Acharya Pingala

Dated to the final centuries BCE the was elaborated in 10th century CE by Halayudha in his commentary, the Mṛtasañjīvinī. According to Indian tradition and historical accounts, Maharshi Pingala is described as the younger brother of Pāṇini, the famous Sanskrit grammarian of Vyākaraṇa. Others traditions identify him with Patanjali, the 2nd-century BCE scholar who authored the Mahābhāṣya.

• Name — Pingala • Born — c. 3rd or 2nd century BCE • Era — Maurya or post-Maurya • Main Interests — Sanskrit prosody, Indian mathematics, Sanskrit grammar • Notable Ideas — Mātrāmeru, Binary numeral system, Meru Prastāra • Major Works — Author of the "Chandaḥśāstra" (also called Pingala Sutras), the earliest known treatise on Sanskrit prosody, Creator of Pingala's formula

The Chandaḥśāstra presents a recursive method to systematically enumerate metres by generating all possible combinations of light (laghu) and heavy (guru) syllables for a verse of n syllables, producing a binary representation. Pingala systematized Sanskrit metrics using six algorithmic procedures known as Pratyayas: • Prastāra (permutation/expansion): A systematic, recursive table generating all combinations of laghu (light, short) and guru (heavy, long) syllables for an n -syllable meter. The two syllable types can be represented as binary symbols, so the procedure corresponds mathematically to enumerating all 2^n 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 (2^n) for a meter of length n. • 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). Metrical combinations generated via Prastāra for length n • Word length (n characters) — Total meters (2^n) — 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 zero, using the Sanskrit word śūnya to refer to the number. His binary system increases from left to right, rather than right to left as in modern binary notation. 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 values. 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.

Acharya Pingala is recorded with the occupation mathematician. Classed in the genre mathematics. Recorded as mathematician, linguist, academic and writer. Associated with Vedic period.