One day while Sachin Tendulkar was building his new home opposite where I live, there was an effort to move many tons of marble.
I know you want to know about that.
Check my essay in the November issue of Caravan: Mr Tendulkar's Neighbourhood: Living next door to Sachin.
Comments always welcome.
October 30, 2011
Your public key, please
Question: where oh where can you read about large primes and the "p" language in the same essay?
Answer: My new Mint column. On air last Friday October 28.
I need to tell you that reliable sources tell me that this is the first piece of writing in human history that makes mention of Skipjack, Ramdulari and Shamir. A fact of which I am inordinately proud.
On that dubious note ... Check Your Public Key, Please.
And as ever, your comments welcome.
Answer: My new Mint column. On air last Friday October 28.
I need to tell you that reliable sources tell me that this is the first piece of writing in human history that makes mention of Skipjack, Ramdulari and Shamir. A fact of which I am inordinately proud.
On that dubious note ... Check Your Public Key, Please.
And as ever, your comments welcome.
October 16, 2011
Walking the Freedom Trail
The October-November 2011 issue of Conde Nast Traveller is an "India Special". Nevertheless, it carries an essay I wrote after a visit to South Africa last May, perhaps because it includes some musing I did on connections to India in Cape Town's beautiful Kirstenbosch Botanical Gardens.
I haven't figured out if the article is online, so the text is below. They called it "Walking the Freedom Trail", different from the title I gave it.
And yes, your comments welcome.
***
Trail to Somewhere
It's a grainy black-and-white photograph, one that practically screams "1960s". No really identifiable face in it, just human forms. Most are in a large crowd, a few off to the side by themselves. In my imagination, one of those figures by himself has his arm up, hand in a fist, shouting something stirring and passionate that, these years later, isn't really identifiable either.
It's in Johannesburg's Apartheid Museum, the photograph. The caption says it was taken at a Cape Town rally, tens of thousands strong. Dating from March 1960, this was one of the early demonstrations against apartheid's perverse pass laws, the mood also fanned that morning by the infamous Sharpeville massacre of a few days before. It was led by Philip Kgosana, then a young student-activist with the Pan African Congress, who lived in a teeming shantytown outside Cape Town. After some hard-nosed negotiation, the police told him that the "Justice" Minister, FC Erasmus, had agreed to meet a small delegation.
"This was an exciting moment," Kgosana would write years later. "Never in the history of South Africa had an 'agitator' forced a racist minister to succumb to an undesirable appointment." Of course, when Kgosana and his colleagues arrived to meet Erasmus, he didn't show up. So much for "justice". Instead, the police arrested them and put them on trial for incitement to public violence.
Days after I saw the photograph, I waited in a convenience store in Pretoria, idly wondering if this was the first time I would meet a figure from a museum exhibit. Kgosana walked in right on time, a stocky man with gentle eyes. Nobody paid him any attention. "Hey guys," I wanted to shout at the shoppers desultorily examining biltong and wafer packets, "this man is in your Apartheid Museum! He's one reason you're here today!"
I held my tongue.
It must be a sign of how far South Africa has come that Kgosana lives today not in a shantytown, but in a comfortable bungalow in a once-whites-only suburb of the city that, more than every other in the country, personified white domination over black. But it's also a sign of that long journey that this figure from the resistance to white rule is now just another 70-something year-old, anonymous in this place. And if people warn you about violent crime in South Africa in these post-apartheid times, the reality also is that this is the continent's most dynamic economy, a vibrant nation with a host of attractions for every kind of tourist.
How far removed from protests against apartheid.
As Kgosana drove me home, I thought: a country subsides into a long night of oppression and brutality. It takes decades of struggle to emerge from that darkness. The guides on the way are heroes large and not-so-large: the Mandelas, but the Kgosanas too. There's an eruption of euphoria in 1994, at journey's end. But inevitably the ordinariness of daily life returns. Which is as it should be.
And that's how Kgosana saw his country. "Sure there's crime in South Africa," he said as we spoke in his comfortable living room, echoing other conversations in other living rooms. "Of course it bothers me. But where did we come from? You can't forget that."
True. After convulsions like South Africa has seen, there will necessarily be a time of uncertainty and unrest. You don't emerge from darkness directly into utopia. Yet if some heroes return to obscurity, as Kgosana has, perhaps that is itself a sign of some normalcy.
Speaking of normalcy and utopias, we visited the Kirstenbosch Botanical Gardens.
I'll say this: give me one botanical garden, you've given me them all. Oh sure, this expanse of verdant vegetation outside Cape Town is picture-perfect gorgeous. There are elegant bird of paradise flowers (Strelitzia) in dazzling colours, enormous trees, an aroma garden with plant smells familiar and otherwise, and plenty more to gladden a horticulturist's heart. All very impressive, but not quite the place I might have visited on my own.
Then again, that's because I did not know, ahead of time, about the Braille Trail.
Across from the aroma garden, marked by ropes, the Braille Trail meanders into the trees and out again. Just another walk, unless you get into the spirit and use it as it was designed to be used. The name is, of course, the giveaway. This is a self-guided tour designed specifically for the blind. They hold the rope and walk, stopping at regular boards in Braille and English (for those who can't handle Braille). This way, they learn about the trees among which they stroll, the arboreal scents and surfaces that surround them. They get some idea of what fellow-tourists, the sighted ones, experience.
My 11 year-old and I decided to give it a shot.
We grabbed the rope, shut our eyes tight and started walking. For the first few minutes, it was surprisingly hard to keep my eyes closed, to actually trust that I'd be fine without sight, with just the rope to guide me. So I kept opening then, though just slits, unable to shake the silly thought that somebody was watching to see if I cheated. But when I settled into the experience, the eyes stayed closed, it wasn't silly, and I was astonished by just how sensory and fulfilling it was. The 11 year-old, just as charmed.
Without sight, my ears tuned in even faint sounds. My skin registered the gentle breeze. My nose, a series of forest smells. My fingers, the curving strands of rope, the roughness of tree trunks. Twice, I felt my way to a bench for a sit-down, then back to the trail. Simple things when I look back, but they left me refreshed and curiously humbled by this small taste of what it is to be blind.
But more than that, too. All through, words that Nelson Mandela made famous coursed, maybe incongruously, through my mind. Mine was no long walk, but something about this trail had me musing, maybe incongruously, about the one he wrote about:
"I have walked that long road to freedom. I have tried not to falter … I have taken a moment here to rest, to steal a view of the glorious vista that surrounds me, to look back on the distance I have come. But I can only rest for a moment, for with freedom come responsibilities, and I dare not linger, for my long walk is not ended."
Mandela: always thought-provoking. At trail's end, I paid silent tribute to whoever designed this walk, amazed by their singular thoughtfulness. Did they realize it would be so meaningful for the sighted too? After all, walking it with my eyes shut had me contemplating themes and metaphors like darkness and light, new ways of seeing, concern for your fellow human, going bravely into the great unknown. All in about 45 minutes. Long walk, for sure.
And for an Indian, such themes resonate. For they must have meant something, over a century ago, to an Indian whose insight and fibre were forged here. Here, in apartheid South Africa, decades before it woke to freedom and hope.
They remember Gandhi in this country too.
Constitution Hill in Johannesburg, now home to the country's highest court, used to be the site of a notorious prison. The cell where Gandhi was once detained is now a memorial to the man. A serene place to remember him, it has photographs, accounts of his arrests and meetings with various leaders, his writings and even a BBC interview from some months before he died, playing over the buzz of construction next door. This must be one of the few places in the world where you can actually hear Gandhi's thin voice.
From here, Gandhi crossed the ocean to a land yearning for freedom. With a unique cocktail of courage, morality, political savvy and empathy, and with a unprecedented cast of giants among men, he guided India to a new dawn in 1947. Darkness to light, you might say. Our own Braille Trail.
"I can't promise you sunlight and roses," I imagine Philip Kgosana saying that day to the gathered thousands in Cape Town, perhaps to his country itself. "But walk the trail with me. Stay the course. I can promise you hope."
For new dawns, on either side of an ocean, perhaps hope is all you need.
I haven't figured out if the article is online, so the text is below. They called it "Walking the Freedom Trail", different from the title I gave it.
And yes, your comments welcome.
Trail to Somewhere
It's a grainy black-and-white photograph, one that practically screams "1960s". No really identifiable face in it, just human forms. Most are in a large crowd, a few off to the side by themselves. In my imagination, one of those figures by himself has his arm up, hand in a fist, shouting something stirring and passionate that, these years later, isn't really identifiable either.
It's in Johannesburg's Apartheid Museum, the photograph. The caption says it was taken at a Cape Town rally, tens of thousands strong. Dating from March 1960, this was one of the early demonstrations against apartheid's perverse pass laws, the mood also fanned that morning by the infamous Sharpeville massacre of a few days before. It was led by Philip Kgosana, then a young student-activist with the Pan African Congress, who lived in a teeming shantytown outside Cape Town. After some hard-nosed negotiation, the police told him that the "Justice" Minister, FC Erasmus, had agreed to meet a small delegation.
"This was an exciting moment," Kgosana would write years later. "Never in the history of South Africa had an 'agitator' forced a racist minister to succumb to an undesirable appointment." Of course, when Kgosana and his colleagues arrived to meet Erasmus, he didn't show up. So much for "justice". Instead, the police arrested them and put them on trial for incitement to public violence.
Days after I saw the photograph, I waited in a convenience store in Pretoria, idly wondering if this was the first time I would meet a figure from a museum exhibit. Kgosana walked in right on time, a stocky man with gentle eyes. Nobody paid him any attention. "Hey guys," I wanted to shout at the shoppers desultorily examining biltong and wafer packets, "this man is in your Apartheid Museum! He's one reason you're here today!"
I held my tongue.
It must be a sign of how far South Africa has come that Kgosana lives today not in a shantytown, but in a comfortable bungalow in a once-whites-only suburb of the city that, more than every other in the country, personified white domination over black. But it's also a sign of that long journey that this figure from the resistance to white rule is now just another 70-something year-old, anonymous in this place. And if people warn you about violent crime in South Africa in these post-apartheid times, the reality also is that this is the continent's most dynamic economy, a vibrant nation with a host of attractions for every kind of tourist.
How far removed from protests against apartheid.
As Kgosana drove me home, I thought: a country subsides into a long night of oppression and brutality. It takes decades of struggle to emerge from that darkness. The guides on the way are heroes large and not-so-large: the Mandelas, but the Kgosanas too. There's an eruption of euphoria in 1994, at journey's end. But inevitably the ordinariness of daily life returns. Which is as it should be.
And that's how Kgosana saw his country. "Sure there's crime in South Africa," he said as we spoke in his comfortable living room, echoing other conversations in other living rooms. "Of course it bothers me. But where did we come from? You can't forget that."
True. After convulsions like South Africa has seen, there will necessarily be a time of uncertainty and unrest. You don't emerge from darkness directly into utopia. Yet if some heroes return to obscurity, as Kgosana has, perhaps that is itself a sign of some normalcy.
Speaking of normalcy and utopias, we visited the Kirstenbosch Botanical Gardens.
I'll say this: give me one botanical garden, you've given me them all. Oh sure, this expanse of verdant vegetation outside Cape Town is picture-perfect gorgeous. There are elegant bird of paradise flowers (Strelitzia) in dazzling colours, enormous trees, an aroma garden with plant smells familiar and otherwise, and plenty more to gladden a horticulturist's heart. All very impressive, but not quite the place I might have visited on my own.
Then again, that's because I did not know, ahead of time, about the Braille Trail.
Across from the aroma garden, marked by ropes, the Braille Trail meanders into the trees and out again. Just another walk, unless you get into the spirit and use it as it was designed to be used. The name is, of course, the giveaway. This is a self-guided tour designed specifically for the blind. They hold the rope and walk, stopping at regular boards in Braille and English (for those who can't handle Braille). This way, they learn about the trees among which they stroll, the arboreal scents and surfaces that surround them. They get some idea of what fellow-tourists, the sighted ones, experience.
My 11 year-old and I decided to give it a shot.
We grabbed the rope, shut our eyes tight and started walking. For the first few minutes, it was surprisingly hard to keep my eyes closed, to actually trust that I'd be fine without sight, with just the rope to guide me. So I kept opening then, though just slits, unable to shake the silly thought that somebody was watching to see if I cheated. But when I settled into the experience, the eyes stayed closed, it wasn't silly, and I was astonished by just how sensory and fulfilling it was. The 11 year-old, just as charmed.
Without sight, my ears tuned in even faint sounds. My skin registered the gentle breeze. My nose, a series of forest smells. My fingers, the curving strands of rope, the roughness of tree trunks. Twice, I felt my way to a bench for a sit-down, then back to the trail. Simple things when I look back, but they left me refreshed and curiously humbled by this small taste of what it is to be blind.
But more than that, too. All through, words that Nelson Mandela made famous coursed, maybe incongruously, through my mind. Mine was no long walk, but something about this trail had me musing, maybe incongruously, about the one he wrote about:
"I have walked that long road to freedom. I have tried not to falter … I have taken a moment here to rest, to steal a view of the glorious vista that surrounds me, to look back on the distance I have come. But I can only rest for a moment, for with freedom come responsibilities, and I dare not linger, for my long walk is not ended."
Mandela: always thought-provoking. At trail's end, I paid silent tribute to whoever designed this walk, amazed by their singular thoughtfulness. Did they realize it would be so meaningful for the sighted too? After all, walking it with my eyes shut had me contemplating themes and metaphors like darkness and light, new ways of seeing, concern for your fellow human, going bravely into the great unknown. All in about 45 minutes. Long walk, for sure.
And for an Indian, such themes resonate. For they must have meant something, over a century ago, to an Indian whose insight and fibre were forged here. Here, in apartheid South Africa, decades before it woke to freedom and hope.
They remember Gandhi in this country too.
Constitution Hill in Johannesburg, now home to the country's highest court, used to be the site of a notorious prison. The cell where Gandhi was once detained is now a memorial to the man. A serene place to remember him, it has photographs, accounts of his arrests and meetings with various leaders, his writings and even a BBC interview from some months before he died, playing over the buzz of construction next door. This must be one of the few places in the world where you can actually hear Gandhi's thin voice.
From here, Gandhi crossed the ocean to a land yearning for freedom. With a unique cocktail of courage, morality, political savvy and empathy, and with a unprecedented cast of giants among men, he guided India to a new dawn in 1947. Darkness to light, you might say. Our own Braille Trail.
"I can't promise you sunlight and roses," I imagine Philip Kgosana saying that day to the gathered thousands in Cape Town, perhaps to his country itself. "But walk the trail with me. Stay the course. I can promise you hope."
For new dawns, on either side of an ocean, perhaps hope is all you need.
Silence in the places of men
The current issue of Himal magazine (October-November 2011) is titled "Dust of the road: Trips, travel and journeys". It carries an essay I did based on a road trip through Karnataka, and then a stay in Shillong.
Take a look: Silence in the places of men.
And of course, comments always welcome.
Take a look: Silence in the places of men.
And of course, comments always welcome.
The Primes keep rolling on
Days and long nights on the road escorting 80 spirited kids around the sights of Agra and Delhi, and I'm tired even 5 days after our return. Minimal time for and access to the Web means this blog has been neglected for some time now. Let's see if I can pick up the pieces.
First, some writing that's on air.
My Mint mathematics column went live on Friday, October 14. In deference to vociferous demand, it manages to mention both Arshanapalayam and Dumbledore. There's a passing reference to Euclid thrown in, too.
See The primes keep rolling on.
Comments welcome.
And in case the link doesn't work, the text of the essay is below.
***
The primes keep rolling on
A possibly strange beast featured in my last column. No, I don't mean kitties named Aziz or Cleo. They aren't strange. What I'm referring to is the factorial. To jog your memory, the factorial of a positive number is what you get when you multiply all the numbers between 1 and itself.
Thus 4! ("four factorial") = 1 x 2 x 3 x 4 = 24.
Writing that, I can visualize eyes glazing over. Sure, I used this in the last column as an example of what recursion does well, but maybe you're wondering, why? Of what possible use is it to multiply numbers in this crazy way? Why subject us to this stuff?
Good question. We should all keep mathematicians, and those who try to write about mathematics, on their toes, especially when they try to foist evidently obscure operations on us. So yes, what is this factorial business?
Actually, it isn't that obscure.
Imagine you have four framed photographs of different relatives -- Arshanapalayam, Bob, Cherukuri and Dumbledore -- that you want to put in a row on your wall. You worry, because they are all fussy and prone to taking offence about where in the row they appear. My advice, of course, would be to fling all the photos out: equal opportunity offence. But that's not practical. Besides, it doesn't lend itself easily to mathematical intervention. So you decide that each day, you'll shuffle the positions of the photos. How many days before you must repeat an arrangement?
In other words, in how many different ways can you order the photographs on the wall?
Start with the first place. Any of the four photographs -- A, B, C or D -- can go there, meaning there are four ways of filling it. Once you've done so, let's say with C, go to the second spot. For which -- since you've used up C -- you have three frames available (A, B, D). That is, for each way of choosing a frame for the first spot, there are three ways of filling the second. Thus 4 x 3, or 12 ways of filling both the first two. Say you choose A for the second spot. You have two frames left (B, D), and either can occupy the third spot. So for each of the 12 ways of filling the first 2 slots, there are 2 ways to fill the third. Thus 4 x 3 x 2, or 24 ways, to fill those three. In our case, say you choose D for spot 3. Which leaves you with just B for the last slot -- or, only one way to fill it.
Thus your answer: there are 4 x 3 x 2 x 1, or 4!, or 24 ways to order the photographs. One of those, CADB, is what we chose above.
How many of the 24 will offend which of your relatives is also not a question that lends itself easily to mathematical intervention. But what you do have is 24 days of fresh arrangements, before you repeat.
If you had 5 framed photographs -- cousin Ekalavya joins the grumpy gang of four -- you'd have 120 arrangements.
Not much to write home about, even if it buys peace with fussy relatives? Then consider how the great Greek mathematician Euclid used factorials to prove that there is no such thing as a largest prime number.
Primes, remember, are numbers that have no factors other than 1 and themselves. For example, 3, 7, 19 and 31 are primes. Whereas 21 (3 x 7) and 10 (2 x 5) are not -- they are products of other primes.
So is there a largest such number?
Let's assume there is one. Amazingly enough, Euclid showed that this assumption undermines itself -- that there is no largest prime.
For simplicity, let's say the largest prime is 5. Consider, said Euclid, the number 5! + 1 = (1 x 2 x 3 x 4 x 5) + 1 = 121. Is 121 prime?
Well, you can't divide it by a prime less than the largest -- 5 -- because you get a remainder of 1 each time (try it). Thus either 121 is itself prime, or it is divisible by a prime larger than 5. Whichever it is, we've found a prime larger than 5. (In fact, 121 is the square of 11, a prime). And since the identical reasoning works with any prime, not just 5, this proves that there is no largest prime. Our initial assumption, that there is one, is wrong.
In other words, there is an infinite number of primes.
(As an aside, this mechanism of making an assumption and showing it is wrong is a favourite mathematical proof technique, called "reductio ad absurdum").
Such an elegant use of factorials, I always thought. Intuitive and understandable. Why can't life itself be like that? Instead, we worry about arranging and rearranging photos.
First, some writing that's on air.
My Mint mathematics column went live on Friday, October 14. In deference to vociferous demand, it manages to mention both Arshanapalayam and Dumbledore. There's a passing reference to Euclid thrown in, too.
See The primes keep rolling on.
Comments welcome.
And in case the link doesn't work, the text of the essay is below.
The primes keep rolling on
A possibly strange beast featured in my last column. No, I don't mean kitties named Aziz or Cleo. They aren't strange. What I'm referring to is the factorial. To jog your memory, the factorial of a positive number is what you get when you multiply all the numbers between 1 and itself.
Thus 4! ("four factorial") = 1 x 2 x 3 x 4 = 24.
Writing that, I can visualize eyes glazing over. Sure, I used this in the last column as an example of what recursion does well, but maybe you're wondering, why? Of what possible use is it to multiply numbers in this crazy way? Why subject us to this stuff?
Good question. We should all keep mathematicians, and those who try to write about mathematics, on their toes, especially when they try to foist evidently obscure operations on us. So yes, what is this factorial business?
Actually, it isn't that obscure.
Imagine you have four framed photographs of different relatives -- Arshanapalayam, Bob, Cherukuri and Dumbledore -- that you want to put in a row on your wall. You worry, because they are all fussy and prone to taking offence about where in the row they appear. My advice, of course, would be to fling all the photos out: equal opportunity offence. But that's not practical. Besides, it doesn't lend itself easily to mathematical intervention. So you decide that each day, you'll shuffle the positions of the photos. How many days before you must repeat an arrangement?
In other words, in how many different ways can you order the photographs on the wall?
Start with the first place. Any of the four photographs -- A, B, C or D -- can go there, meaning there are four ways of filling it. Once you've done so, let's say with C, go to the second spot. For which -- since you've used up C -- you have three frames available (A, B, D). That is, for each way of choosing a frame for the first spot, there are three ways of filling the second. Thus 4 x 3, or 12 ways of filling both the first two. Say you choose A for the second spot. You have two frames left (B, D), and either can occupy the third spot. So for each of the 12 ways of filling the first 2 slots, there are 2 ways to fill the third. Thus 4 x 3 x 2, or 24 ways, to fill those three. In our case, say you choose D for spot 3. Which leaves you with just B for the last slot -- or, only one way to fill it.
Thus your answer: there are 4 x 3 x 2 x 1, or 4!, or 24 ways to order the photographs. One of those, CADB, is what we chose above.
How many of the 24 will offend which of your relatives is also not a question that lends itself easily to mathematical intervention. But what you do have is 24 days of fresh arrangements, before you repeat.
If you had 5 framed photographs -- cousin Ekalavya joins the grumpy gang of four -- you'd have 120 arrangements.
Not much to write home about, even if it buys peace with fussy relatives? Then consider how the great Greek mathematician Euclid used factorials to prove that there is no such thing as a largest prime number.
Primes, remember, are numbers that have no factors other than 1 and themselves. For example, 3, 7, 19 and 31 are primes. Whereas 21 (3 x 7) and 10 (2 x 5) are not -- they are products of other primes.
So is there a largest such number?
Let's assume there is one. Amazingly enough, Euclid showed that this assumption undermines itself -- that there is no largest prime.
For simplicity, let's say the largest prime is 5. Consider, said Euclid, the number 5! + 1 = (1 x 2 x 3 x 4 x 5) + 1 = 121. Is 121 prime?
Well, you can't divide it by a prime less than the largest -- 5 -- because you get a remainder of 1 each time (try it). Thus either 121 is itself prime, or it is divisible by a prime larger than 5. Whichever it is, we've found a prime larger than 5. (In fact, 121 is the square of 11, a prime). And since the identical reasoning works with any prime, not just 5, this proves that there is no largest prime. Our initial assumption, that there is one, is wrong.
In other words, there is an infinite number of primes.
(As an aside, this mechanism of making an assumption and showing it is wrong is a favourite mathematical proof technique, called "reductio ad absurdum").
Such an elegant use of factorials, I always thought. Intuitive and understandable. Why can't life itself be like that? Instead, we worry about arranging and rearranging photos.
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