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# Phi

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### Topic Elements

simple explanation

From Wikipedia the free encyclopedia

## Use as a symbol

The lowercase letter φ (or often its variant ϕ) is often used to represent the following:

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:

ϕ

|

ψ

{\displaystyle \langle \phi |\psi \rangle }

.
The golden ratio

1
+

5

2

{\displaystyle {\tfrac {1+{\sqrt {5))}{2))\approx }

1.618033988749894848204586834... 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 standard 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). It can denote:
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.)[citation needed]
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
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 Φ is often used as shorthand for a generic act. (Also in lowercase.)[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. [dubious – discuss]
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 is 14 units.

## Computing

In Unicode there are multiple forms of the phi letter:

Character Name Correct appearance Your browser LaTeX Usage
U+03A6 GREEK CAPITAL LETTER PHI

Φ

{\displaystyle \Phi \ \!}

Φ

Φ

{\displaystyle \Phi }

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57165455
f24" style="vertical-align: -0.338ex;width:1.678ex;height:2.176ex" />
Used in Greek texts
U+03C6 GREEK SMALL LETTER PHI

or

φ

φ

{\displaystyle \varphi }

or

ϕ

{\displaystyle \phi }

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30316670
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Used in Greek texts
U+03D5 GREEK PHI SYMBOL

ϕ

{\displaystyle \phi \ \!}

0577802
bc4ffa026a29ab35c365" style="vertical-align: -0.671ex;margin-right: -0.387ex;width:1.773ex;height:2.509ex" />
ϕ (ϕ)

ϕ

{\displaystyle \phi }

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Used in mathematical and technical contexts. (Italicized.)
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" or "open" form) and less often with a glyph more like the representative glyph shown for U+03D5 (ϕ the "stroked" or "closed" form). Unicode makes an effort to distinguish the two by generally calling the loopy form "small letter phi" or "small phi" and by calling the stroked form "phi symbol" but this is not exclusively true on all variants.

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 (
apla

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

φ

{\displaystyle \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 the upper- and lowercase phi character entity references are Φ (Φ) and φ (φ) respectively.

In LaTeX the math symbols are \Phi (

Φ

{\displaystyle \Phi \ \!}

) \phi (

ϕ

{\displaystyle \phi \ \!}

) and \varphi (

φ

{\displaystyle \varphi \ \!}

).

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

Character information
Preview Φ φ
.mw-parser-output span.smallcaps{font-variant:small-caps}.mw-parser-output span.smallcaps-smaller{font-size:85%}Unicode name GREEK CAPITAL LETTER PHI GREEK SMALL LETTER PHI COPTIC CAPITAL LETTER FI COPTIC SMALL LETTER FI
Encodings decimal hex decimal hex decimal hex decimal hex
Unicode 934 U+03A6 966 U+03C6 11434 U+2CAA 11435 U+2CAB
UTF-8 206 166 CE A6 207 134 CF 86 226 178 170 E2 B2 AA 226 178 171 E2 B2 AB
Numeric character reference Φ Φ φ φ
Named character reference &Phi; &phi;

Character information
Preview ɸ
Unicode name LATIN SMALL LETTER PHI LATIN SMALL LETTER TAILLESS PHI
Encodings decimal hex decimal hex
Unicode 632 U+0278 11383 U+2C77
UTF-8 201 184 C9 B8 226 177 183 E2 B1 B7
Numeric character reference ɸ ɸ

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

## simple explanation

Phi (/faɪ/; uppercase Φ lowercase φ or ϕ; Ancient Greek: ϕεῖ pheî [pʰé͜e]; Modern Greek: φι fi [fi]) is the 21st letter of the Greek alphabet.

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]. In traditional Greek numerals phi has a value of 500 (φʹ) or 500 000 (͵φ). The Cyrillic letter Ef (Ф ф) descends from phi.

As with other Greek letters lowercase phi (encoded as the Unicode character U+03C6 φ GREEK SMALL LETTER PHI) 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 U+03D5 ϕ GREEK PHI SYMBOL.

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