Published on: 6/22/2014IST
Finally, find out: was the zero really reinvented in India?
A new collection of essays examines the most popular stories Indians tell each other about the nation's history.
One day, in March 2010, the editor of the newspaper I worked for summoned me to his room.
‘What do you know about Swiss watches?’ he asked me out of the corner of his mouth as he typed away on his laptop, no doubt ripping some poor reporter somewhere.
‘My father used to collect some of the cheaper brands once upon a time. And I thought he was nuts for wasting his time and money on them.’
‘But you’ve heard of Omega and Rolex and TAG Heuer and all that?’
‘Yes, of course.’
‘Great. You’re going to Basel this year...’
A few days later, I was on a plane from Delhi to Zurich to cover Basel World, the world’s largest and most important annual watch and jewellery fair. Since then, watches have become something of a personal obsession. Each year, I travel to Switzerland at least half a dozen times to attend fairs, visit factories and interview watchmakers. And, on average, I publish approximately 200 pages worth of watch editorial each year.
I have also started behaving exactly like my father. I can spend hours outside a watch store, just looking through the display window and whimpering softly. I’ve even accumulated something of a fledgling collection. But none of them is particularly expensive or rare, though there is one HMT automatic, an NASL-03, that I am particularly proud of.
During one of these trips to Switzerland, I once interviewed a master watchmaker who actually had his brand named after himself. This, by itself, is not all that rare in the Swiss watch business; in fact, it is the norm. Almost every watch brand, barring a few such as Rolex, is named after the original founder. The Swiss are quite proud of this, and try to flaunt the age of their brands as widely as possible. But a successful watch brand named after a watchmaker who is still alive? That is pretty rare. Most Swiss brands are named after founders who died centuries ago.
The industry tends to be full of either grim, inscrutable automatons or flamboyant showmen who lie through their teeth. This fellow, one of the greatest watchmakers alive right now, was neither. In fact, he seemed something of a romantic and an amateur philosopher. ‘I love your country,’ he said. ‘You Indian guys are so intelligent, so smart. You discovered so many mathematical things.’ For the next thirty minutes or so, he spoke about the Fibonacci series, the golden ratio and other such mathematical curiosities. In the end, he returned to the subject of India: ‘But all this was made possible only because of you Indians. You guys invented the zero! Without the zero...’ He threw up his arms, shrugged and exhaled loudly. Nothing, he seemed to say. Without the zero, there would be nothing.
A few days later, I spotted him having lunch in a restaurant, surrounded by a bevy of Asian women. I was still in a daze after our glorious interview, so I quickly Googled up the French Wikipedia page for the ‘inventor of the zero’ – Aryabhatta, of course – and handed my phone over to him. I admit I was hoping to impress him with ‘Indian heritage’.
When I looked over a few minutes later, the phone was just lying on the table in front of the watchmaker; he was busy snacking on a woman’s ear while she giggled appreciatively. They did not notice when I gingerly walked over and retrieved my phone, with the Aryabhatta page untouched.
India’s claim to the invention of the zero is perhaps the most widely used – and abused – ‘India fact’. It appears on every single list of facts I have gathered in the course of my research. It is so popular that it has graduated from fact to dogma and then all the way to the butt of jokes. It is also one of those rare facts that is repeated with complete credulity in both Indian and international literature. A December 2012 news article on the Scientific American website about the 125th anniversary of the Indian mathematician Srinivasa Ramanujan, was subtitled: ‘India, home of the number zero, ends a year-long math party in unique fashion’.
Everyone, everywhere, it seems, is in broad agreement that the zero was invented in India.
Or was it? The Sceptical Patriot is not one to be fazed by national and international repetition. We must find the true story.
And what an intriguing story it proves to be.
***
In Gwalior, there is a fort commonly known as Gwalior Fort. Next to the fort is the small Chaturbuja temple. Inside the temple is a statue with four arms but no face. It did once have a face, but it has since been vandalised. There are two inscriptions in this temple. One is engraved over the main door. The other is inscribed into an indentation, roughly square in shape, on the left wall of the sanctum sanctorum as you enter it, to Lord Vishnu’s right.
The temple had fallen into ruin long before the first archaeologists began studying it in the late nineteenth century. The inscription over the main door lay unnoticed even after initial excavations. It was first noticed, copied down and translated into English in 1883 by our old friend and expert Indologist, Eugen Julius Theodor Hultzsch. The second one inside the sanctum sanctorum had been transcribed before, but Hultzsch copied it down again anyway.
Hultzsch seems surprised at the quality of the prose in the inscription over the door:
The first inscription consists of 27 Sanskrit verses and must have been composed by an ingenious pandit, who was well versed in alamkara. His extravagant hyperboles will appear startling and amusing even to one accustomed to the usual kavya style.
The second inscription from the tablet next to Vishnu is not so great. It is written, Hultzsch says, in ‘incorrect Sanskrit prose’.
But then, history and discovery are eccentric muses. Sometimes they care not for art and aesthetics. The first inscription that impressed Hultzsch has passed into the annals without emitting even a low whimper. Nice, but meh.
The second, shoddy inscription, on the other hand, is one of the most important records in the history of mathematics. If there is any record in all of India that is fully deserving of generating and maintaining its own cannon of India facts, this is it. There should be entire museums complete with multimedia displays and gift shops dedicated to this inscription.
So what does this piece of inscription say? Does it reveal the name of a mysterious king? Give a concrete date for a historical episode that experts had argued over for decades? Does it tell the future, then, in some Nostradamic way?
Bewareth thee the phone that is all touch but no buttons. For children will buyeth expensive apps...
No. It is merely an inscription informing one of a donation that has been made to this temple. It goes like this:
Om! Adoration to Vishnu! In the year 933, on the second day of the bright half of Magha...the whole town gave to the temple of the nine Durgas...a piece of land belonging to the village of Chudapallika...270 royal hastas in length, and 187 hastas in breadth, a flower-garden, on an auspicious day...
Then, a little later, the transcription says:
And on this same day, the town gave to these same two temples a perpetual endowment to the effect...for the requirements of worship, 50 garlands of such market flowers as available at the particular season.
There is more to this second Gwalior inscription. But these lines are the relevant bits.
So, what is so groundbreaking about these lines?
Simple. The numbers in them. Especially the two measures in hastas and the number of flower garlands. Inscribed in 876 CE, this inscription is the oldest text anywhere in India in which the zero is used in exactly the way we use it today. (The inscription itself refers to year 933 in the Saka calendar. In case you’re wondering.) And not just because the zero in 270 hastas or 50 garlands looks like the modern zero -- it does; it looks like a small circle. But also because it is used in the way it is, both as a placeholder for no value and a number in its own right.
There is broad agreement amongst researchers that the inscription at the Chaturbuja Temple in Gwalior is one of the earliest records anywhere of the modern zero. In February 2007, Bill Casselman, a professor at the University of British Columbia, wrote a brief essay titled ‘All for Nought’ for the website of the American Mathematical Society. In the essay, he talks of a journey he made to Gwalior to have a look at the inscriptions. He wrote:
What is surprising about these numbers is that they are so similar to what modern civilization uses currently. The more you learn about how our current number symbols developed – transmitted from the Hindus to the Persians, then to Mediterranean Islam, and differently in East and West – the more remarkable this appears...
What the Gwalior tablet shows is that by 876 CE our current place-value system with a base of 10 had become part of popular culture in at least one region of India.
So, are we done with this chapter, then? Pats on back all around, 10/10 for this ‘India fact’? Also, what is all that confusing talk of placeholders and numbers and usages?
Alas, that is the problem with the history of the zero. It is much more complicated than a little circle that stands for nothing.
And this is why establishing India’s ownership of the zero will take a little more sceptical enquiry, one that will take us far, far away from that little abandoned temple in Gwalior.
***
William Gladstone
William Ewart Gladstone was one of the greatest politicians in British history. He became prime minister not once or twice but four times. And he left the British government with a legacy of liberal thinking that continues to influence it in direct and indirect ways to this day.
Gladstone was also a Homer fanatic. He read, reread and re- reread works by the great Greek epic poet, first as a student of the classics, and then just for the pure awesome heck of it.
Then, suddenly, during yet another reading of the Greek epics, Gladstone noticed something strange. In all of Homer’s work, not once was there a reference to the colour blue. Not once. Never. Despite several mentions of seas and skies and other things we would normally associate with the blue colour, Homer never actually used the word ‘blue’ in his work.
Gladstone came to the conclusion that this was because Homer and most other Greeks of his period were colour-blind. Their eyes simply didn’t register the colour blue.
Since then, other researchers have disproved this theory and come up with many of their own. The German philosopher Lazarus Geiger took Gladstone’s analysis and extended it further, across several other great epic poems and religious texts of many other religions around the world. Geiger made a stunning discovery: Blue scarcely made an appearance anywhere.
He wrote in his 1880 book History and Development of the Human Race:
If we consider the nature of the books to which this observation applies, the idea of chance must here be excluded. Let me first mention the wonderful, youthfully fresh hymns of the Rigveda, the discovery of which amidst the mass of Indian literature seems destined to become as important to the present century in awakening a sense of genuine antiquity as the revival of Greek antiquity at the threshold of modern times was to that period in arousing the sense of beauty and artistic taste. These hymns, consisting of more than 10,000 lines, are nearly all filled with descriptions of the sky. Scarcely any other subject is more frequently mentioned; the variety of hues which the sun and dawn daily display in it, day and night, clouds and lightning, the atmosphere and the ether, all these are with inexhaustible abundance exhibited to us again and again in all their magnificence; only the fact that the sky is blue could never have been gathered from these poems by anyone who did not already know it himself.
Which is Geiger’s roundabout way of saying that while the Rigveda refers to the sky several times, it never actually calls it ‘blue’.
I first came across all this analysis by Gladstone and Geiger on an episode of Radiolab, my favourite radio show/podcast in the whole world. Produced by a New York public radio station, Radiolab explores one topic each episode through the medium of fascinating stories. The whole Gladstone bit came up during an episode called ‘Colours’.
Now if you’re wondering what all this has to do with the concept of zero...well, it doesn’t have anything to do with it directly. But indirectly, I wanted to bring up the complicated notion of identity.
Let us assume, for a moment, that the Greeks actually didn’t have a real word for the colour blue. Does this mean that they never saw the sky or noticed its colour? Absolutely not. Unless the Greeks didn’t have a sky, or had one but in purple. This seems unlikely. Awesome, but unlikely.
So did someone have to invent blue for them? Think about it. (I am trying to.) Blue was all around them all the time. They just didn’t have a name for it. Or find the need to. Until one day somebody decided that the colour of the sky deserved a name. And that name would be: blue. Or whatever was the local language equivalent for blue.
But would it make any sense to call this bright individual the inventor of blue? After all, it is not like the stuff wasn’t around till he/she came along and decided to call it something. It was there all along. All our inventor managed to do was to give it a name and an identity.
When we talk about the ‘invention’ of zero, we’re faced with a similar problem. How do you invent a number?
Now, the term ‘zero’ itself can mean many things. But for the purposes of this enquiry, just two or three of them should suffice.
First of all, zero stands for nothing. A void. Nothingness. An absence of anything. So if ‘one’ represents a single instance of something, ‘zero’ represents no instances of that thing. But, like blue, this is not really something you would expect someone to have invented.
Mrs Caveman: How many mastodon kebabs did I cook?
Mr Caveman: Three?
Mrs Caveman: How many did you eat?
Mr Caveman: Three?
Mrs Caveman: How many do I have left for myself now, you greedy pig?
Mr Caveman: I cannot answer that question because I am yet to develop a sense of nothingness or a term for this sense.
Mrs Caveman: Damn! Every single time...
And even if someone did invent the idea of nothingness and a term for it, it seems ludicrous to try to fix a time or place for it. Also, chances are that this sense of ‘nothing’ developed in many places simultaneously.
So let’s skip that definition of zero.
There are two more.
One of the best, most concise histories of the zero I’ve read anywhere is an online essay titled, would you believe it, ‘A History of Zero’, written by two professors of mathematics at the University of St Andrews.
Professors JJ O’Connor and EF Robertson write about the other two uses of zero:
One use is as an empty place indicator in our place-value number system. Hence, in a number like 2106 the zero is used so that the positions of the 2 and 1 are correct. Clearly 216 means something quite different. The second use of zero is as a number itself in the form we use it as 0. There are also different aspects of zero within these two uses, namely the concept, the notation, and the name.
To me, both of these ideas are somewhat less abstract than the notion of nothing. And therefore, in a sense, more ‘inventable’. Alas, in the very next paragraph the good professors write:
‘Neither of the above uses has an easily described history.’ Ugh.
So let us start, then, at one of the earliest systems of writing in the world: the Babylonian cuneiform. Did they have a sophisticated understanding of mathematics? And if so, how and when did they start representing zeroes in their texts?
Babylonian Clay Tablet, with annotations
By 3000 BCE, more or less around the time the Indus Valley civilization was establishing itself, the Babylonians had developed a system of positional numbering very similar to the number system we use today. This means that they wrote long numbers with digits in the ‘one’s place’, then ‘ten’s place’, and so on. Except that while we use a base-10 system, the Babylonians used base-60. We have 10 digits in our number system, the Babylonians had 60. (Well 59 actually. They didn’t have a zero for a long time.)
The easiest way to explain the difference between base-10 and base-60 without making you want to throw this book against a wall/spouse in frustration is to ask you to look at your clock or watch. Credit the Babylonians and history’s propensity for memory, but to this day we still measure time on a base-60 system like the Babylonians. So, if I told you to add 1 hour and 34 minutes to 2 hours and 40 minutes, you’d effortlessly carry out a base-60 addition and tell me 4 hours and 14 minutes.
So how did the Babylonians indicate numbers like, say, 602? In the beginning, they did this by just leaving an empty space between the symbols for 2 and 600 to indicate that there was nothing in the ‘ten’s place’. After hundreds of years of doing this, sometime around 700 BCE (but perhaps earlier), the Babylonians started indicating these empty spaces with a special symbol to indicate positions with no numbers.
OH MY GOD, THE BABYLONIANS INVENTED THE ZERO BEFORE INDIA!
No! No! No! [Slap across the face.] Calm down.
This symbol – often two wedges but sometimes one or three – only indicated a zero as far as the first of the definitions that Professors O’Connor and Robertson outlined above: as an empty place indicator. The Babylonians still didn’t think of the zero as a digit by itself.
In fact, if you were to go back in time and ask a Babylonian to multiply the ‘wedge symbol by 10’, he’d probably laugh at your ignorance and then behead you just to be safe.
The wedge symbol was something of a half-zero. But not a zero in the modern sense.
Now, one popular embellishment of the ‘India invented zero’ fact is that without the zero it would have been impossible for mankind to accomplish complicated mathematics. One version even says: ‘If Indians had not invented the zero, man would have never walked on the moon.’
Cough.
So did that mean that Babylonian mathematicians fumbled about like little children, crippled with zero-lessness, cursed to spend their whole lives adding and subtracting and struggling to make a career out of it?
Not at all. The Babylonians, it turns out, were kickass at math. (Just like everybody in your class, right? I know the pain.)
They could do all kinds of cool things with fractions and binomial equations and even quadratic equations. But nothing, perhaps, indicates their mathematical ability more than a small round tablet that is part of Yale University’s Babylonian Collection. Around eight centimetres in diameter, the tablet is commonly referred to as object number YBC 7289, and has a calculation inscribed into it in the Babylonian cuneiform script.
According to this tablet, dated between 1800 and 1600 BCE, the Babylonians calculated the square root of 2 as 1.41421296. I just punched in square root of 2 on my laptop’s calculator and I got 1.41421356. Almost 4,000 years ago, the Babylonians could calculate the square root of 2 to within five decimal places of modern computers! The accuracy would remain unmatched for thousands of years. So they got along quite swimmingly without the modern notion of zero.
Over time, the Babylonian tendency to use a symbol to denote an empty placeholder would spread east and westwards. The Greeks, great mathematicians themselves, perhaps took one small step for zero-kind: Some records suggest that they used a circular symbol as a placeholder. However, we are still centuries away from zero being used as an actual number.
O’Connor and Robertson write:
The scene now moves to India where it is fair to say the numerals and number system was born which have evolved into the highly sophisticated ones we use today. Of course that is not to say that the Indian system did not owe something to earlier systems and many historians of mathematics believe that the Indian use of zero evolved from its use by Greek astronomers. As well as some historians who seem to want to play down the contribution of the Indians in a most unreasonable way, there are also those who make claims about the Indian invention of zero which seem to go far too far.
These are the biases and tendencies that make nailing down ‘India facts’ such as this one so difficult. So many commentators are driven not by a need to reveal the truth but to drive home a point. Swirl in some patriotism, racism or cultural chauvinism – and you have the perfect environment that breeds unsubstantiated cultural legend.
Indian: India invented the zero! And you know what Gandhi said when they asked him about Western civilization?
Non-Indian: Blah blah. You guys didn’t invent anything. And you killed Gandhi...
Indian: HOW DARE YOU!!!
Etcetera, etcetera.
Thankfully, amongst all this biased nonsense, there are always a few historians going about their job in a comparatively honest and ‘truthful’ way. (I say ‘comparative’ because no one is ever truly bias-free.)
And, at this point, we shall stop pontificating and leap onto their robust shoulders for the rest of this enquiry.
Perhaps the first celebrity mathematician in Indian history was Aryabhatta. Nobody really knows where he was born. Suggestions for his birthplace range from Kodungallur in Kerala to Dhaka in Bangladesh. In fact, much of what we know about the life of Aryabhatta has been pieced together over the last century and a half, like a messy jigsaw puzzle. For centuries, it was believed that there were two Aryabhattas. This was only sorted out as recently as the 1920s. There is greater agreement about the fact that he spent at least some of his time as a mathematician and astronomer working in or around modern-day Patna, then called Kusumapura. And it is in Kusumapura that Aryabhatta is believed to have written his most famous work: the mathematical and astronomical handbook Aryabhatiya.
Several translations of the Aryabhatiya were prepared in the early years of the twentieth century. One popular translation, published by Walter Eugene Clark, a professor of Sanskrit at Harvard, opens with an engrossing monologue. The monologue starts as follows:
In 1874 Kern published at Leiden a text called the Aryabhatiya which claims to be the work of Aryabhata, and which gives... the date of the birth of the author as 476 CE. If these claims can be substantiated, and if the whole work is genuine, the text is the earliest preserved Indian mathematical and astronomical text bearing the name of an individual author, the earliest Indian text to deal specifically with mathematics, and the earliest preserved astronomical text from the third or scientific period of Indian astronomy.
Clark does sound a tiny bit sceptical, doesn’t he? That is only because this was a period when forgeries of ancient manuscripts were rampant. Not because the forgers wanted to rewrite history but because they wanted to make money.
Of course, there is little doubt now that the Aryabhatiya was an authentic work that was widely quoted and criticised through the ages, perhaps even in Aryabhatta’s own lifetime. Today the work is lionised and put up on a pedestal; but other ancient Indian scholars seem to have treated it with much less veneration. Indeed, one great source of verification for the Arybhatiya’s authenticity and age is widespread reference to it in other treatises and writings.
Aryabhatta appears to have been something of a prodigy. In verse 10 of the third section – a section titled ‘Kalakriya’, the reckoning of time – he writes:
When three yugappdas and sixty times sixty years had elapsed (from the beginning of the yuga) then twenty-three years of my life had passed.
Twenty-three seems to be a young age, in any era, for a work of this historical importance. In fact, it seems a pity that the Aryabhatiya isn’t more widely read, not so much for its didactic value but for the sake of curiosity and enjoyment. It is a remarkably short work. Clark’s extremely accessible translation is around eighty-two pages long, and anyone with a decent school education in mathematics should be able to make most of their way through it.
According to the Internet, the Aryabhatiya has been credited with everything from modern mathematics to commerce, business and even quantum mechanics. Which may be over-chickening the biryani a little bit, as my grandmother used to say.
The Aryabhatiya is, however, at least partly responsible for the global use of the base-10 system. Developed to a certain fullness in India, the system was later taken by the Arabs, along with Indian numerals, and propagated throughout the world. One of the Aryabhatiya’s most frequently quoted verses is the second verse from the ‘Ganitapada’, or mathematics, section. Clark translates it thus:
The numbers eka [one], dasa [ten], sata [hundred], sahasra [thousand], ayuta [ten thousand], niyuta [hundred thousand], prayuta [million], koti [ten million], arbuda [hundred million], and vrnda [thousand million] are from place to place each ten times the preceding.
Boom. The decimal system outlined in a single verse. Five hundred years later, the great Persian mathematician and polymath Abu Rayhan Al Biruni would repeat the content of this verse almost word for word in his Indica, a compendium of Indian religion and philosophy.
But does the Aryabhatiya refer to a zero? At all?
Kind of. Like the Babylonians, Aryabhatta also suggests using a placeholder, called kha, whenever there is no digit in a certain place in a number. So Aryabhatta would write 2,106 as two-one-kha-six. But he still didn’t use kha as a number itself.
Early researchers tended to call the kha Aryabhatta’s version of the zero numeral. But this view seems to have changed since then. Instead, credit for pushing the idea of zero even further than Aryabhatta is given to another ancient Indian mathematician, Brahmagupta, who lived around a century later.
Brahmagupta
Around 630 CE, Brahmagupta wrote the Brahma-Sphuta- Siddhanta. One thing is immediately clear from this book: something had changed drastically in the way ancient Indian mathematicians dealt with the zero. It had gone from simply being a place-value holder or a null-value indicator in Aryabhatta’s time, to becoming a proper numeral in its own right.
Almost. Brahmagupta writes:
The sum of zero and a negative number is negative, the sum of a positive number and zero is positive, the sum of zero and zero is zero.
Also:
A negative number subtracted from zero is positive, a positive number subtracted from zero is negative, zero subtracted from a negative number is negative, zero subtracted from a positive number is positive, zero subtracted from zero is zero.
So far so good. But then he begins to waver.
A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero.
This makes little sense. But Brahmagupta’s leap of thinking, which has him operating with the zero as a numeral, and not just an indicator of nothing or a placeholder, is phenomenal. As Connor and Robertson write: ‘...it is a brilliant attempt from the first person that we know who tried to extend arithmetic to negative numbers and zero.’
Given that Brahmagupta had already started thinking in these terms, the Gwalior inscription can be seen as the sign of a society at large slowly adopting cutting-edge mathematical ideas. Three centuries after Aryabhatta, and two after Brahmagupta, badly written temple donation inscriptions were using the zero not just as a null placeholder but exactly as we would use it today.
Now, if we put all these pieces together, a fairly unambiguous narrative begins to emerge.
India was certainly not unique in using a rudimentary form of the zero as a placeholder or as an indicator of null-value. Aryabhatta certainly did outline the decimal system in his work, and incorporated a kha into the decimal system. But he still didn’t really think of it as a numeral by itself. This had changed by the time of Brahmagupta, who was doing all kinds of nifty business with a zero. And then there is the Gwalior business. Surely India can then stake a claim for outstanding innovation in, if not invention of, applied and theoretical zero sciences?
Except for one more inscription. (I swear, no more in this chapter.) And here we need to bring back another one of our old friends: George Coedes of Sri Vijaya fame.
Right up until 1931, the Gwalior inscription wasn’t just the oldest instance of a zero numeral used in the modern sense in India, but in the whole world. In that year, Coedes published a paper in which he talked about an inscription in a ruined temple in Sambor in Cambodia. The inscribed tablet, that Coedes called K-127, said this in Old Khmer: ‘Chaka parigraha 605 pankami roc...’ which stands for: ‘The Chaka era has reached 605 on the fifth day of the waning moon...’ And it used a dot for the zero.
The date of this inscription? 683 CE.
Sure, it is not a little circular loop. But there is now widespread agreement that this is perhaps the oldest existing zero anywhere in the world, predating the Gwalior inscription by around two centuries.
So close. So close. If Gwalior had maintained its primacy, this chapter could have ended on a slightly more satisfying note.
Still, it is pleasing to know that this ‘India fact’ is not without substance. It is, in fact, quite agreeably watertight if one is willing to loosen the definitions of ‘invention’ just a little bit. Also who is to say that the Cambodians didn’t get their idea of the zero numeral from India? Entirely possible, given the huge sphere of influence Indian culture, religion and scholarship had in Southeast Asia around the time K-127 was carved.
But I also think that the story of the zero shows how invention in the ancient world was hardly a matter of eureka moments or light bulbs going off. Those guys sat and thought about things for a long time. They shared their ideas widely. They published widely. They criticised each other. It was as if they cared for the knowledge itself, and not the credit associated with discovering things.
How bizarre.
Excerpted with permission from The Sceptical Patriot: Exploring the Truths Behind the Zero and Other Glories by Sidin Vadukut. Published by Rupa Publications.
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