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 ...
.
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 pe ...
(c. 9th century BC to 4th century BC), it represented an
aspirated voiceless bilabial plosive (), 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 (), 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 app ...
().
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 (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 those ...
, 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, ...
character ) is used as a mathematical or scientific symbol. Some uses, such as the
golden ratio, 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 ...
* 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 ...
to represent
wave functions 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 chemistry, ...
, 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:
.
* 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 Φ
''n''(''x'') of algebra.
* The number of phases in a power system in
electrical engineering
Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
, for example 1ϕ for
single phase, 3ϕ for
three phase.
* In algebra,
group or
ring homomorphism
In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
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 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 ...
, 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.
G ...
). 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, 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
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
** 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 ''s ...
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 r ...
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, J ...
.
* 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 ...
.
* 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.
* 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 ...
.
* In
aircraft flight mechanics
Aircraft flight mechanics are relevant to fixed wing ( gliders, aeroplanes) and rotary wing (helicopters) aircraft. An aeroplane (''airplane'' in US usage), is defined in ICAO Document 9110 as, "a power-driven heavier than air aircraft, deriving i ...
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.
* The
Veblen function
In mathematics, the Veblen functions are a hierarchy of normal functions (continuous strictly increasing functions from ordinals to ordinals), introduced by Oswald Veblen in . If φ0 is any normal function, then for any non-zero ordinal α, φ� ...
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 concern ...
.
*
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, used to account for statistical variabilities in materials and construction methods.
* The symbol for a
voiceless bilabial fricative 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
Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. ...
, 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
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
Site most often refers to:
* Archaeological site
* Campsite, a place used for overnight stay in an outdoor area
* Construction site
* Location, a point or an area on the Earth's surface or elsewhere
* Website, a set of related web pages, typicall ...
, 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
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 physics, with subscripts distinguishing the two.
* In
philosophy
Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. ...
, Φ is often used as shorthand for a
generic act. (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 mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
.
* In
Lacanian algebra, Φ stands for the imaginary phallus and also represents phallic signification; −Φ stands in for castration.
The
diameter symbol
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for ...
in engineering,
⌀
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for ...
, 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 cons ...
is 14
units.
* 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, ...
, 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
at U+03D5.
For use as a phonetic symbol in
IPA
IPA commonly refers to:
* India pale ale, a style of beer
* International Phonetic Alphabet, a system of phonetic notation
* Isopropyl alcohol, a chemical compound
IPA may also refer to:
Organizations International
* Insolvency Practitioners A ...
, 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, the upper- and lowercase phi
character entity reference
Character or Characters may refer to:
Arts, entertainment, and media Literature
* ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk
* ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...
s 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
(
), and
\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)
*
中
*
Psi and phi type figurine
*
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
References
External links
*
*
*
*
{{DEFAULTSORT:Phi (Letter)
Greek letters