Phi (letter)

Encyclopedia

**Phi**pronounced ˈfaɪ or sometimes ˈfiː in English, and ˈfi in modern Greek, is the 21st letter of the Greek alphabet

Greek alphabet

The Greek alphabet is the script that has been used to write the Greek language since at least 730 BC . The alphabet in its classical and modern form consists of 24 letters ordered in sequence from alpha to omega...

. In modern Greek, it represents [f], a voiceless labiodental fricative

Voiceless labiodental fricative

The voiceless labiodental 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 labiodental fricative:...

. In Ancient Greek it represented [pʰ], an aspirated

Aspiration (phonetics)

In phonetics, aspiration is the strong burst of air that accompanies either the release or, in the case of preaspiration, the closure of some obstruents. To feel or see the difference between aspirated and unaspirated sounds, one can put a hand or a lit candle in front of one's mouth, and say pin ...

voiceless bilabial plosive

Voiceless bilabial plosive

The voiceless bilabial plosive is a type of consonantal sound used in many spoken languages. The symbol in the International Phonetic Alphabet that represents this sound is , and the equivalent X-SAMPA symbol is p...

(from which English ultimately inherits the spelling "ph" in words derived from Greek). In the system of Greek numerals

Greek numerals

Greek numerals are a system of representing numbers using letters of the Greek alphabet. They are also known by the names Ionian numerals, Milesian numerals , Alexandrian numerals, or alphabetic 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) is often used to represent the following:- The golden ratioGolden ratioIn mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874989...

in mathematics, art, and architecture. - Euler's totient functionEuler's totient functionIn number theory, the totient \varphi of a positive integer n is defined to be the number of positive integers less than or equal to n that are coprime to n In number theory, the totient \varphi(n) of a positive integer n is defined to be the number of positive integers less than or equal to n that...

*φ*(*n*) in number theory; also called*Euler's phi function*. - In algebra, groupGroup (mathematics)In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

or ringRing (mathematics)In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...

homomorphismHomomorphismIn abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...

s - In probability theoryProbability theoryProbability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

, is the probability density functionProbability density functionIn probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

of the normal distribution. - In probability theoryProbability theoryProbability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

, is the characteristic functionCharacteristic function (probability theory)In probability theory and statistics, the characteristic function of any random variable completely defines its probability distribution. Thus it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative...

of a random variable*X*. - An angleAngleIn geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

, typically the second angle mentioned, after*θ*(thetaThetaTheta is the eighth letter of the Greek alphabet, derived from the Phoenician letter Teth...

). Especially:- The argument of a complex numberComplex numberA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

. - The phase of a wavePhase (waves)Phase in waves is the fraction of a wave cycle which has elapsed relative to an arbitrary point.-Formula:The phase of an oscillation or wave refers to a sinusoidal function such as the following:...

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 angleDihedral angleIn geometry, a dihedral or torsion angle is the angle between two planes.The dihedral angle of two planes can be seen by looking at the planes "edge on", i.e., along their line of intersection...

s in the backbones of proteinProteinProteins are biochemical compounds consisting of one or more polypeptides typically folded into a globular or fibrous form, facilitating a biological function. A polypeptide is a single linear polymer chain of amino acids bonded together by peptide bonds between the carboxyl and amino groups of...

s. - Internal or effective angle of friction.

- The argument of a complex number
- Electric potentialElectric potentialIn classical electromagnetism, the electric potential at a point within a defined space is equal to the electric potential energy at that location divided by the charge there...

in physics. - The work functionWork functionIn solid-state physics, the work function is the minimum energy needed to remove an electron from a solid to a point immediately outside the solid surface...

in electronics. - A shorthandShorthandShorthand is an abbreviated symbolic writing method that increases speed or brevity of writing as compared to a normal method of writing a language. The process of writing in shorthand is called stenography, from the Greek stenos and graphē or graphie...

representation for an aromatic functional groupFunctional groupIn organic chemistry, functional groups are specific groups of atoms within molecules that are responsible for the characteristic chemical reactions of those molecules. The same functional group will undergo the same or similar chemical reaction regardless of the size of the molecule it is a part of...

in organic chemistryOrganic chemistryOrganic chemistry is a subdiscipline within chemistry involving the scientific study of the structure, properties, composition, reactions, and preparation of carbon-based compounds, hydrocarbons, and their derivatives...

. - The fugacityFugacityIn chemical thermodynamics, the fugacity of a real gas is an effective pressure which replaces the true mechanical pressure in accurate chemical equilibrium calculations. It is equal to the pressure of an ideal gas which has the same chemical potential as the real gas. For example, nitrogen gas ...

coefficient in thermodynamicsThermodynamicsThermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

. - The ratio of free energyThermodynamic free energyThe thermodynamic free energy is the amount of work that a thermodynamic system can perform. The concept is useful in the thermodynamics of chemical or thermal processes in engineering and science. The free energy is the internal energy of a system less the amount of energy that cannot be used to...

destabilizations of protein mutantMutantIn biology and especially genetics, a mutant is an individual, organism, or new genetic character, arising or resulting from an instance of mutation, which is a base-pair sequence change within the DNA of a gene or chromosome of an organism resulting in the creation of a new character or trait not...

s in phi value analysisPhi value analysisPhi value analysis is an experimental protein engineering method used to study the structure of the folding transition state in small protein domains that fold in a two-state manner. Since the folding transition state is by definition a transient and partially unstructured state, its structure is...

. - In cartographyCartographyCartography is the study and practice of making maps. Combining science, aesthetics, and technique, cartography builds on the premise that reality can be modeled in ways that communicate spatial information effectively.The fundamental problems of traditional cartography are to:*Set the map's...

and navigationNavigationNavigation is the process of monitoring and controlling the movement of a craft or vehicle from one place to another. It is also the term of art used for the specialized knowledge used by navigators to perform navigation tasks...

, latitudeLatitudeIn geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...

. - In combustion engineering, equivalence ratio. The ratio between the actual fuel air ratio to the stoichiometric fuel air ratio.
- A sentence in first-order logicFirst-order logicFirst-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...

. - PorosityPorosityPorosity or void fraction is a measure of the void spaces in a material, and is a fraction of the volume of voids over the total volume, between 0–1, or as a percentage between 0–100%...

in geology and hydrologyHydrologyHydrology is the study of the movement, distribution, and quality of water on Earth and other planets, including the hydrologic cycle, water resources and environmental watershed sustainability...

. - Strength (or resistance) reduction factor in structural engineeringStructural engineeringStructural engineering is a field of engineering dealing with the analysis and design of structures that support or resist loads. Structural engineering is usually considered a specialty within civil engineering, but it can also be studied in its own right....

, used to account for statistical variabilities in materials and construction methods. - The symbol for a voiceless bilabial fricativeVoiceless bilabial fricative-See also:* List of phonetics topics...

in the International Phonetic AlphabetInternational Phonetic AlphabetThe International Phonetic Alphabet "The acronym 'IPA' strictly refers [...] to the 'International Phonetic Association'. But it is now such a common practice to use the acronym also to refer to the alphabet itself that resistance seems pedantic...

(using the straight line variant character) - In economics, this is usually an additive term.
- In flight dynamicsFlight dynamicsFlight dynamics is the science of air vehicle orientation and control in three dimensions. The three critical flight dynamics parameters are the angles of rotation in three dimensions about the vehicle's center of mass, known as pitch, roll and yaw .Aerospace engineers develop control systems for...

, the roll angle. - In philosophyPhilosophyPhilosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...

, 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 fluxMagnetic fluxMagnetic flux , is a measure of the amount of magnetic B field passing through a given surface . The SI unit of magnetic flux is the weber...

and electric flux in physics, with subscripts distinguishing the two. - The cumulative distribution functionCumulative distribution functionIn probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...

of the normal distribution in mathematicsMathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

and statisticsStatisticsStatistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

. - In philosophyPhilosophyPhilosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...

, Φ is often used as shorthand for a generic act. (Also in lower-case.) - The number of phases in a power system in electrical engineeringElectrical engineeringElectrical engineering is a field of engineering that generally deals with the study and application of electricity, electronics and electromagnetism. The field first became an identifiable occupation in the late nineteenth century after commercialization of the electric telegraph and electrical...

, for example 1Φ for single phase, 3Φ for three phase. - A common symbol for the parametrization of a surface in vector calculusCalculusCalculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

. - In Lacanian algebra, Φ stands for the imaginary phallus and also represents phallic signification. -Φ stands in for castration.

The diameter

Diameter

In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...

symbol in engineering

Engineering

Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

, ⌀, 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

Circle

A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

is 14 units

Units of measurement

A unit of measurement is a definite magnitude of a physical quantity, defined and adopted by convention and/or by law, that is used as a standard for measurement of the same physical quantity. Any other value of the physical quantity can be expressed as a simple multiple of the unit of...

.

## Computing

In UnicodeUnicode

Unicode is a computing industry standard for the consistent encoding, representation and handling of text expressed in most of the world's writing systems...

, there are multiple forms of the phi letter:

Character | Name | Correct appearance | Your browser | Usage |
---|---|---|---|---|

GREEK CAPITAL LETTER PHI | used in Greek texts | |||

GREEK SMALL LETTER PHI | or | used in Greek texts (contrary to what is shown here, it is normally not italicized) | ||

GREEK PHI SYMBOL | used in mathematical and technical contexts | |||

LATIN SMALL LETTER PHI | used in IPA to symbolise a voiceless bilabial fricative Voiceless bilabial fricative -See also:* List of phonetics topics... |

In some older fonts that are not yet compatible with Unicode

Unicode

Unicode is a computing industry standard for the consistent encoding, representation and handling of text expressed in most of the world's writing systems...

3.0 from 1998, the U+03D5 GREEK PHI SYMBOL might be represented by the "loopy" symbol instead. This is no longer a correct representation. The U+03C6 GREEK SMALL LETTER PHI may be presented as either the "stroked" glyph

Glyph

A glyph is an element of writing: an individual mark on a written medium that contributes to the meaning of what is written. A glyph is made up of one or more graphemes....

, but preferably as the "loopy" glyph.

In HTML/XHTML

XHTML

XHTML is a family of XML markup languages that mirror or extend versions of the widely-used Hypertext Markup Language , the language in which web pages are written....

, the upper and lower case phi character entity reference

Character entity reference

In the markup languages SGML, HTML, XHTML and XML, a character entity reference is a reference to a particular kind of named entity that has been predefined or explicitly declared in a Document Type Definition . The "replacement text" of the entity consists of a single character from the Universal...

s are

`Φ`(Φ) and

`φ`(φ) respectively.

In LaTeX

LaTeX

LaTeX is a document markup language and document preparation system for the TeX typesetting program. Within the typesetting system, its name is styled as . The term LaTeX refers only to the language in which documents are written, not to the editor used to write those documents. In order to...

, the math symbols are

`\Phi`(),

`\phi`(), and

`\varphi`().

## See also

- F, f - Ef (Latin)FF is the sixth letter in the basic modern Latin alphabet.-History:The origin of ⟨f⟩ is the Semitic letter vâv that represented a sound like or . Graphically, it originally probably depicted either a hook or a club...
- Ф, ф - Ef (Cyrillic)
- Psi and phi type figurinePsi and phi type figurinePsi- and phi- type figurines date back to 1450-1100 BC in Mycenaean Greece. They were made of ceramic and terracotta and were found in tombs and settlement areas. They got their names from their shape and a resemblance to the Greek letters of psi and phi .-References:*...
- Փ (Armenian letterArmenian alphabetThe Armenian alphabet is an alphabet that has been used to write the Armenian language since the year 405 or 406. It was devised by Saint Mesrop Mashtots, an Armenian linguist and ecclesiastical leader, and contained originally 36 letters. Two more letters, օ and ֆ, were added in the Middle Ages...

)