Beautiful Numbers

While reading the The Da Vinci Code a few years back, I came across this passage:

He felt himself suddenly reeling back to Harvard, standing in front of his “Symbolism in Art” class, writing his favorite number on the chalkboard.


Langdon turned to face his sea of eager students. “Who can tell me what this number is?”

A long-legged math major in back raised his hand. “That’s the number PHI.” He pronounced it fee.

“Nice job, Stettner,” Langdon said. “Everyone, meet PHI.”

“Not to be confused with PI,” Stettner added, grinning. “As we mathematicians like to say: PHI is one H of a lot cooler than PI!”

Langdon laughed, but nobody else seemed to get the joke.

Stettner slumped.

“This number PHI,” Langdon continued, “one-point-six-one-eight, is a very important number in art. Who can tell me why?”

Stettner tried to redeem himself. “Because it’s so pretty?”

Everyone laughed.

“Actually,” Langdon said, “Stettner’s right again. PHI is generally considered the most beautiful number in the universe.”

The laughter abruptly stopped, and Stettner gloated.

As Langdon loaded his slide projector, he explained that the number PHI was derived from the Fibonacci sequence – a progression famous not only because the sum of adjacent terms equaled the next term, but because the quotients of adjacent terms possessed the astonishing property of approaching the number 1.618 – PHI!

Despite PHI’s seemingly mystical mathematical origins, Langdon explained, the truly mind-boggling aspect of PHI was its role as a fundamental building block in nature. Plants, animals, and even human beings all possessed dimensional properties that adhered with eerie exactitude to the ratio of PHI to 1.

– From the Da Vinci Code, by Dan Brown

That passage set me thinking about other numbers considered pretty or at least very interesting. As an Indian I can point to quite a few numbers that we can be proud of. At the top of the list is Aryabhata’s invention of the most famous number of them all – 0, which by helping establish the place value system and the decimal number system made innumerable mathematical and scientific discoveries possible and practical. Just imagine having to write a number like 999,999 in the Roman Numeral system. You would need to write CM XC IX CMXCIX. And if you weren’t aware that adding the “overline” multiplies a number by 1000, then you would have to struggle significantly more to represent a number such as 999,999.

As an ex-IIT’ian I have had a fascination for numbers and so have many of my classmates. Both during and after life at IIT I have seen my friends use one particular number quite often – 1729. I myself used 1729 as my page id when I was building my hostel’s website back in the days when you needed to sign up for a free web-page at sites like GeoCities. A few years after graduation my friend asked me to unlock his bicycle. The code – 1729. A few more years later another friend sent out an email saying that his previous email id had been handed out to several mailing lists and he was receiving a lot of spam. So he changed his email id to something that had the number 1729 in it. If you are not very mathematically inclined you might think of 1729 as a very weird number to be fascinated with. But there is history behind it. 1729 is in fact called the Hardy-Ramanujan number, following a very famous conversation between G. H. Hardy and Srinivasa Ramanujan.

Hardy used to visit him, as he lay dying in hospital at Putney. It was on one of those visits that there happened the incident of the taxi-cab number. Hardy had gone out to Putney by taxi, as usual his chosen method of conveyance. He went into the room where Ramanujan was lying. Hardy, always inept about introducing a conversation, said, probably without a greeting, and certainly as his first remark: “I thought the number of my taxi-cab was 1729. It seemed to me rather a dull number.” To which Ramanujan replied: “No, Hardy! No, Hardy! It is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways.”

That is the exchange as Hardy recorded it. It must be substantially accurate. He was the most honest of men; and further, no one could possibly have invented it.

– Foreword by C. P. Snow, to G. H. Hardy’s “A Mathematician’s Apology (Canto)

For those trying to figure out what Ramanujan meant, 1729 = 123 + 13 = 103 + 93. Not only is this number an Indian favorite (Ramanujan was Indian), but mathematicians worldwide recognize it for the brilliance and simplicity of the discovery. This number is also referred to as a Taxicab number due to the associated incident, though the unique property of this number was actually discovered by Bernard Frénicle de Bessy.

There is another number that piqued my interest, however, when I was preparing for the Indian National Mathematics Olympiad in 1994. I came across a number that was referred to as the Kaprekar Number – 6174. In later years I came to know that the information was inaccurate, because this number was called Kaprekar Constant, and Kaprekar Numbers referred to a separate category of numbers. A Kaprekar Number is a number that is thus defined:

A Kaprekar number for a given base is a non-negative integer, the representation of whose square in that base can be split into two parts that add up to the original number again.

As the Wikipedia article states, 45 is a Kaprekar number because 45 = 20 + 25 and 452 = 2025. These numbers were discovered by another Indian mathematician Dattaraya Ramchandra Kaprekar, who had a penchant for discovering several results in number theory and was very well known as a recreational mathematician. Funny what people come up with during their free time!

But back to the Kaprekar constant – 6174. Again, this falls wholly into the category of an unremarkable-looking number. But there is a lot more to it. Arrange the digits of the number in descending order: 7641. Arrange its digits in ascending order: 1467. Subtract the two: 7641 – 1467 = 6174. This happens to be the only 4-digit number that exhibits this property. If you think that is surprising, there is more. Take any 4 digit number with at least 1 digit different from the rest. Repeat the operation of subtracting the ascending order of digits from the descending order. After a finite number of iterations you will hit 6174!! I was so impressed with this number that I couldn’t rest till I had established the proof of this. Yutaka Nishiyama has a well-documented proof, which is much more rigorous than what I came up with (plus I am too lazy to type out my proof in HTML here).

There are other Kaprekar constants when you change the number of digits to 3 (495) or something else.

I am sure there are several other numbers that have even more quirky properties. Having had an affinity towards mathematics in general since a young age and towards number theory in particular since I was 15, I know that I am missing out on such a huge treasure by pursuing a career in something so far removed from mathematics.

Haunting Photos

As a kid I remember watching the Friday night 10:00 PM TV program on Doordarshan called “The World This Week”. It was a very good program – a weekly news update about events outside India, telecast in a very western manner, significantly more engrossing than the rather staid staple news programs of DD. An enduring picture that remained stuck in my mind for years was that of a girl running naked from a site attacked with Napalm. A few months back I turned to my favourite hunting ground, the internet, for finding out who that person was and what happened to her. It wasn’t difficult – Wikipedia had a story about her and National Public Radio actually had her speaking. Phan Thị Kim Phúc, or Kim Phuc or simply Phuc lives in Toronto with her husband and two children. She runs an organization called Kim Phuc Foundation International, which aids children who are war victims.

Kim Phuc, AP Photo by Nick Ut
Kim Phuc, AP Photo by Nick Ut

From her talk on NPR:

On June 8, 1972, I ran out from Cao Dai temple in my village, Trang Bang, South Vietnam; I saw an airplane getting lower and then four bombs falling down. I saw fire everywhere around me. Then I saw the fire over my body, especially on my left arm. My clothes had been burned off by fire.

I was 9 years old but I still remember my thoughts at that moment: I would be ugly and people would treat me in a different way. My picture was taken in that moment on Road No. 1 from Saigon to Phnom Penh. After a soldier gave me some drink and poured water over my body, I lost my consciousness.

Several days after, I realized that I was in the hospital, where I spent 14 months and had 17 operations.

It was a very difficult time for me when I went home from the hospital. Our house was destroyed; we lost everything and we just survived day by day.

Kim Phuc later recuperated and became the symbol of the war. Richard Nixon once doubted the authenticity of this Pulitzer Prize winning photograph – it was so iconic.

A few years after The World This Week I caught another picture, this time on a rerun of a story from the National Geographic. This picture is now famous as the Afghan Girl and it became the inspiration for people to volunteer and help out at the Afghan refugee camps.

Sharbat Gul, the Afghan Girl - photo by Steve McCurry, National Geographic June 1985
Sharbat Gul, the Afghan Girl - photo by Steve McCurry, National Geographic June 1985

At the time of this photograph the author of the story in National Geographic Debra Denker did not know the name of this girl with arguably the most haunting pair of eyes ever seen. The focus of the article being more on the ravages of the Afghan war of that time, the identity of this person was not paid much attention to. However, the photo was so captivating that many years later, after the 9/11 attacks in 2001, a team from National Geographic made an earnest attempt to track her down. Cathy Newman reported the new story in April 2002, with a second photograph, again by Steve McCurry.

Sharbat Gul, 1985 and 2002, Photo by Steve McCurry, National Geographic
Sharbat Gul, 1985 and 2002, Photo by Steve McCurry, National Geographic

The name of the girl is Sharbat Gula or Sharbat Gul, depending on whether you read the story or listen to the narrative about how they tracked her down. Though Sharbat Gula means rose sharbat (sharbat being a drink of water with some sweet things added), I am tempted to believe that the name was actually Sharbat Gul, because not only is Gul a more authentic surname, but her husband’s name is Rahmat Gul too.

Sharbat, when National Geographic photographed her a second time was living with her husband and 3 daughters. Though the face had aged over the 17 years, the eyes still had that same piercing look. This time National Geographic ensured that Sharbat and her family received the aid.

Doing an Asok

For all the money a consultant makes, the job is often thankless. Nothing exemplifies this better than today’s Dilbert:

There are some parallels and some non-parallels between Asok and me:

  • Asok, like me is an IIT graduate. Worse still, like me he is trained to sleep only on national holidays

  • I must have flunked the course where they taught us to reheat tea by holding the cup to my forehead

  • But I did meet the prerequisite for getting in

  • Asok is an intern, hence his position at the company is not permanent. As a consultant my privileges at my client are pretty similar, except that a consultant probably earns about 4 times more than an intern

  • I must have also flunked this other course, since I am still nutty about a lot of things:

My kinship with Asok often reminds me of this classic scene from Lage Raho Munna Bhai:

Yes, just keep doing it hoping that some day things will be better.

Sketch Me Up

I was looking for Photoshop-free options to create pencil sketches from photos. Photoshop and most other tools apply desaturation filters to different extents to create these effects. These are a couple of nice online tools that I found:

Let me know how you like them

Sporting Gestures

I had written about Nadal and Federer at this year’s Australian Open, applauding the spirit displayed by Nadal after his victory. Today I came across an article in the Guardian (yes, I read all the British and Australian newspapers whenever India does well in Cricket, just to see what other countries think of it) that talked about the author’s top 10 favourite sporting gestures on the field. While a few readers have commented that Nadal’s act should have made the list, I was happy to see Andrew Flintoff consoling Brett Lee figure in the top 10 – that was something I had appreciated in my post.

I will be keeping my posts short till 24th April. The 7-day work weeks clubbed with a work-related repetitive stress injury has severely hampered my capacity to write here or to work on Aquoid.


The Baddy Blog

No – this is not a bad blog by any stretch of imagination. It is actually a Badminton player’s blog. Saina Nehwal’s, to be precise. Unfortunately it is not a blog in the classical style – there is no RSS / Atom feed or a place to enter comments. Nevertheless, it is a very enjoyable read. More so because I really love playing Badminton myself and Saina is an excellent player (10th in the world at the moment) and a very good writer. The writing style is direct and witty, describing aspects of daily training and with a peek into some interactions that she has on a day to day basis.

Saina’s rise to the top is a very inspiring story in itself. As an 8-year old she would start her day at 6:00 AM each day and ride pillion on her father’s scooter for 20km. From an American perspective 20km is a trifling distance – 12.5 miles, but this is a pretty long ride in India where your scooters don’t go faster than 40-50 kmph (25-30 mph). There were also financial hardships like kit costs, training costs etc. Luckily she managed to find sponsorships starting in 2002, which ameliorated the situation to a large extent.

I do hope Saina finds great professional and personal success – we need people like her to put India on the map of world sports.

Sinful Places

You have probably heard of the quote: “Capitalism civilizes greed just like marriage civilizes lust”. I wanted to get other such civil analogies, when I came across this site. Add those places to your travel plans! Oh, and by the way, if you know the rest of the analogies please let me know.

Quel dommage!

Did you know that a company called Despair, Inc has copyrighted the frownie (the “:-(” that you put on your text messages and chats)? They sell “Motivational products and posters for pessimists, underachievers, and the chronically unsuccessful” and call their products “Demotivators”. They start off with a picture of their COO:

Just Condescending
Just Condescending

Check their site out – it is a masterpiece of gloom, something that Marvin the Paranoid Android could find great use for. Of all their demotivators, I found this one the most apt for me and my profession:

If you're not a part of the solution, there's good money to be made in prolonging the problem.
Consulting, indeed!

I looked at their website for copyright information to ensure that I was not doing anything wrong by putting their stuff on my page and here is what I came across:

I want to put your images on my homepage without crediting you or acknowledging you in any way, so that I can do my small part to violate the copyrights of your photographers and whoever else might have a commercial interest in your intellectual property. How cool is that?

It is okay with us provided you promise to throw an online tantrum when we ask you politely to stop.

On the F.A.Q. for Despair, Inc.