Meru Prastara

Meru Prastara is basically a triangular array of numbers (arithmetic triangle), consisting of binomial coefficients often known as Pascal's Triangle. The term Meru in ‘Meru Prastara’ stands for Mount Meru in Sanskrit literature. Thus, Meru Prastara is “steps of Mount Meru”. The mention of Meru Prastara was initially found in the ancient text on Sanskrit prosody called Chandaḥśhastra, which was written by Pingalacharya around 200- 300 BCE. The last chapter of the book contains a mathematical treatment of Sanskrit metres. which offers the notion of triangular tabulation of numbers.

This triangle tabulation of numbers has travelled a long way before came into light as Pascal’s triangle.  Initially, around 200 BCE Indian text, the method of construction of a triangle was mentioned. Later, in China (Yang Hui's triangle (c. 1238-1298), in Italy "Tartaglia's Triangle," named after Italian algebraist Tartaglia, born Niccolò Fontana (1499-1557), and Pascal's Triangle, which came into existence in Europe after the publication of Pascal's book Traité du Triangle arithmétique in 1665.

Construction of Meru Prastara

Pingala, in Chandahshastra discussed some algorithms for arranging Sanskrit syllables Laghu(L)- short syllables (अ),i(इ),u(उ),ṛ(ऋ),ḷ(लृ) and Guru(G)- long syllables ā(आ),ī(ई),ū(ऊ),ṝ(ऋृ),e(ए), ai(ऐ),o(ओ),au(औ))  which forms a basis for writing Vedic metre. Sanskrit Metres may be broadly divided into three categories. First, according to the number of syllables in the metre (varnavrttas) and other, number of syllabic instants or matras (morae) - time measure (matravṛttas) and third ganas - groups of syllables (Singh, 2001). In the concluding verse of Chandaḥśāstra, a cryptic statement is given, which provides the notion for the construction of Varna Meru(Meru Prastara), Pingala lists six Pratyayas (problems and answers) for Varnavrttas in the Chandahshastra. One of the problems is Prastara Pratyaya (algorithm), which offered an account for the binary representation of syllabic metres.  La-Ga- kriya Pratyaya elaborates on the number of combinations generated by taking "n" syllables at a time. The letters "L" and "G" stand for the laghus and gurus in a specific metre. Pingala's aphorism “परेपूर्णम्” (6.34), परेपूर्णमिति (6.35), this is too compressed  and cryptic can be understood  only with the help of commentary, In the tenth century, Halayudha, an Indian mathematician elaborated that statement in his text Mṛtasañjīvani, a commentary on Chandaḥśāstra and derived a triangular table called Meru Prastara, The explanation as follows,

उपरिष्टादेकं चतुरस्त्रकोष्ठं लिखित्वा तस्याधस्ता दुभयतोऽर्धनिष्क्रान्तं कोष्ठद्वयं लिखेत्| तस्याप्यधस्तात्त्रयं तस्याप्यधस्तात्त्रयं यावद्भिमतं स्थानमिति मेरुप्रस्तारः| तस्यप्रथमे कोष्ठे एकसंख्यां व्यवस्थाप्य लक्षणमिदंप्रवर्तयेत| तत्र परे कोष्ठे यद्वृत्तसंख्याजातं तत् पूर्वकोष्ठयोःपूर्णंनिवेशयेत्| तत्रोभयोः कोष्ठकयोरेकैकमङ्कं दद्यात् मध्ये कोष्ठे तु परकोष्ठद्वयाङ्कमेकी- कृत्य पूर्णं निवेशयेदिति पूर्णशब्दार्थः | चतुर्थ्यां पङ्क्तावपि पर्यन्तकोष्ठयो- रेकैकमेवस्थापयेत् |मध्यमकोष्ठ योस्तु परकोष्ठद्वयाअङ्कमेकी कृत्य पूर्णं त्रिसन्ख्यारूपं स्थापयेत्|…….

“First write one square cell at the top. Below it, write two cells, extending halfway on both sides. Below that three, below that again, four, until the desired number of places (are obtained.) Begin by writing 1 in the first cell. In the other cells, put down the sum of the numbers of the two cells above it. ⋅⋅⋅”

According to Halayudha, In the triangular array of numbers, the second row represents the expansion of a meter with one syllable, and the third row, the expansion of the metre with two syllables, and so on. It expresses the basic rule for the construction of Meru Prastara, the number of metrical forms of r Gurus (or Laghus) in the Prastara of metre of n syllables is the binomial coefficient nCr. In terms of the binomial coefficients, the recursive relation is as follows:    nCr = n-1Cr-1   + n-1Cr.

The triangular array of numbers can be represented in the form of binomial sequence (L+G)n. The first row of the triangular array represents (L+G)0, similarly, the second row represents (L+G)1, and so on.

(L+G)0 = 1

(L+G)1 = 1G +1L

(L+G)2 = 1G2 +2GL +1 L3

(L+G)3 = 1 G3 + 3G2L + 3 GL2 +1 L3

(G+L) n = 1 Gn + n Gn-1L .…... + n.GLn-1 +1.Ln. This technique of expanding expression that has been raised to some finite powers, of summation of laghus and gurus, also leads to the developments of the binomial theorem.

The Agni Purana also mentions Meru Prastara and states Pingala’s rule for its formation.  The rule is also given by Virahanka in Vrittajatisamucchaya, Trivikarma and Bhatta Narayana ( the commentators of Vrttaratnakara of Kedara) also in the work of Prakrta Pingala, Moreover,  Gopala (c.1135)in his commentary on Vrittajatisamucchaya have illustrated the rule with figures of Meru Prastara. La-Ga-Kriya has also been described by Acharya Bharata in his Natyashastra. Another form of figures of numbers which also follows the same recursive formula, called Khand Meru (in Lilavati )and  Suci Prastara has been given by Varahamihira, Janasrya, Virahanka, Jayadeva, Jaykirti, Kedara, Trivikarma, Hemachanda.

Mathematics of Meru Prastara

Meru Prastara illustrates many concepts of elementary mathematics as well as higher-level mathematics, the number which forms a triangle are all natural numbers, The first row of Meru Prastara represents 1, and the next row represents the sum of two values from the previous row, and this forms Triangular numbers, numbers associated with a triangular array of dots. The nth number of the sequence can be represented by T(n) = n(n+1)/2 (Subramanium, K.B.,1994). The resulting triangular sequence is 1,3,6,10,15,21,28,36…so forth. The sum of the first n triangular number is (n (n+1)( n+2))/6 which represents the nth Tetrahedral number (Asiru, M.A.,2008). The sum of two values from the previous row in the Meru Prastara gives Tetrahedral numbers through the figurate numbers (Willson, R., 2015). The resulting tetrahedral sequence of numbers is 1,4,10,20,35,56…....so forth. Moreover, this triangular array of numbers also depicts powers of 2 and powers of 11.

The diagonal number of Meru Prastara denotes Fibonacci numbers. This popular sequence comprised of 0, 1, 1, 2, 3, 5, 8,13….so on, the recursive relation Un =Un-1+ Un-2 yields the nth term in the series. Although the name L.Fibonacci(1140) is attached to the series of numbers, the sequence was well known in India prior to him, in the process of expansion of matravrttas.  In the expansion of matra vṛttas, the number of variations of one mora is one, the number of variations of two morae is 2, 3 morae are 3, 4 morae are 5, and so on, which ultimately gives such a pattern of numbers. According to Pingala’s rule, miśrau ca (“and the two mixed”) is also meant for the expansion of matravrttas, The rule for the formation of numbers of variation of matravrttas was continued to be given in the later works on metrics, and various revised series have been mentioned in the works of mathematicians, including Abhinavgupta, Achrya Bharata, Acharya Virahanka, Acharya Gopala, Acharya Hemachandra, Bhatt Narayana. Acharya Hemachandra Suri(1150CE) gives the rule of formation of matra vrittas which is अङ्कान्त्योपान्त्य  योगः परे परेमात्रा मात्राणाम्| stating that “the sum of the last and the last but one number [of variations] is [that] of the matra-vrtta coming next”. In response to his own rule of variation of matra vṛttas, Hemachandra proposed the "statement 1,2,3,5,8,13,21,34 and in this way forward". Thus, It can be seen that the so-called Fibonacci numbers resulted from the variations of matravṛttas in the Indian subcontinent. While in Europe, Fibonacci’s Liber Abaci in 1202 describes the sequence in his book to explain the Indian number system and its mathematics which he learned from Arabian teachers.

Moreover, the elements and patterns of Meru Prastara provide some ideas of Permutations and combinations, which are prerequisites for modern combinatorics and computation, Indian prosody texts not only describe Meru Prastara, It also explore numerous recursive algorithms, computation of Binomial coefficients, the application of repeated partial sums of sequences and the formula for summing a geometric series and many more.

Meru Prastara and Binomial coeffiecients

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