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{{Short description|3rd–2nd century BC Indian mathematician and poet}}
{{For|the subtle energy channel described in yoga|Nadi (yoga)}}
{{Infobox scholar
| image = 
| caption = 
| name = Pingala
| birth_date = c. 3rd or 2nd century BCE<ref name=plofker55/>
| era = [[Maurya period|Maurya]] or post-Maurya
| main_interests = [[Sanskrit prosody]], [[Indian mathematics]], [[Sanskrit grammar]]
| notable_ideas = ''[[Fibonacci number#History|Mātrāmeru]]'', [[Binary number#History|Binary numeral system]].
| major_works = Author of the  "''{{IAST|Chandaḥśāstra}}''" (also called ''Pingala Sutras''), the earliest known treatise on [[Sanskrit prosody]], Creator of Pingala's formula.
| influences = 
| influenced = 
}}
[[Acharya]] '''Pingala'''<ref>{{cite journal|title=The So-called Fibonacci Numbers in Ancient and Medieval India|last=Singh|first=Parmanand|url=http://www.sfs.uni-tuebingen.de/~dg/sdarticle.pdf|journal=[[Historia Mathematica]]|year=1985|publisher=[[Academic Press]]|volume=12|issue=3|page=232|doi=10.1016/0315-0860(85)90021-7|access-date=2018-11-29|archive-date=2019-07-24|archive-url=https://web.archive.org/web/20190724230820/http://www.sfs.uni-tuebingen.de/~dg/sdarticle.pdf|url-status=dead}}</ref> ({{Langx|sa|पिङ्गल|translit=Piṅgala}}; c. 3rd{{En dash}}2nd century [[Common Era|BCE]])<ref name=plofker55>{{cite book|first=Kim|last=Plofker|author-link=Kim Plofker|title=Mathematics in India|title-link= Mathematics in India (book) |pages=[https://books.google.com/books?id=DHvThPNp9yMC&pg=PA55 55–56] |year=2009|publisher=Princeton University Press|isbn=978-0-691-12067-6}}</ref> was an ancient Indian poet and [[Indian mathematics|mathematician]],<ref>{{Cite web|title=Pingala – Timeline of Mathematics|url=https://mathigon.org/timeline/pingala|access-date=2021-08-21|website=Mathigon|language=en}}</ref> and the author of the ''{{IAST|Chandaḥśāstra}}'' ({{Langx|sa|छन्दःशास्त्र|lit=A Treatise on Prosody}}), also called the ''Pingala Sutras'' ({{Langx|sa|पिङ्गलसूत्राः|lit=Pingala's Threads of Knowledge|translit=Piṅgalasūtrāḥ}}), the earliest known treatise on [[Sanskrit prosody]].<ref>{{cite book|author=Vaman Shivaram Apte|title=Sanskrit Prosody and Important Literary and Geographical Names in the Ancient History of India|url=https://books.google.com/books?id=4ArxvCxV1l4C&pg=PA648|year=1970|publisher=Motilal Banarsidass |isbn=978-81-208-0045-8|pages=648–649}}</ref>

The ''{{IAST|Chandaḥśāstra}}'' is a work of eight chapters in the late [[Sūtra]] style, not fully comprehensible without a commentary. It has been dated to the last few centuries BCE.<ref>R. Hall, ''Mathematics of Poetry'', has "c. 200 BC"</ref><ref>[[Klaus Mylius|Mylius]] (1983:68) considers the Chandas-shāstra as "very late" within the Vedānga corpus.</ref> In the 10th century CE, [[Halayudha]] wrote a commentary elaborating on the ''{{IAST|Chandaḥśāstra}}''. According to some historians [[Maharishi|Maharshi]] Pingala was the brother of [[Pāṇini]], the famous [[Vyākaraṇa|Sanskrit grammarian]], considered the first [[Linguistic description|descriptive linguist]]''.<ref name="FPencyclo">[[Pāṇini#FPencyclo|François & Ponsonnet (2013: 184)]].</ref>'' Others identify him as [[Patanjali]], the 2nd century CE scholar who authored Mahabhashya.

==Combinatorics==
The ''{{IAST|Chandaḥśāstra}}'' presents a formula to generate systematic enumerations of [[Metre (poetry)|metres]], of all possible combinations of [[Sanskrit prosody#Light and heavy syllables|light (''laghu'') and heavy (''guru'') syllables]], for a word of ''n'' syllables, using a recursive formula, that results in a partially ordered [[binary numeral system|binary]] representation.<ref>Van Nooten (1993)</ref> Pingala is credited with being the first to express the [[combinatorics]] of [[Sanskrit prosody|Sanskrit metre]], e.g.:<ref>{{Cite journal |last=Hall |first=Rachel Wells |date=February 2008 |title=Math for Poets and Drummers |url=https://www.jstor.org/stable/25678735 |journal=Math Horizons |publisher=[[Taylor & Francis]] |volume=15 |issue=3 |pages=10{{en dash}}12 |doi=10.1080/10724117.2008.11974752 |jstor=25678735 |s2cid=3637061 |access-date=27 May 2022 }}</ref>
* Create a syllable list ''x'' comprising one light (''L'') and heavy (''G'') syllable
* Repeat till list ''x'' contains only words of the desired length ''n''
** Replicate list ''x'' as lists ''a'' and ''b''
*** Append syllable ''L'' to each element of list ''a''
*** Append syllable ''G'' to each element of list ''b''
** Append lists ''b'' to list ''a'' and rename as list ''x''

{| class="wikitable"
|+ Possible combinations of ''Guru'' and ''Laghu'' syllables in a word of length ''n''<ref>{{Cite web |last=Shah |first=Jayant |title=A History of Pingala's Combinatorics  |url=https://web.northeastern.edu/shah/papers/Pingala.pdf}}</ref>
|-
!Word length (''n'' characters)!!Possible combinations
|-
| 1 || G L
|-
| 2 || GG LG GL LL
|-
| 3 || GGG LGG GLG LLG GGL LGL GLL LLL
|-
|}

Because of this, Pingala is sometimes also credited with the first use of [[0|zero]], as he used the [[Sanskrit]] word ''[[Śūnyatā|śūnya]]'' to explicitly refer to the number.<ref>{{harvtxt|Plofker|2009}}, pp. 54–56: "In the Chandah-sutra of Pingala, dating perhaps the third or second century BC, [...] Pingala's use of a zero symbol [śūnya] as a marker seems to be the first known explicit reference to zero. ... In the Chandah-sutra of Pingala, dating perhaps the third or second century BC, there are five questions concerning the possible meters for any value "n". [...] The answer is (2)<sup>7</sup> = 128, as expected, but instead of seven doublings, the process (explained by the sutra) required only three doublings and two squarings – a handy time saver where "n" is large. Pingala's use of a zero symbol as a marker seems to be the first known explicit reference to zero."</ref> Pingala's binary representation increases towards the right, and not to the left as modern [[binary numbers]] usually do.<ref>{{Cite book|title=The mathematics of harmony: from Euclid to contemporary mathematics and computer science|first1=Alexey|last1=Stakhov|author1-link=Alexey Stakhov|first2=Scott Anthony|last2=Olsen|isbn=978-981-277-582-5|year=2009|publisher=World Scientific |url=https://books.google.com/books?id=K6fac9RxXREC}}</ref> In Pingala's system, the numbers start from number one, and not zero. Four short syllables "0000" is the first pattern and corresponds to the value one. The numerical value is obtained by adding one to the sum of [[place value]]s.<ref>B. van Nooten, "Binary Numbers in Indian Antiquity", Journal of Indian Studies, Volume 21, 1993, pp. 31–50</ref> Pingala's work also includes material related to the [[Fibonacci numbers]], called ''{{IAST|mātrāmeru}}''.<ref>{{cite book |title = Toward a Global Science | author = Susantha Goonatilake |publisher = Indiana University Press |year = 1998 |page = [https://archive.org/details/towardglobalscie0000goon/page/126 126] |isbn = 978-0-253-33388-9 |url = https://archive.org/details/towardglobalscie0000goon |url-access = registration |quote = Virahanka Fibonacci. }}</ref>

==Editions==
* [[Albrecht Weber|A. Weber]], ''Indische Studien'' 8, Leipzig, 1863.
* Janakinath Kabyatittha & Brothers, ''Pingala Chhanda Sutram'', Calcutta, 1931.<ref>{{Cite book |url=http://archive.org/details/ChhandaSutra-Pingala |title=Chhanda Sutra – Pingala}}</ref>
* Nirnayasagar Press, ''Chand Shastra'', Bombay, 1938.<ref>{{Cite book |last=Pingalacharya |url=http://archive.org/details/in.ernet.dli.2015.327579 |title=Chand Shastra |date=1938}}</ref>

==Notes==
{{Reflist}}

==See also==
{{col div|colwidth=40em}}
* [[Chandas]]
* [[Sanskrit prosody]]
* [[Indian mathematics]]
* [[Indian mathematicians]]
* [[Binomial theorem#History|History of the binomial theorem]]
* [[List of Indian mathematicians]]
{{colend}}

==References==
* Amulya Kumar Bag, 'Binomial theorem in ancient India', ''Indian J. Hist. Sci.'' 1 (1966), 68–74.
* George Gheverghese Joseph (2000). ''The Crest of the Peacock'', p.&nbsp;254, 355. [[Princeton University Press]].
* [[Klaus Mylius]], ''Geschichte der altindischen Literatur'', Wiesbaden (1983).
* {{Cite journal
| doi = 10.1007/BF01092744
| volume = 21
| issue = 1
| pages = 31–50
| last = Van Nooten
| first = B.
| title = Binary numbers in Indian antiquity
| journal = Journal of Indian Philosophy
| date = 1993-03-01
| s2cid = 171039636
}}

==External links==
* ''[https://web.archive.org/web/20120616225617/http://www.sju.edu/~rhall/Rhythms/Poets/arcadia.pdf Math for Poets and Drummers]'', Rachel W. Hall, [[Saint Joseph's University]], 2005.
* ''[https://web.archive.org/web/20120716224803/http://www.sju.edu/~rhall/Multi/rhythm2.pdf Mathematics of Poetry]'', Rachel W. Hall
*
''[https://archive.org/details/eWNd_pingala-krita-chhandah-sutram-the-prosody-of-pingala-by-kapil-dev-dwivedi-2013-b Internet Archive]'', The Prosody of Pingala

{{Indian mathematics}}

{{Authority control}}

[[Category:Fibonacci numbers]]
[[Category:Ancient Indian mathematicians]]
[[Category:Ancient Sanskrit grammarians]]
[[Category:Indian Sanskrit scholars]]
[[Category:2nd-century BC mathematicians]]