In the legend the Sage reveals himself to be Lord Krishna and tells the King that he doesn't have to pay the debt at once, but can pay him over time, just serve rice to pilgrims every day until the debt is paid off). This story reveals the magic of numbers – Binary exponential expansion along with an essential life skill, humility. The story doesn’t quite end here. The discovery of Binary numbers was also quite by accident.
WED, JUL 31, 2024 | UPDATED 13:51 PM IST
https://timesofindia.indiatimes.com/blogs/voices/sanskrit-poets-colonialism-and-the-curse-of-fibonacci/
Author Pankaj Jha is an Ivy League graduate educated in three continents- US, UK, and India. He is a senior consultant to investment banks on wall street after working for almost two decades on wall street. But his passion is working closely with schools, teachers, and students to help them achieve their best.
Calling it unprecedented would be too dramatic! But the nefarious impact of Colonialism is not only limited to human rights and equality and degradation of wealth but has also had a profound effect on lesser talked about topics like manipulation of discovery, and proper credit accorded to people who made it.
A central trailing theme that has created a false impression with many that Mathematics is only for people interested in Stem. In fact, our rich history shows that people associated with liberal arts have made important contributions. Mathematics is about creativity and originality, which can also come from poets!
Poetry and Mathematics
Many fundamental mathematical discoveries have their origins in the composition of poetry. The counting that came from poetry led to important mathematical theorems that were first discovered in the context of poetry. The rhythm of Sanskrit poetry makes it extremely mathematical and lends itself naturally to mathematical questions.
Rhythm of Poetry – Stressed and Unstressed Syllables
Stress is the emphasis that falls on certain syllables and not others; the arrangement of stresses within a poem is the foundation of poetic rhythm. The pattern of stressed and unstressed parts of words is known as the meter. It is the arrangement of words in regularly measured, patterned, or rhythmic lines or verses. Working out which syllables in a poem are stressed is known as scansion; once a metrical poem has been scanned, it should be possible to see the meter.
Sanskrit Poetry adds a new dimension of length.
In Sanskrit poetry, stressed syllables are called long syllables, and unstressed syllable is called short syllable. The reason is that the long syllable is longer than the short syllable when you pronounce it. In fact, a long syllable in Sanskrit is twice as long as a short syllable when you pronounce it. When you read Sanskrit poetry, a long syllable lasts two beats of time, and a short syllable lasts one beat of time.
Sanskrit Poems lends itself to mathematical questions
This simple setup leads to exciting mathematical possibilities and questions. These questions came from ancient poets, and they analyzed them in great detail. Often writing about these mathematical questions in poetry. For instance, a natural question that ancient Sanskrit poets were to ask when composing poetry is that a long syllable lasts two beats and a short syllable lasts one beat, so how many rhythms and poetic meters can one construct in eight beats? These questions were raised even in 300 BC.
One possibility would be Long, Long, Long, Long. Another would be Short, Short, Short……Short (eight times), or you can mix them, say Short, Long, Long, Short, Long, or Long, Long, Short, Short, Long. So what is the total number of ways you can fill eight beats with longs and shorts? A natural mathematical question that came into poetry back in 300 BC.
We ask how many ways we can write the number n (n= 8 in our case ) as a sum of 1s and 2s.
You can write 2 + 2 + 2 + 2, which is the Long, Long, Long, Long
or
1 + 1 + 1 +……..+1 that is 8 shorts Or various combinations in between, with order mattering as it is poetry!
A way mathematician would approach the question was the same way that the poet would approach the question back in 300 BC: to work out the first few examples, understand the pattern, formulate the theorem, and then prove the theorem.
Finding the pattern
Suppose there is only one beat to fill; then there is only one way: a short (one beat). There is 1 way.
If two beats, then we can have Long( 2 beat) or Short( 1 beat) Short (1 beat) so two ways
If you have three beats left, then you can do three Shorts or Long (2 beat) Short (1 beat), or you can do Short ( 1 beat) Long(2 beats), so 3 ways.
If you have four beats left, then four Longs or Long, Short, Short in three different ways, or Two longs. Giving all together 5 ways.
Similarly, we would have eight ways when you have five beats left.
It is written in this beautiful Sanskrit poem translation, which says Write down numbers 1 and 2, and then the following line says that every subsequent number you obtain should be by adding the two previous numbers.
Then if you look at the nth number, that will give you the poetic meters of numbers having n beats
So let’s try out this approach for filling 8 beats
The number of rhythms having 8 beats
We take the eight element of the sequence: 1, 2, 3, 5, 8, 13, 21, 34, 55,…
There are 34 rhythms having 8 beats. These are called the Virahanka numbers, after the linguist Virahanka (c. 700 A.D.), who first documented them and proved their recurrence property.
It is the first reference of these numbers that were first written about. Virahanka wrote about them in a complete way and also why these sequences gave the right answers.
These numbers were also written about by Gopala (c.1135) and Hemachandra (c. 1150).
Of course, they are ubiquitously known as the Fibonacci numbers in the West after the Italian mathematician who wrote about them about 500 years later. So Fibonacci was not even the first, second, or third person to talk about these numbers. Here is the irony of it all! Even in India, textbooks, math students, math teachers, and mathematicians call them the Fibonacci numbers!
The Virahanka numbers have a journal devoted to them. They are important in Poetry, where they first arose, visual arts and architecture, and nature’s art. Counting the number of petals in a daisy show numbers 13, 24, and 34. These are all numbers in the Virahanka sequence.
The point of it all
It is vital to preserve our history and culture and wean away the effects of Colonialism as it has had many adverse effects. If the notion that Sanskrit poets have made important contributions to mathematics had persisted across history, it would have attracted many talented and creative people to this field rather than pushing them away. The hope is that articles like these move our authorities to fix the harm that Colonialism has caused, and proper recognition is given to folks who have made significant contributions in mathematics and other areas.
Another article
Manish at Sepia Mutiny has an interesting entry on Fibonacci numbers which in fact should be called Hemecandra numbers.
The Fibonacci series is the set of numbers beginning with 1, 1 where every number is the sum of the previous two numbers. The series begins with 1, 1, 2, 3, 5, 8, 13, and so on. They were known in India before Fibonacci as the Hemachandra numbers. And the ratio of any two successive Fibonacci numbers approximates a ratio, ~1.618, called the golden section or golden mean.
It’s long been known that the Fibonacci series turns up freqently in nature. The numbers of petals on a daisy and the dimensions of a section of a spiral nautilus shell are usually Fibonacci numbers. For plants, this is because the fractional part of the golden mean, a constant called phi (0.618), is the rotation fraction (222.5 degrees) which yields the most efficient and scalable packing of circular objects such as seeds, petals and leaves.
But Bhargava points out that the series also shows up in the arts. Sanksrit poetry, tabla compositions and tango, to name a few examples, use the series to find the number of possible combinations of single and double-length beats within a stanza.[Sepia Mutiny: Hemachandra numbers everywhere]
Fibonacci himself wrote that he had studied Indian numbers and did not come up with the number series. Donald Knuth also wrote about this
Before Fibonacci wrote his work, the sequence Fn had already been discussed by Indian scholars, who had long been interested in rhythmic patterns that are formed from one-beat and two-beat notes. The number of such rhythms having n beats altogether is Fn+1; therefore both Gospala (before 1135) and Hemachandra (c. 1150) mentioned the numbers 1, 2, 3, 5, 8, 13, 21, … explicitly.
One more
https://mathenchant.wordpress.com/2021/11/16/why-names-matter/
Why Names Matter
5 Replies
I just went through my lesson plan for an upcoming lecture on number-sequences and replaced the name “Fibonacci” by the name “Hemachandra”. By the time you finish reading this essay, you’ll know why I did it, and if you’re a teacher, I hope you’ll do it too. [Note added on November 19: I might now go back again and change “Hemachandra” to “Virahanka”; see the Endnotes.]
To the extent that we can reconstruct the story of the famous sequence
1,2,3,5,8,13,21,…
from historical sources, the tale starts with the ancient Indian poet and mathematician Pingala (a contemporary of Euclid’s, give or take a century). For Pingala, these numbers arose from exhaustive consideration of the rhythmic possibilities of Sanskrit poetry. If you want a six-beat poetic phrase built out of short (1-beat) syllables and long (2-beat) syllables, how many possibilities are there? The answer turns out to be 13, so that’s the sixth term of Pingala’s sequence. Likewise, if one is playing the tabla, there are 13 different six-beat drumming patterns one can build from 1-beat and 2-beat components. (The 1-beat and 2-beat components are often rendered vocally as “dhin” and “dha” respectively, so that the two most dissimilar six-beat patterns would be the leisurely “dha, dha, dha” and the rapid-fire “dhin-dhin-dhin-dhin-dhin-dhin”.)
In a similar way, Pingala noted that the number of seven-beat patterns is 21, so that was the seventh terms of his sequence.
Pingala’s description of how one can compute new terms of the sequence is a bit terse (“The two are combined”), but later Indian scholars interpreted the phrase to mean that we compute a new term by combining (that is, adding) the two previous terms. For instance, to find the number of eight-beat patterns, we would add the number of six-beat patterns (13) to the number of seven-beat patterns (21), obtaining 34 as the number of eight-beat patterns.
WHY WE COMBINE
To see why this combination process gives us the right answer, notice that we can divide the eight-beat patterns into two categories: those that begin with “dha” and those that begin with “dhin”.
How many drumming patterns of the first kind are there? Each of them consists of a 2-beat “dha” followed by six more beats which can be built of dhas and dhins in any manner, so the number of eight-beat patterns that start with “dha” is equal to the total number of unconstrained six-beat patterns, which is 13.
And how many eight-beat drumming patterns of the second kind are there? Each of them consists of a 1-beat “dhin” followed by seven more beats which can be built of dhas and dhins in any manner, so the number of eight-beat patterns that start with “dhin” is equal to the total number of unconstrained seven-beat patterns, or 21.
Combining the two cases (which together account for all possible eight-beat patterns) we get 13+21, or 34. I believe that the mathematician and tabla player Manjul Bhargava plays all 34 of these patterns at the end of his informative video “Poetry, Daisies and Cobras”, but I’m not sure. Can any tabla aficionados enlighten me?
The terms of the sequence increase exponentially. For instance, the number of hundred-beat patterns is 573,147,844,013,817,084,101 — that’s far too many patterns for us to ever hope to list, yet we can know precisely how many patterns there are in that stupefyingly large collection of possibilities. I find that a bit magical.
Various Indian mathematicians wrote about the problem of counting rhythmic patterns in the centuries that followed Pingala. Perhaps the most notable was the twelfth-century polymath Hemachandra, who was like Pingala a poet and mathematician and also a linguist, philosopher, and political theorist. Hemachandra described the sequence around the year 1140, and he explained the rule for computing terms more clearly than earlier writers had, but he didn’t attach his own name or anyone else’s to the sequence.
THE FIB IN “FIBONACCI”
Several decades later, an Italian kid named Leonardo was born into a merchant family, got exposed to the mathematics of North Africa (which included much of what had been learned in India and the Arab world in the preceding centuries), traveled far and wide to pick up as much math as he could (see Endnote 1), and wrote a popular book about mathematics featuring a somewhat artificial problem about rabbits. The rabbits weren’t intended to be the main attraction, but the exponentially-growing sequence governing the rabbit population (1,2,3,5,8,13,21,…) struck Europeans as new and different, and a meme was born.
Leonardo was known as “Fibonacci” (“Bonacci” being his father’s name and “figlio” being Italian for “son”), so you might suppose that the term “Fibonacci sequence” came into prominence not long after Leonardo wrote about the sequence in his book. But in fact nobody used the term “Fibonacci sequence” until a century and a half ago, when the French mathematician Edouard Lucas decided to name the sequence after his Italian forerunner. (See Endnote 2.)
This is not the only instance of Europeans naming things after themselves or after other Europeans. It seems to be a thing Europeans like to do. Perhaps the most dramatic instance of this tendency is the way the Italian explorer Amerigo Vespuci succeeded in attaching his name to fifteen million square miles of Earth’s surface. But Leonardo himself gave credit where credit was due, at least as regards the main idea of his book: in championing the system for representing numbers that the West uses today, he rightfully credited India and the Arab world. So perhaps it’s time to imitate Leonardo rather than Lucas (see Endnote 3), especially now that people in the U.S. and elsewhere are seeking a more just reckoning with the past as part of a path toward a more just future.
We could do worse than start by decommissioning the phrase “Fibonacci numbers”.
MATH IS ALREADY GLOBAL; HOW CAN WE SPREAD THE NEWS?
There’s never been a better time for terminological make-overs than the current era of widely-available internet search. If you’re worried that renaming the sequence 1,2,3,5,8,13,21,… would cause massive confusion, consider that anyone can look up “Hemachandra sequence” to find out what it means in a matter of seconds. Likewise, search engines a decade or two from now will incorporate the knowledge that “Fibonacci number” is an old-fashioned term for “Hemachandra number”, so a search for either term will yield matches to both.
Perhaps others more historically versed than I will argue that “Pingala number” or some other phrase from India would be more appropriate than “Hemachandra number”. Let’s have lots of arguments like this, and have them in public! (See Endnote 4.)
Mathematics is a deeply global enterprise, and its true history shows ideas circulating around the planet. As Pingala noticed, it’s when we combine things that exponential growth occurs. When young Manjul Bhargava went back to the original writings of Carl Friedrich Gauss and found new ways to apply ideas that had mostly languished in obscurity, was he doing Indian mathematics or German mathematics? Neither, of course. He was doing mathematics, and he was doing what great mathematicians have done for centuries: taking good ideas wherever they can find them. Simplistic phrases like “imposition” and “appropriation” don’t do justice to this rich process of intercultural dialogue. For instance, what can one say about the Chinese mathematician Xu Guangqi who took some forgotten Chinese mathematics and falsely presented it to the Chinese royal court as being of European origin? (See Endnote 5.) Stories like these resist tidy categorization. And it is precisely these sorts of messy stories that are the most effective antidote to a reductive view of mathematics. If we mathematicians want non-mathematicians to see our subject in the way that we do, then the history of our discipline needs to take a more accurate measure of the contributions that people from all over the world have made to contemporary mathematics.
Why on earth should we present a view of mathematics that falsely stresses the contributions of Europeans when the truth is both more interesting and more reassuring?! If our terminology remains Eurocentric, can we blame people for concluding that our discipline needs to be decolonized? Plus: if we want to attract people from around the world to our discipline, shouldn’t we highlight its global nature, especially since that way of putting a “spin” on things is actually truer than the story we’re implicitly telling now via the names we give things?
Math belongs to everyone, but not everyone is getting access to it. There’s hard work to be done in making things right, especially in the realm of education, but there’s also easy work to be done. What I propose is comparatively easy: let’s credit mathematical ideas to the people and civilizations that gave rise to them. Fibonacci and Pascal and others had a good run of it, with their sequences and their triangles and whatnot, and they did important work that mathematical historians should recognize. But it’s time to bring others out from the shadows of history into the light to receive the recognition they deserve. Let’s get to work.
Thanks to Michael Somos.
One more
Fibonacci or Hemachandra Numbers
जयकृष्णः | ജയകൃഷ്ണൻ History: India October 20, 2004 1 Minute
Manish at Sepia Mutiny has an interesting entry on Fibonacci numbers which in fact should be called Hemecandra numbers.
The Fibonacci series is the set of numbers beginning with 1, 1 where every number is the sum of the previous two numbers. The series begins with 1, 1, 2, 3, 5, 8, 13, and so on. They were known in India before Fibonacci as the Hemachandra numbers. And the ratio of any two successive Fibonacci numbers approximates a ratio, ~1.618, called the golden section or golden mean.
It’s long been known that the Fibonacci series turns up freqently in nature. The numbers of petals on a daisy and the dimensions of a section of a spiral nautilus shell are usually Fibonacci numbers. For plants, this is because the fractional part of the golden mean, a constant called phi (0.618), is the rotation fraction (222.5 degrees) which yields the most efficient and scalable packing of circular objects such as seeds, petals and leaves.
But Bhargava points out that the series also shows up in the arts. Sanksrit poetry, tabla compositions and tango, to name a few examples, use the series to find the number of possible combinations of single and double-length beats within a stanza.[Sepia Mutiny: Hemachandra numbers everywhere]
IMPORTANT VIDEO
https://www.youtube.com/watch?embeds_referring_euri=https%3A%2F%2Fmathenchant.wordpress.com%2F&source_ve_path=Mjg2NjQsMTY0NTAz&v=siFBqH-LaQQ&feature=youtu.be
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