It may be that phi originated as the letter qoppa (Ϙ, ϙ), and initially represented the sound /kʷʰ/ before shifting to Classical Greek [pʰ].[2] In traditional Greek numerals, phi has a value of 500 (φʹ) or 500,000 (͵φ). The Cyrillic letter Ef (Ф, ф) descends from phi.
Like other Greek letters, lowercase phi (encoded as the Unicode character U+03C6φGREEK SMALL LETTER PHI) is used as a mathematical or scientific symbol. Some uses[example needed] require the old-fashioned 'closed' glyph, which is separately encoded as the Unicode character U+03D5ϕGREEK PHI SYMBOL.
Use as a symbol
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In lowercase
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The lowercase letter φ (or its variant, ϕ or ɸ) is often used to represent the following:
In spherical coordinates, mathematicians usually refer to ϕ as the polar angle (from the z-axis). The convention in physics is to use ϕ as the azimuthal angle (from the x-axis).
In combustion engineering, fuel–air equivalence ratio. The ratio between the actual fuel-air ratio to the stoichiometric fuel-air ratio.
In granulometry, sedimentology, and soil engineering, φ is a logarithmic unit of sediment grain size, defined such that a change of 1 φ in grain size corresponds to a factor of 2 in grain diameter.
Strength (or resistance) reduction factor in structural engineering, used to account for statistical variabilities in materials and construction methods.
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 linguistics, φ-features denote features such as case, number and gender in which adjectives and pronouns agree with nouns.
In philosophy, Φ is often used as shorthand for a generic act. (Also in lowercase.)[citation needed]
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.[6][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.
A clock signal in electronics is often called Phi or uses the symbol.[7]
In emulsion and suspension science, Φ often denotes the volume fraction of the dispersed phase.[8]
Intended for use in modern (monotonic) Greek texts. Used italicized in mathematical and technical contexts when the "straight-line" variant glyph is preferred.[9]
In ordinary Greek text, the character U+03C6 φ is used exclusively, though 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.[9]
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 LaTeX, the math symbols are \Phi (), \phi (), and \varphi ().
The Unicode standard includes the following variants of phi and phi-like characters:
U+0278ɸLATIN SMALL LETTER PHI
U+03A6ΦGREEK CAPITAL LETTER PHI (Φ)
U+03C6φGREEK SMALL LETTER PHI (φ)
U+03D5ϕGREEK PHI SYMBOL (ϕ, ϕ, ϕ)
U+1D60ᵠMODIFIER LETTER SMALL GREEK PHI
U+1D69ᵩGREEK SUBSCRIPT SMALL LETTER PHI
U+1DB2ᶲMODIFIER LETTER SMALL PHI
U+2C77ⱷLATIN SMALL LETTER TAILLESS PHI
U+2CAAⲪCOPTIC CAPITAL LETTER FI
U+2CABⲫCOPTIC SMALL LETTER FI
U+1D6BD𝚽MATHEMATICAL BOLD CAPITAL PHI
U+1D6D7𝛗MATHEMATICAL BOLD SMALL PHI
U+1D6DF𝛟MATHEMATICAL BOLD PHI SYMBOL
U+1D6F7𝛷MATHEMATICAL ITALIC CAPITAL PHI
U+1D711𝜑MATHEMATICAL ITALIC SMALL PHI
U+1D719𝜙MATHEMATICAL ITALIC PHI SYMBOL
U+1D731𝜱MATHEMATICAL BOLD ITALIC CAPITAL PHI
U+1D74B𝝋MATHEMATICAL BOLD ITALIC SMALL PHI
U+1D753𝝓MATHEMATICAL BOLD ITALIC PHI SYMBOL
U+1D76B𝝫MATHEMATICAL SANS-SERIF BOLD CAPITAL PHI
U+1D785𝞅MATHEMATICAL SANS-SERIF BOLD SMALL PHI
U+1D78D𝞍MATHEMATICAL SANS-SERIF BOLD PHI SYMBOL
U+1D7A5𝞥MATHEMATICAL SANS-SERIF BOLD ITALIC CAPITAL PHI
U+1D7BF𝞿MATHEMATICAL SANS-SERIF BOLD ITALIC SMALL PHI
U+1D7C7𝟇MATHEMATICAL SANS-SERIF BOLD ITALIC PHI SYMBOL