Misrau Cha and Fibonacci


Professor Amartya Sen, never tires of pointing out the irony of the fact that even through the Indian political economist of Mauryan times – Kautilya (or Chanakya) lived a good seventeen centuries before the Italian, Machiavelli of the Medici-time Venice, and covered much the same ground as him – he is known as the “Machiavelli of the East” and not the other way round.

Perhaps it has something to do with who holds the power to propagate, something that is contentious and often gets lost in the politics of the times. But in a deeper sense, it is how histories are structured around real and imagined milestones. Once at a workshop on the so-called “epistemologies of the south” when the inevitable question came up – on whether the “east” (south) will ever catch up with the “west” (north) in terms of knowledge, a casual remark had left me wondering  – that the accumulation of knowledge is really a universal continuum. If for example, China were to dominate the world for the next 500 years, they will be able to project (out of the same existing milestones) that so much of it originated and evolved in Chinese minds. On the side, the compass metaphor: how east-west becomes north-south, always leaves me somewhere between perplexed and amused.

Anyhow, I love both the myth as well as the historicity of the Italy – India connection, and herein are three wonderful examples, and it is, as you would make out from the title – the third, that is the real reason for this essay.

The first is from the century, just at the beginning of the Common Era. Emperor Augustus had consolidated his hold over Egypt, and the Red Sea ports had become part of the trading infrastructure of the Roman Empire, especially for their trade primarily with India, and also secondarily with China. The trade was substantial enough for Pliny the Elder to warn the Roman senate on various occasions – that the empire’s gold reserves run perilously low, warning of a kind of balance of payments crisis of those days. They say at peak times, one fleet would depart daily from either end. India took all its payments in gold those days – in exchange for the textiles, spices, dyes (and wild animals for the circus) that Rome liberally imported. Some of this is corroborated by ancient documentation of the time – equivalent to today’s letters of credit issued by reputed traders on either side. Today historians do not deny that this was big-time trade, but its magnitude is often contested, especially depending on your political orientation. For the Indian right-wing, this is yet another thread to tie-up India’s mythical supremacy over the world in ancient times, in yet another way.

The second amuses me no end, and involves the most comical creature – the Water Buffalo. Among India’s vast cattle population today, the water buffalo is the most characteristic – jet black, wide expressionless face, trundling casually, holding up the traffic of the latest BMWs on India’s brand new highways as they criss-cross past its ancient villages, or on a summer day – lolling in the filth of the muddy village pond. However, its milk has that extra “content” prized even by the Indian gold medal winning athletes and sportsmen. Sometime in the seventh century (yes!), the Indian Water Buffalo reached Italy (via East-European herdsmen, as is best understood) and it was from the Water Buffalo’s milk that the Italians were able to finally master the making of Mozzarella Cheese that really makes the Italian Pizza the Italian Pizza. Geneticists do verify that the Mediterranean Water Buffalo indeed originated around that time from Indian stock. Ironically, I heard this story for the first time from an Italian chef in Johannesburg, and this does not seem to have currency in India as yet, and amusing because I can just imagine a Haryanvi politician, if he knew, speechifying – “arre bhaiya, duniya bhar mein jo pijja bane hai, bhains ki doodh ka bane hai. Wah, kitne garv ki baat hai, maari bhains” (brothers, in the world today, all pizzas are made with our water buffalo milk. Wow! We are so proud of you, O Water Buffalo).

But coming to Fibonacci …

In 1202 CE, when Leonardo Fibonacci of Pisa was about 32 years old, he wrote a book called “Liber Abaci”. The title roughly translates to – The Book of Calculations. While it has many amazing things to say about numbers in general, it has remained in the popular memory, as well as that of specialist mathematicians – for what have come to be known as “Fibonacci Numbers” or “Fibonacci Series” or “the Fibonacci Sequence”.  Very simply (and non-mathematicians please don’t get intimidated) it is the sequence of numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 … ad infinitum, where every number (after the first 1, actually) is the sum of the previous 2 numbers. What is so special with this series or sequence? It is precisely this that despite its simplicity, it brings to you some of the amazing properties that numbers can possess. Just to enumerate a few (among many others!):

As you go further down this sequence, the ratio of two successive numbers is indeed the Golden Ratio (1:1.618) that has been known to recur in aesthetics as the perfect proportion. Even in computational sciences, the fastest search technique was the “binary search” – that is sub-dividing an array into two equal parts till the search is completed, till someone discovered that sub-dividing into the Fibonacci proportion is faster.

The most amazing was the observation that many natural phenomena (arrangements of branches on trees, leaves on a stem, cones on a pine etc.) seem to follow the Fibonacci sequence, and somewhat exciting was a vague calculation that the number of rabbits in the universe (because of their penchant for reproducing rather enthusiastically) would follow the Fibonacci sequence as well. Over the years, some of these claims from the biological world have been found to be somewhat exaggerated, and even the petal / stem arrangements of only some species are known to follow the precepts of the thirteenth century Italian, only some of the time.

To those who know the “Pascal’s Triangle” that very quickly enumerates the number of possible combinations (examples – 2 elements give 3 combinations: a, b, ab and 3 gives 7: a, b, c, ab, ac, bc, abc etc.) Fibonacci numbers wonderfully connect the number of elements to the number of their possible combinations. But let us leave it at that!

But as President Obama once said half in jest I suppose, on the eve of a visit to India – I want to visit this great country because, whatever, the world has done, India has always done before!

Actually, just fifty years before Fibonacci, an Indian Jain scholar – Hemachandra (1089 – 1172) from Dhanduka, Gujarat had written a comprehensive treatise on the sequence of these magic numbers, and had referred to the earlier works of, among others: Gopala (c.1135), Halayudha (c.950), Virahanka (c.700) and also leading to the oldest known reference to a study of this sequence to a master linguist – mathematician who was a contemporary of either Panini (500 BCE) or Patanjali (200 BCE) and went by the name – Pingala, and who gives the final (or shall we say the earliest!) connect between these magic numbers and yet another manifestation of aesthetics – that is in the sound of words.

Sanskrit poetry is ancient. Even the Vedas (best estimate 1500 BCE) are precise in their versification and rhythm. The fact that they had to be remembered orally for nearly two millennia required them to be compact and well-defined to their very last syllable. The phonetics of Sanskrit, like any other language, recognizes that there are short and long (unstressed and stressed) syllables. In Sanskrit they make the additional rule that one long syllable = two short syllables. As is required in poetry that every line must have an equal number of syllables, Pingala went on to discover that the possible combinations of long & short syllables in a given line actually follow the Fibonacci sequence. The same has been found to be true for music as well – for the number of stressed and unstressed beats. And thus it was how the magic of this series was first discovered – at least as we know it now.

But coming back to Fibonacci …

Did Fibonacci write, unknowing of this ancient past, or was he trying to lay claim to an idea as his own? Actually – No!

The “Liber Abaci” was actually his exposition of Indian arithmetic to Europe, and the first known explanation of the decimal system (with zero) to the west, that we have continued to use to this day, universally. The chapter on what came to be known as the “Fibonacci” sequence was to give an example of how the numbers that arose out of the decimal system have wonderful properties, as demonstrated by Indian mathematicians that came before him.

The term “Fibonacci sequence” was first used by another European mathematician, Edouard Lucas, only in the nineteenth century. Of course, as is well known, Fibonacci like other Europeans had studied Indian mathematics, not by reading about them directly from the Sanskrit texts, but from the seminal Persian / Arabic translations of Al-Khwarizmi and Al-Khindi of c.830 to 850 CE. Even today, the numbers that we universally use are known as the Hindu-Arabic numerals. The “zero” they say came from India (most probably Aryabhatta c. 500 CE), but for many centuries the symbol for the other end – that of “infinity” – was the Arabic Aleph-cypher.

Thus, whenever we look closely, how any wonderful idea originated, evolved and flourished, and came down to us today, more often than not, it has had very universal roots. But this of course is just one small story – of Pingala and Fibonacci, and all those that came between and betwixt, that showed us what numbers could mean, in so many different ways, to all of us.

And what is the “Misrau Cha” in the title of this essay? It was the title of Pingala’s treatise that means – Mixture of the Two – meaning the combination of long and short syllables – and expounding then, of what we call now, as the Fibonacci sequence. At least for now, Pingala can lay claim to be known as “Fibonacci of the East” complete with the customary seventeen (or so) centuries that separated them.

© Johannesburg, November 2013

 

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