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Business News/ Opinion / A pi day miracle, may be?
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A pi day miracle, may be?

Imagine searching over 72 mn sq. km for a Boeing that's 65m long when intact. A needle in a haystack would be easier to find.

If the search for the MH370 is successful and we find that plane, with all its passengers alive, that astonishing miracle would be the best possible way to celebrate Pi Day 2014. Photo: ReutersPremium
If the search for the MH370 is successful and we find that plane, with all its passengers alive, that astonishing miracle would be the best possible way to celebrate Pi Day 2014. Photo: Reuters

I’ve been trying to write this column in between much heartbreaking news. In particular, another sickening Maoist attack, and the mystery of a missing passenger plane.

Because this column will appear on Pi Day, I want it to say things about that curious and beloved number pi. But I want it to be topical as well. No connection to murderous Maoists, but is there a link between pi and the search for a plane?

But first: Pi Day? Indeed: you’re reading this on 14 March, which you could write as 3.14. Think of that as a number and note that it approximates pi (3.1415926358...). Which is why today is celebrated, whimsically, as Pi Day.

Though there’s nothing remotely whimsical about a vanished plane.

So now consider the starting point of the search: where do we search? We know MH370 had flown for about 40 minutes after take off, we know it had enough fuel to reach Beijing—4,500km away—and for about an hour longer, we know its cruising speed and we know where we last heard from its pilots. Put all that together.

Starting from that last-heard-from point, the plane had fuel enough to travel nearly 4,800km. That is, draw a circle of radius 4,800km centred there: in theory, the plane could be anywhere inside it.

And how big is this circle? Bring on pi: You remember from school that a circle’s area is pi multiplied by the square of its radius. Thus the area of this circle is, get ready, over 72 million sq. km.

To put that in perspective, that’s about one-seventh the area of the planet, about 24 times the area of India. (In fact, this circle takes in much of the eastern half of India).

Imagine searching in that expanse, plenty of which is open sea, for a Boeing that’s 65m long when intact. A needle in a haystack would be easier to find.

But of course the search won’t be that extensive. It would be stupid to even try. Instead, searchers use other clues to trim the search area.

For one example, there was a report that a worker on an oil rig about 700km east of where MH370 was last heard from saw a burning object in the sky. No word yet on whether that was the plane. But here’s how searchers might consider this. Last heard from, it was at 35,000ft. If we can assume the worker saw it 1,000ft above him, that means it had descended 34,000ft while flying 700km east. If it held the same trajectory, an easy calculation, familiar from elementary geometry, shows that it would have crashed into the sea nearly 21km further east from the rig. So search there. Or at any rate, search in a circle whose radius is 21km, centred on the oil rig. Its area? Use pi again: about 1,385 sq. km. Not a patch on that earlier, far vaster circle.

As it turns out, two areas of sea have been searched. First, the South China Sea, east of the Thai-Malay peninsula, which the plane was flying over. Second, the Malacca Strait, to the west of the peninsula, because one report said military radar spotted the plane there. These are relatively limited expanses of water. But especially in a channel such as the Malacca Strait, currents can move as fast as 50km each day.

As days pass, even this basic consideration of currents tells us that if the plane did crash, any debris could be swept pretty far pretty fast.

Those circles again: Almost 8,000 sq. km to search on the first day, about 31,500 on the second day (when the radius is 100km), etc. By the time you read this—six days into the search—we’re talking about nearly 285,000 sq. km.

As you see, even seemingly gentle ocean currents swiftly compound the difficulty of the search. Though again, there are factors that limit the area. The currents don’t flow in every direction, and that cuts down the circle considerably. And the Malacca Strait is itself less than 50km wide at its narrowest point. So instead of a circle, you have what looks like a pizza slice of water to search in.

There are many other aspects of this search: surveying and navigation, for example.

Modern GPS makes these simple, of course. But at their heart is a process cartographers have used for centuries, called triangulation. Restricted to the surface of the Earth before the coming of airplanes and satellites, they drew accurate maps anyway—using triangulation. In turn, triangulation uses angles and trigonometry: again, pi is involved.

Tenuous connections for Pi Day, you think? Yet that’s almost my point: pi is so ubiquitous that we don’t even notice it.

Still, forget the calculations. If the search is successful and we find that plane today, with all its passengers alive, that astonishing miracle would be the best possible way to celebrate Pi Day 2014.

Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. A Matter of Numbers will explore the joy of mathematics, with occasional forays into other sciences.

To read Dilip D’Souza’s previous columns, go to www.livemint.com/dilipdsouza

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Published: 13 Mar 2014, 07:23 PM IST
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