Unraveling Pingala Contribution to Mathematics

Pingala Contribution to Mathematics
Pingala Contribution to Mathematics

The most interesting Pingala Contribution to Mathematics is Combinatorics. Combinatorics is a field of mathematics that deals with counting decision-making or arranging things in a particular way you probably know this by a different name called permutations and combinations in your high school.

Which is used in fields like statistical physics, to study atomic Behavior, evolutionary biology, to study the combination of chromosomes, or machine learning, to study the exhaustive number of possibilities for making a decision.

It is quite easy to understand combinatorics, say you have three T-shirts and two jeans how many different ways can you dress up? your results are quite easy it’s six.

Let’s take a more complex problem imagine a cricket field where you have 25 common fielding positions and only nine fielders counting out the Wicket keeper and the bowler so you have nine fielders in 25 positions and then you have right-handed and left-handed batsman and right-handed and left-handed Bowers and round the Wicket with all these combinations how many exhaustive Fielding setups can we have?

Don’t try to count, it’s going to run into lacks and crores for this kind of complex problem. Combinatorics is used combinatorics is also used in DNA sequencing while studying human genetic variations there are many other fields where combinatorics plays a very crucial role.

All this is to help you understand how important combinatorics are in different fields of Science and Technology now let’s see how it originated from Sanskrit let’s take the Bhavani Ashtakam poem.History of Sanskrit language & Founder of sanskrit

Again just like how we arrange the fielders on a cricket field to get the best result same way syllables are to be arranged in such a beautiful way that they sound very pleasant to hear. For this Arrangement, Pingolacharya gave six algorithms Prastara, Nashtam, Uddhishtam, Lagakriya, Sankhya, and Adhvayoga.

Pingala binary numeral system

The first one Prastara is what we saw in the binary numeral system and now let us see Lagakriya which defines the combinations.

We have to go back to Pingolas Chandra Sastra again this is a lengthy algorithm and I’m going to cover it directly in English.

First, draw one square, and just below that draw two more squares in such a way that half of it overlaps with the above just like it’s shown below in the picture

Then repeat the process at the third row again with three squares and repeat the process with four squares shown in the below picture and continue the process to a certain level say for example four levels here we assume. You are reading the Pingala Contribution to Mathematics.

Pingala Contribution to Mathematics

In the topmost Square write 1 and then in the next level write one and one again and in the third level write one in the extreme boxes and leave the middle one empty the middle one should be a sum of the above two so here one plus one is two. How is Sanskrit related to computer?

Same way for the third level write one in the extremes and add the number above into the boxes like one plus two is three, two plus one is three. This arrangement of numbers is called Meru Prasthara.

Pingala Contribution to Mathematics

I was speechless when I first read this because this is called Pascal’s triangle in today’s world written by Pingala back in 200 BC and reinvented by Pascal in the 17th century.

I’ll explain how Meru Prasthara became Pascal’s triangle but that’s not the important point what’s rather important is to understand the mathematical features of this arrangement of numbers

Firstly the name Meru Prasthara metaphorically represents the mahama yantra of Sri chakra let’s not get much into this topic but I’ll briefly explain what it is in a little while

Pingala Contribution to Mathematics

Let us understand why Pingala gave this numerical arrangement of numbers. let’s take a simple problem so if you have five coins how many ways we can get only two heads and four tails, in how many different ways can I get these combinations?

We can just write it down we can just look up this Meru Prasthara table to solve this problem.

Pingala Contribution to Mathematics

We have six coins so we have to go to row number six R6 and then the first box represents zero heads and six tails that’s in one way. you can have zero heads in all tails in just one way so that’s one.

The second box says six so basically the six is one head and five tails in six different ways Kitchen items name in Sanskrit

The third one says two heads and four tails in 15 different ways. So the answer to the question is it is in 15 different ways that we can get two heads and four tails.

So basically Meru Prasthara is just a lookup table of numbers in terms of how can we play with different combinations. Now let’s exactly replace the six-coin problem with a six-letter word. Imagine that you would like to compose a six-letter word as part of a poem where you want only two gurus and four Laghus.Shree Yantra

So you want a six-letter word just like the coin problem go to the sixth row and then the third box is going to give you two gurus and four laghus combinations or vice versa. So that is in 15 different ways that you can write such kind of a word which has two gurus and four Lagus. you can read. You can also read 13 interesting facts about Sanskrit language

In the previous algorithm, 2 to the power of n gives you all the possibilities of writing a word here in a six-letter word 2 power 6 is 64. That means you can write in 64 different ways irrespective of how many Laghus or how many gurus are there a six-letter word can be written in 64 different ways.

but if you specifically want two Laghus and four gurus or vice versa then this table is to look up and understand how many ways you can write the expected combination of Laghus and gurus and that is the purpose of Meru prasthara.

This is the worksheet of an unknown Samskrutam poet writing down the Meru Prasthara almost in the year 755. You see the combinations of Guru Lagus. How they wrote to identify different words and their combinations. It was incredibly surprising for me to see this manuscript which is 1300 years old and this was way back in the year approximately 755 A.D.This is one thousand years before Pascal’s quote-unquote invented Pascal’s triangle. Is Sanskrit still spoken?

Pingala binomial theorem

Meru Prasthara has a lot of applications in different areas of mathematics. Here is one example of the binomial theorem. Meru Prasthara is nothing but the arrangement of the binomial coefficients, a plus b whole square is equal to a square plus 2 a B plus b square.

You see the coefficients can be directly read from the table the same way for a plus b whole Cube and again A plus b to the power of 4 you can directly read the binomial coefficients from the Meru prasthara.

binomial theorem
binomial theorem

Pingala mathematician fibonacci

Another very important mathematical feature of Meru Prasthara is if you add up the diagonals it gives out the Fibonacci series which is in Golden Ratio. these series are called as Fibonacci series because an Italian mathematician Fibonacci in the 13th century wrote a book called Liber Abaci

Fibonacci series

This book is all about introducing the Indian way of arithmetic operations to Europe. Thanks to Fibonacci he gave credit to Indians and he did not claim any of the work as his own.

Fibonacci was the first one to introduce the Indian way of mathematics or the Hindu numeral system that the whole world is using today. He is the first one to introduce that to Europe.

The Fibonacci series is very special because we see it directly visible for example a sunflower arranges all its seeds in a Fibonacci Sequence so that it can accommodate the maximum number of seeds in its womb, The same way the cauliflower also follows a Fibonacci Sequence it a huge topic if you’re interested read about Fibonacci series

Fibonacci series in nature

Now you know that the Fibonacci series came from Meru Prasthara like this many mathematical patterns are coming out of Meru Prasthara even today. Mathematicians are discovering new patterns concerning Meru Prasthara or Pascal’s triangle.

Which is more commonly called today in Hinduism the SRI Chakram Maha Meru Yantra is revered as a symbol of universal motherhood and a source of energy.

SRI Chakram Maha Meru Yantra

SRI chakram is considered the single source that disseminates different levels of energy. Metaphorically meru prasthara also disseminates different kinds of mathematical patterns maybe that’s the reason the names are alike.

How Pingalas meru Prasthara became Pascal’s triangle

let’s see how Pingalas meru Prasthara became Pascal’s triangle. He wrote it in 200 BCE. It was eventually translated by Omar Khan a Persian mathematician in the 11th century, a Chinese mathematician Yang Hui in the 13th century, Niccolo Tartaglia Italian mathematician in the 16th century, and finally by Blaise Pascal from France in the 16th century.

How Pingalas meru Prasthara became Pascal's triangle
how Pingalas meru Prasthara became Pascal’s triangle

This is a classic example of how knowledge originated in India and traveled across the world contributing to the betterment of humanity. let me be very clear on one aspect here I’m not saying that the human intellect is a property of Indians no that’s absolutely nonsense my only point is that credit should be given to where it rightfully belongs period.

If we roll back to Shiva’s sutras that’s where we started so Shiva’s sutras gave birth to the four Vedas (Rig, Sama, Atharva, Yajur ), and these four Vedas are accompanied by six Vedangas (siksha, niruktha,vyakarana, kalpa, jyotisha and chhamdas ) written by Pingalacharya in which he gave six combinatorial algorithms of which we could cover only two and couple of these were translated by different mathematicians across the globe.

which gave rise to binary number systems, algebra, combinatorics, infinite series, and so on, and has applications in the fields of electronic Computing, statistical physics, genetical analysis, machine learning, and so on.

So this is the end-to-end picture of how Shiva sutras which are the very basis of human speech an art form some scrutam that has given rise to all these mathematical and technological advancements that we have today. You can read about Why learn Sanskrit | What is sthita prajna?