loopy" or "open" form), and less often with a glyph more like the representative glyph shown for U+03D5 (ϕ, the "stroked" or "closed" 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 those in the Porson family (used widely in editions of classical Greek texts), have a "stroked" glyph in this position (), most other typefaces have "loopy" glyphs. This also applies to the "Didot" (or "''apla''") typefaces employed in most Greek book printing (), as well as 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 code point U+0278, LATIN SMALL LETTER PHI, because only the stroked glyph is considered correct in this use. 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 Extensible HyperText Markup Language (XHTML) is part of the family of XML markup languages. It mirrors or extends versions of the widely used HyperText Markup Language (HTML), the language in which Web pages are formulated. While HTML, prior ...
, the upper- and lowercase phi character entity references are Φ (Φ) and φ (φ), respectively. In
LaTeX Latex is an emulsion (stable dispersion) of polymer microparticles in water. Latexes are found in nature, but synthetic latexes are common as well. In nature, latex is found as a milky fluid found in 10% of all flowering plants (angiosperms ...
, 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:


See also

* F, f: Ef (Latin) * Ф, ф: Ef (Cyrillic) * *
Psi and phi type figurine Tau-, Psi- and phi- type figurines date back to 1450–1100 BC in Mycenaean Greece. They were typically small (about 10cm high), made of terracotta, although a group of ivory figurines has been found, and were found in tombs, shrines and settle ...
* Փ (Armenian) *
Deposition (geology) Deposition is the geological process in which sediments, soil and rocks are added to a landform or landmass. Wind, ice, water, and gravity transport previously weathered surface material, which, at the loss of enough kinetic energy in the flu ...
*
Golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...


References


External links

* * * * {{DEFAULTSORT:Phi (Letter) Greek letters