Editing Pi (letter)

{{about|a Greek letter|the number symbolized by this letter|Pi}}
{{Short description|Greek letter}}
{{Greek Alphabet|letter=pi}}
'''Pi''' ({{IPAc-en|'|p|aɪ|audio=LL-Q1860 (eng)-Flame, not lame-Pi (letter).wav}}; [[Ancient Greek]] {{ipa|/piː/}} or {{ipa|/peî/}}, uppercase '''Π''', lowercase '''π''', [[cursive]] '''ϖ'''; {{langx|el|πι}}) is the sixteenth letter of the [[Greek alphabet]], representing the [[voiceless bilabial plosive]] {{IPA|el|p|IPA}}.  In the system of [[Greek numerals]] it has a value of 80. It was derived from the [[Phoenician alphabet|Phoenician]] letter [[Pe (Semitic letter)|Pe]] (<span style="background-color: white;">[[File:Phoenician pe.svg|16px]]</span>). Letters that arose from pi include [[Latin alphabet|Latin]] [[P]], [[Cyrillic script|Cyrillic]] [[Pe (Cyrillic)|Pe]] (П, п), [[Coptic alphabet|Coptic]] pi (Ⲡ, ⲡ), and [[Gothic alphabet|Gothic]] [[pairthra]] (𐍀).<ref>{{Cite web |title=Pi Symbol in Greek Alphabet |url=https://www.greeksymbols.net/pi-symbol |access-date=2023-02-07 |website=greeksymbols.net |language=en}}</ref>

== Uppercase Pi ==
The uppercase letter Π is used as a symbol for:
* In textual criticism, ''[[Codex Petropolitanus (New Testament)|Codex Petropolitanus]]'', a 9th-century [[uncial]] [[codex]] of the [[Gospel]]s, now located in [[St. Petersburg, Russia]].
* In [[legal shorthand]], it represents a [[plaintiff]].<ref>{{Cite web |last=sinclair |date=2024-01-22 |title=The Top Legal Symbol Keyboard Shortcuts You Should Know |url=https://lawrank.com/top-legal-keyboard-shortcuts/ |access-date=2025-01-18 |website=LawRank |language=en-US |quote=If you work with plaintiffs, you may choose to replace the word “plaintiff” with a symbol in some of your writing. The plaintiff symbol looks like the Greek letter pi (π).}}</ref>
* In [[Mathematical finance]], it represents a portfolio.

[[Greek letters used in mathematics, science, and engineering|In science and engineering]]:
* The [[product (mathematics)|product]] operator in mathematics, indicated with [[Multiplication#Capital pi notation|capital pi notation]] {{Math|Π}} (in analogy to the use of the capital [[Sigma]] {{Math|Σ}} as [[summation]] symbol).<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Product |url=https://mathworld.wolfram.com/Product.html |access-date=2025-01-18 |website=mathworld.wolfram.com |language=en}}</ref>
* The [[osmotic pressure]] in chemistry.<ref>{{Cite web |date=2015-01-18 |title=13.7: Osmotic Pressure |url=https://chem.libretexts.org/Bookshelves/General_Chemistry/Map:_General_Chemistry_(Petrucci_et_al.)/13:_Solutions_and_their_Physical_Properties/13.07:_Osmotic_Pressure |access-date=2025-01-18 |website=Chemistry LibreTexts |language=en}}</ref>
* The [[viscous stress tensor]] in continuum mechanics and fluid dynamics.

== Lowercase Pi ==
The lowercase letter {{pi}} is used as a symbol for:
* The mathematical real [[Transcendental number|transcendental]] (and thus [[Irrational number|irrational]]) [[Pi|constant π]]&nbsp;≈&nbsp;3.14159..., the ratio of a circle's [[circumference]] to its [[diameter]] in [[Euclidean geometry]],<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Pi |url=https://mathworld.wolfram.com/Pi.html |access-date=2025-01-18 |website=mathworld.wolfram.com |language=en |quote=The constant pi, denoted {{pi}}, is a real number defined as the ratio of a circle's circumference C to its diameter}}</ref> exactly half of the circle constant. The letter "{{pi}}" is the first letter of the Greek words {{wikt-lang|grc|περιφέρεια}} 'periphery' and {{wikt-lang|grc|περίμετρος}} 'perimeter', i.e. the circumference.
* The [[prime-counting function]] in mathematics.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Prime Counting Function |url=https://mathworld.wolfram.com/PrimeCountingFunction.html |access-date=2025-01-18 |website=mathworld.wolfram.com |language=en |quote=The prime counting function is the function {{pi}}(x) giving the number of primes less than or equal to...}}</ref><ref>{{Cite web |title=DLMF: §27.12 Asymptotic Formulas: Primes ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory |url=https://dlmf.nist.gov/27.12 |access-date=2025-01-31 |website=dlmf.nist.gov}}</ref>
* [[Homotopy group]]s  in algebraic topology.
* Dimensionless parameters constructed using the [[Buckingham π theorem]] of [[dimensional analysis]].
* The [[hadron]] called the [[pion]] (pi meson).<ref>{{Cite web |title=Pi meson {{!}} subatomic particle {{!}} Britannica |url=https://www.britannica.com/science/pi-meson |access-date=2025-01-18 |website=www.britannica.com |language=en |quote=Also known as: pion}}</ref>
* Often [[inflation]] rate in macroeconomics.<ref>{{Cite web |title=Fisher Equation |url=https://corporatefinanceinstitute.com/resources/economics/fisher-equation/ |access-date=2025-01-18 |website=Corporate Finance Institute |language=en-US |quote=π – the inflation rate}}</ref>
* Sometimes [[Profit (economics)|profit]] in microeconomics.
* A type of [[chemical bond]] in which the p orbitals overlap, called a [[pi bond]].<ref>{{Cite web |title=Pi bond {{!}} Double Bond, Electron Sharing & Hybridization {{!}} Britannica |url=https://www.britannica.com/science/pi-bond |access-date=2025-01-18 |website=www.britannica.com |language=en}}</ref>
* The natural projection on the [[tangent bundle]] on a [[manifold (mathematics)|manifold]].
* The unary operation of [[Projection (relational algebra)|projection]] in [[relational algebra]].<ref>{{Cite web |date=2024-03-25 |title=What are the Unary Operations in Relational Algebra? |url=https://www.geeksforgeeks.org/what-are-the-unary-operations-in-relational-algebra/ |access-date=2025-01-18 |website=GeeksforGeeks |language=en-US |quote=The projection operation is a unary operation that is performed on a relation. A projection operation is used to retrieve all the values of one or more attributes. It is denoted by π.}}</ref>
* Policy in [[reinforcement learning]].
[[File:Polyamory Pride Flag.svg|thumb|right|The earliest [[polyamory]] [[pride flag]] design, created by Jim Evans in 1995, in which the lowercase letter π stands for the first letter of polyamory.]]
* [[Polyamory]] (in the earliest polyamory [[pride flag]] design, created by Jim Evans in 1995, pi stands for the first letter of polyamory).<ref>{{Cite web|url=https://www.thespinoff.co.nz/society/22-04-2019/pretty-poly-why-non-monogamous-relationships-are-all-the-rage/|title=Pretty poly: Why non-monogamous relationships are all the rage|first=Melody|last=Thomas|date=April 22, 2019|access-date=March 24, 2021|archive-date=February 26, 2021|archive-url=https://web.archive.org/web/20210226022106/https://thespinoff.co.nz/society/22-04-2019/pretty-poly-why-non-monogamous-relationships-are-all-the-rage/|url-status=live}}</ref><ref>{{Cite web|url=https://www.goodhousekeeping.com/life/g39944774/lgbtq-flags/|title=21 LGBTQ Flags and What They Symbolize|first=Lizz|last=Schumer|date=May 16, 2022|website=Good Housekeeping}}</ref><ref>{{Cite web|url=https://www.lgbtqnation.com/2022/06/polyamorous-flag-look-like/|title=What does the polyamorous flag look like?|first=Matthias|last=Walsh|website=LGBTQ Nation|date=17 June 2022 }}</ref><ref>{{cite web|title=Jim Evans' Polyamory Pride Flag|url=http://homepage.isomedia.com/~jene/flag.html|archive-url=https://web.archive.org/web/20141110021410/http://homepage.isomedia.com/~jene/flag.html|archive-date=November 10, 2014}}</ref>

==History==
An early form of pi was <span style="background-color: white;">{{GrGl|Pi archaic}}</span>, appearing almost like a [[gamma]] with a hook.<ref>{{Cite book |last=Thompson |first=Edward Maunde |url=https://www.cambridge.org/core/books/an-introduction-to-greek-and-latin-palaeography/4E5810810D8AEE5ED461584112ACA4FE |title=An Introduction to Greek and Latin Palaeography |date=2013 |orig-date=1912|publisher=Cambridge University Press |isbn=978-1-108-06181-0 |series=Cambridge Library Collection - Classics |location=Cambridge |doi=10.1017/cbo9781139833790}}</ref><ref>{{Cite book |last=Faulmann |first=Karl |title=Schriftzeichen und Alphabete aller Zeiten und Völker |date=2000 |publisher=Augustus |isbn=978-3-8043-0374-4 |edition=Repr. nach d. Wiener Ausg. 1880, Neuausg |location=München|lang=German}}</ref>

== Variant pi ==

'''Variant pi''' or "pomega" (<math>\varpi\,\!</math> or ϖ) is a [[glyph]] variant of lowercase pi sometimes used in technical contexts. It resembles a lowercase [[omega]] with a [[Macron (diacritic)|macron]], though historically it is simply a [[cursive]] form of pi, with its legs bent inward to meet. It was also used in the [[Greek minuscule|minuscule script]]. It is a symbol for:
* [[Angular frequency]] of a [[ocean surface wave|wave]] in [[fluid dynamics]] (angular frequency is usually represented by <math>\omega</math> but this may be confused with [[vorticity]] in a fluid dynamics context).
* [[Longitude of the periapsis|Longitude of pericenter]] in [[celestial mechanics]].<ref>{{cite web |url=http://scienceworld.wolfram.com/physics/Pomega.html |title=Pomega |work=Eric Weisstein's World of Physics |publisher=wolfram.com}}</ref>
* [[Comoving distance]] in cosmology.<ref>[http://odin.physastro.mnsu.edu/~eskridge/astr225/week14.html Outline for Weeks 14&15, Astronomy 225 Spring 2008<!-- Bot generated title -->] {{webarchive|url=https://web.archive.org/web/20100615221333/http://odin.physastro.mnsu.edu/~eskridge/astr225/week14.html |date=2010-06-15 }}</ref>
* [[Single-scattering albedo]] in [[radiative transfer]].
* [[Mean fitness]] of a population in [[biology]].
* [[Weight (representation theory)|Fundamental weights]] of a representation (probably to better distinguish from elements <math>w</math> of the Weyl group, than the usual notation <math>\omega</math>).
* The [[lemniscate constant]].<ref>{{cite journal |last1=Kobayashi |first1=Hiroyuki |last2=Takeuchi |first2=Shingo |title=Applications of generalized trigonometric functions with two parameters |journal=Communications on Pure & Applied Analysis |date=2019 |volume=18 |issue=3 |pages=1509 |doi=10.3934/cpaa.2019072|arxiv=1903.07407|s2cid=102487670 }}</ref>

== Unicode ==

Lower-case pi was fairly common in 8-bit character encodings, for instance it is at {{tt|0xE3}} in [[CP437]] and at {{tt|0xB9}} on [[Mac OS Roman]].
The various forms of pi present in [[Unicode]] are:

* {{unichar|03A0|GREEK CAPITAL LETTER PI|html=}}
* {{unichar|03C0|GREEK SMALL LETTER PI|html=}}
* {{unichar|03D6|GREEK PI SYMBOL|html=}}
* {{unichar|1D28|GREEK LETTER SMALL CAPITAL PI|html=}}
* {{unichar|213F}}
* {{unichar|213C}}
* {{unichar|220F|html=}}
* {{unichar|2210|html=}}
* {{unichar|2CA0}}
* {{unichar|2CA1}}

These are intended for use as mathematical symbols. Text written in the Greek language (i.e. words, as opposed to mathematics) should not come from any of the tables on this page, but instead should use the normal Greek letters, which have different code numbers and often a different appearance. Using the mathematical symbols to display words (or vice versa) is likely to result in inconsistent spacing and a clumsy, mismatched appearance:

* {{unichar|1D6B7}}
* {{unichar|1D6D1}}<ref>{{cite web|url=https://www.fileformat.info/info/unicode/font/calibri/list.htm|title=Unicode characters supported by the Calibri font|work=fileformat.info}}</ref>
* {{unichar|1D6E1}}
* {{unichar|1D6F1}}
* {{unichar|1D70B}}
* {{unichar|1D71B}}
* {{unichar|1D72B}}
* {{unichar|1D745}}
* {{unichar|1D755}}
* {{unichar|1D765}}
* {{unichar|1D77F}}
* {{unichar|1D78F}}
* {{unichar|1D79F}}
* {{unichar|1D7B9}}
* {{unichar|1D7C9}}

== See also ==
{{Wiktionary|Π|π}}
* [[Pe (Cyrillic)|П, п – Pe (Cyrillic)]]
* [[P|P, p – Pe (Latin)]]
* [[Greek letters used in mathematics, science, and engineering#Ππ (pi)]]
*[[Tau]]

== References ==
{{Reflist}}

[[Category:Greek letters]]