Sanskrit Tidbits-16:Metres in Sanskrit,Fibonacci Series,Pascal’s Triangle and Binomial Theorem

So far, you have gathered the following.  There are different types of metres.  Their names depend on the number of syllables (akshara) and the combinations of L and G.   Let me be a bit more specific.  To analyze a pada, I first look at the number of syllables.   This determines the nature or वृत्त of the metre.  Within the वृत्त, I next took at the combinations of L and G.  That determines the जाति.  With different वृत्त-जाति combinations, I have many different kinds of metres.  Ideally, all the padas in the shloka should be similar in structure.  If so, the verse will be called समवृत्त. But Sanskrit poets also found uniformity boring.  Sometimes, assuming four padas, they made the alternate padas similar – the 1st and the 3rd and the 2nd and the 4th.  That kind of structure was called अर्घसमवृत्त.  Sometimes, they didn’t want the padas to match at all.  That kind of structure was called विषमवृत्त.

The net has made learning and use of Sanskrit easier, as it has of many other things.  You will find a complete list of metres at the University of Heidelberg site.  Do visit it.  If you know the name of a metre and want to know about its structure, this site will tell you.  If you have a shloka and want to know what the metre is, the software on this site will tell you.  On this second part, I have had some problems with inputing and fonts, partly because they haven’t completed everything, but do try it out.

Aksharas are short or long, laghu or guru, L or G. We already know that.

If I have one akshara in the pada, the choice is L or G.  There is no other option.  The choices are 2¹ = 2.  If I have 2 aksharas in the pada, the choices are LL, GG, LG and GL.  That is, 2² = 4.  If I have 3 aksharas in the pada, the choice is 2³ = 8, LLL, LLG, LGG, GGG, LGL, GLG, GLL and GGL.  If I have 4 aksharas in the pada, the choice is 2⁴= 16.  I I have n aksharas, the choice is 2ⁿ.  All of us know this now.  If nothing else, we have studied it in school.  This is nothing but the binomial theorem and elementary combinatorial mathematics.  To be even more specific, it is not about the expansion of (a+b)ⁿ, but about the coefficients of that expansion.  With permutations being unimportant, in how many ways can I choose r objects out of n objects? ⁿC_r, no ⁿP_r. However, we are not talking about today, when all of us know this.  We are talking about Pingala in something like the 2nd century BCE.  It is therefore somewhat remarkable that he talked about the binomial theorem then.  Don’t get me wrong.  He didn’t prove the binomial theorem, or name it, or anything like that.  Proof would be the wrong expression to use.  Nor did he talk about the binomial theorem in general terms.  Specifically, he analyzed padas with 4 syllables and gave all these metres different names.

There is 1 pattern with LLLL

There are 4 patterns with LLLG, LLGL, LGLL, GLLL.

There are 6 patterns with LLGG, GGLL, GLGL, LGLG, GLLG and LGGL.

There are 4 patterns with GGGL, GGLG, GLGG, LGGG.

There is 1 pattern with GGGG.

Do you recognize the pattern given above?  This is known as Pascal’s triangle, after the French mathematician Blaise Pascal (1623-1662).  Pascal’s triangle is used to generate the binomial coefficients.  I take the numbers diagonally above, on both sides, and add them to get the number below.  If you don’t know what Pascal’s triangle is, just take a look at the triangle.  It will be clear.  Pingala didn’t construct a triangle.  However, he used the same technique to generate the coefficients.

When I used to interview students for admission (I have given it up now), I sometimes asked them, what is special about the telephone number 0112358134?  Few got the answer right.  It does look like a landline telephone number, with Delhi’s STD code.  Most said, it isn’t a real telephone number.  That’s true of course.  But that’s not the point.  This is the Fibonacci series, named after Leonardo of Pisa, who wrote a book under the name of Fibonacci in 1202.  You get each digit by summing the two preceding digits and Fibonacci series occur abundantly in nature.  (In my telephone number, I have done a little bit of cheating with the last digit of 4.)

There were many Sanskrit writers who followed Pingala – Virahanka (7^th century), Gopala (12^th century), Halayudha (12^th century) and Hemachandra (12^th century). Virahanka’s work no longer exists.  We only have quotations in subsequent works.  I hope I am not going to confuse you. How many patterns can I have with five L aksharas?  You will say LLLLL.  But you haven’t exactly understood what I have asked.  A G akshara is actually twice the maatraa of a L akshara.  Now you have understood the question.  The patterns that I can have with five L aksharas are LLLLL, LLLG, LLGL, LGLL, GLLL, LGG, GLG, GGL.  There are 8 of these.  Understood?  With 1 L akshara, I can only have 1 pattern.  With 2 L aksharas, I can have 2 patterns, LL and G.  With 3 to 6 L aksharas, I can have 8 patterns. (You will have to try this out on your own.)  I am going to skip more of the mathematics.  Read Rachel Hall’s paper.  The point is that this generates the Fibonacci series and the Fibonacci series was known to several authors who wrote on Sanskrit prosody.  Her PPT is also worth taking a look at. Bottom line, there was quite a bit of mathematics in analysis of Sanskrit prosody.  There were no “proofs” in the modern Western sense.  But the notion of “proof” is something we will revisit at a later date.