Is It Even PASCAL'S Triangle?

Is It Even Pascal’s Triangle?

When we read the binomial theorem in mathematics, we come across an interesting term known as Pascal’s Triangle, a triangle that represents the coefficient for any index of the binomial. This, as we have been told, was developed by the marvellous mathematician Blaise Pascal, who also has made his contribution to the fields of physics, fluid mechanics, and depths of applied mathematics. But if I tell you that PASCAL’S TRIANGLE IS ACTUALLY NOT EVEN PASCAL’S?

This comes as a shocking statement to you, right? Now listen to the truth!

Story of Pingala’s Theorem

The shape of the triangle that Pascal’s Triangle forms actually was first seen in Indian Scriptures.

If we look into the scientific texts of the great personalities of India, we will find that this triangle was originally called Meru Prastaar and it was used not for solving binomial expressions for mathematics, but rather to decide the poetic meter on the basis of the number of words, line numbers, syllables. See, ancient India was such an advanced country.

Meru Prastaar was actually a lucid answer to the cryptic code of Pingala’s Theorem in Chanda Sastram where he says - ??? ????????? | This answer was given by Halayudha Bhatt in his book Mritsanjeevani which was composed at around 10th century CE.

Meru Prastaar’s Applications

Just like the modern form of the Meru Prastaar, which we call Pascal’s Triangle, it began with a square with 1 written on it, then on the 2nd row 2 squares, and in the nth row n squares were drawn. On the extreme edges, the boxes were filled with 1. Now, every square was filled up with the sum of numbers in two squares above it.

This may sound like similar to the modern day, but wait, there is a catch! Let’s dive into the fact that how this triangle was a basis to decide poetic meter for the scriptures, and how it logically explained the cryptic theorem of Pingala.

Before entering into the poetic applications, it should be known that a laghu is nothing but a short-form vowel syllable and a guru is a long-form vowel syllable.

OK, so the question that was taken up and resolved by Pingala was to choose a number of meters for a number of guru or laghu syllables, having a certain duration or expanse or prastaar.

This is what Prastaar looks like:

What Represents What

Row Number

So if we take any row from the Meru Prastaar, and then we note the row number, then that assigns the prastaar or span of the quarter - 1, like the second row depicts a quarter of prastaar or span count 1.

Cell Number in a Row

Another important part of the triangle is the position of a cell in the row. Why? Because it represents the number of gurus or laghus in the quarter. The cell position counted from the left is basically the number of gurus + 1.

Let me explain to you with an example:

Have a look at this cell in Pascal’s, pardon, Halayudha’s Meru Prastaar.

So, here’s a number, at 7th Row and 4th Cell from left. Now let’s revise the concept.

It is in the 7th row, so the quarter is of span 7-1=6

Now, it is in the 4th place, hence number of gurus → 4-1=3

Hence number of laghus → 6-3=3

Great.

But did I tell you, what is the number 20 even indicating?

The answer is simple.

It is the number of meters possible for a quarter with prastaar 6 and having 3 gurus and 3 laghus!

What's the logic behind this?

Let me explain.

As I have shown you what a prastaar looks like, you can correlate with the 12th standard problems of putting balls into boxes, right?

Let us consider you have G and L balls to put into boxes.

If you have 6 boxes, like the last example, and you get 3 G and 3 L balls, the permutations to put them with a maximum of one in each box is -

6!/3!x3! = 720/36 = 20

And wallah! A basic application of permutation made you understand the crux of this article. You get 20 meters for such a quarter of prastaar 6, which is nothing but 20 permutations of arranging balls into baskets!

Hence, when they found out the permutations and combinations of the syllables, the poets chose different meters to create a poetic rhythm, where they felt a need to find out the possible amount of rhythms to make. Such was the depth of their research and knowledge.

Thanks for reading this article! For more amazing articles, follow my newsletter!

You can also follow me on LinkedIn Koustav Bhattacharjee - I've mentioned here!

Keep knowing, and keep growing!