{{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]], [[Pascal's Triangle|Meru Prastāra]]
| 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 scholar, poet, grammarian 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> whose master work ''{{IAST|Chandaḥśāstra}}'' ({{Langx|sa|छन्दःशास्त्र|lit=A Treatise on Prosody}}), also called the ''Pingala Sutras'' ({{Langx|sa|पिङ्गलसूत्राः|lit=Pingala's Formulae|translit=Piṅgalasūtrāḥ}}), is the foundational text of ''Chandas'' (prosody and metrics)—one of the six ''Vedāngas'' (auxiiiary science) of traditional Indian scholarship.<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><ref>{{cite book |last=Mylius |first=Klaus |title=Geschichte der altindischen Literatur |publisher=Harrassowitz Verlag |year=1983 |isbn=978-3-447-02324-5 |location=Wiesbaden |page=68}}</ref>

The ''{{IAST|Chandaḥśāstra}}'' is a work of eight chapters written in the late ''[[Sūtra]]'' style, that relies on explanatory commentaries for full comprehension. Dated to the final 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> the was elaborated in 10th century CE by [[Halayudha]] in his commentary, the ''Mṛtasañjīvinī''. According to Indian tradition and historical accounts, [[Maharishi|Maharshi]] Pingala is described as the younger brother of [[Pāṇini]], the famous [[Vyākaraṇa|Sanskrit grammarian]] of ''Vyākaraṇa.''<ref>{{cite book |last=Matilal |first=Bimal Krishna |title=The Word and the World: India's Contribution to the Study of Language |publisher=Oxford University Press |year=1990 |isbn=978-0-19-562515-8 |pages=12–15}}</ref> Others traditions identify him with [[Patanjali]], the 2nd-century BCE scholar who authored the ''Mahābhāṣya''.

==Combinatorics==
The ''{{IAST|Chandaḥśāstra}}'' presents a recursive method to systematically enumerate [[Metre (poetry)|metres]] by generating all possible combinations of [[Sanskrit prosody#Light and heavy syllables|light (''laghu'') and heavy (''guru'') syllables]] for a verse of <math>n</math> syllables, producing a [[binary numeral system|binary]] representation. Pingala systematized Sanskrit metrics using six algorithmic procedures known as ''Pratyayas'':<ref name=shah>{{cite web |last=Shah |first=Jayant |year=2008 |title=A History of Piṅgala's Combinatorics |url=http://www.northeastern.edu/shah/papers/Pingala.pdf |publisher=Northeastern University, Boston | archiveurl=https://web.archive.org/web/20160706044528/http://www.northeastern.edu/shah/papers/Pingala.pdf | archivedate=2016-07-06}}</ref><ref>{{cite journal |last=van Nooten |first=B. |date=1993 |title=Binary Numbers in Indian Antiquity |journal=Journal of Indian Philosophy |volume=21 |issue=1 |pages=31–50 |doi=10.1007/BF01092744}}</ref><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>

* ''Prastāra'' (permutation/expansion): A systematic, recursive table generating all combinations of ''laghu'' (light, short) and ''guru'' (heavy, long) syllables for an <math>n</math>-syllable meter. The two syllable types can be represented as binary symbols, so the procedure corresponds mathematically to enumerating all <math>2^n</math> binary sequences.
* ''Naṣṭa'' (Recovery): An algorithm to determine the exact syllable sequence, given the rank/index number of a meter.
* ''Uddiṣṭa'' (Indexing): The inverse algorithm to find the rank/index number of a given sequence of syllabus.
* ''Laghu-kriyā/Guru-kriyā'' (Weight determination): Computing the number of metres containing a specific count of light or heavy syllables (yielding binomial coefficients).
* ''Saṅkhyā'' (Total calculation): Determining the total number of permutations (<math>2^n</math>) for a meter of length <math>n</math>.
* ''Mātrā-meru/Meru-prastāra'' (Pyramidal arrangement): The triangular array of combinatorial binomial coefficients (later known in Europe as Pascal's triangle) and additive sequences (later known as Fibonacci numbers).<ref>{{cite journal |last=Singh |first=Parmanand |year=1985 |title=The So-called Fibonacci Numbers in Ancient and Medieval India |journal=Historia Mathematica |volume=12 |issue=3 |pages=229–244 |doi=10.1016/0315-0860(85)90021-7}}</ref>

{| class="wikitable"
|+ Metrical combinations generated via ''Prastāra'' for length <math>n</math> <ref name=shah/>
|-
!Word length (<math>n</math> characters)
!Total meters (<math>2^n</math>)!!Combinatorial sequence (''Prasatāra'' order)
|-
| 1 
|2|| G L
|-
| 2 
|4|| GG LG GL LL
|-
| 3 
|8|| GGG LGG GLG LLG GGL LGL GLL LLL
|-
|}

Pingala is also credited with an early explicit use of [[0|zero]], using the [[Sanskrit]] word ''[[Śūnyatā|śūnya]]'' to 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> His binary system increases from left to right, rather than right to left as in modern [[binary numbers|binary]] notation.<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> 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> The combinatorial rules for metres developed by Pingala that were based on moraic time units (''mātrās'') were the mathematical foundation for the sequence known as Fibonacci numbers in the West. This sequence of numbers was later formalized by Indian mathematicians Virahānka (c. 6th–8th century CE) and Hemachandra (c. 1150 CE), centuries prior to Fibonacci.<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><ref>{{cite journal |last=Singh |first=Parmanand |year=1985 |title=The So-called Fibonacci Numbers in Ancient and Medieval India |journal=Historia Mathematica |volume=12 |issue=3 |pages=229–244 |doi=10.1016/0315-0860(85)90021-7}}</ref><ref>{{cite journal |last=Bag |first=Amulya Kumar |year=1966 |title=Binomial theorem in ancient India |journal=Indian Journal of History of Science |volume=1 |issue=1 |pages=68–74}}</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]]