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If you join the midpoints of two sides of a triangle, that line is parallel to and half the length of the third side of a triangle. What does that do for us? Photo: ThinkStock
If you join the midpoints of two sides of a triangle, that line is parallel to and half the length of the third side of a triangle. What does that do for us? Photo: ThinkStock

Bringing geometry to life

There are times when geometry and mathematics can be life-savers and not dull and boring

Studying mathematics at school, my son is in the midst of geometry. You know, those delights we all went through: Pythagoras’s theorem, corresponding angles on parallel lines, congruent triangles, on and on. Fairly logical stuff, I imagine he and his friends think, but also boring and routine.

If you join the midpoints of two sides of a triangle, that line is parallel to and half the length of the third side of a triangle. What does that do for us? And of what practical use is the assertion that a line that cuts across two parallel lines produces various angles, pairs of which are the same? Nothing and none, respectively. Dull stuff.

Or so you’d think. Only, every now and then you run into examples of how these geometrical ideas play out in real life. For example, did you ever wonder how people calculate the heights of tall structures? Like, how do we know that the Qutub Minar is nearly 73m tall? Or that if you stacked up 11 Qutub Minars, the Burj Khalifa, at 828m, would be taller still? Well, if you know about ratios and you were in Delhi or Dubai on a sunny day, you could make those calculations yourself. Or perhaps you’ve wondered how cartographers, especially before planes and spacecraft, managed to map coastlines with remarkable accuracy? They used triangulation, a process which involves triangles, trigonometry and calculations.

And then there are times that geometry and mathematics can be life-savers.

In the early 20th century, the British explorer Ernest Shackleton went on four expeditions to Antarctica. There were scientific reasons for them all, but being the first to reach the South Pole was the overarching goal. In the heroic British tradition, all the expeditions failed: in 1911, it was Norway’s Roald Amundsen who conquered the South Pole for the first time. That left what Shackleton believed was the final untried Antarctica challenge: a crossing of the continent, touching the South Pole.

In 1914, he assembled a team to attempt just that. In that same heroic British tradition, they failed too, when their ship got stuck in pack ice. In November 1915, they abandoned the ship. For months, they camped on the ice, hoping to drift to safety. When it was clear that would not happen, they took to lifeboats and managed to reach the Elephant Island, a small speck of land off an arcing spur of the continent, several hundred miles south-east of Tierra del Fuego.

It wasn’t enough. Their only hope, Shackleton knew, lay in reaching South Georgia Island, another speck of land, but one with some whaling stations. Only, South Georgia is over 700 miles north-east of Elephant Island across some of the world’s wildest seas.

On 24 April 1916, Shackleton and five men clambered into their sturdiest lifeboat and set out for South Georgia.

Consider what they were attempting. They aimed for South Georgia, but couldn’t afford to be off even slightly: miss South Georgia, and the next landfall is South Africa, thousands of miles away. Miss South Georgia, and it’s death for six on the lifeboat and the two dozen on Elephant Island. This is why Shackleton carried supplies for just four weeks: longer than that, and they were all doomed.

But not one man died, and they can thank geometry for that. Shackleton’s navigator, Frank Worsley, used an instrument called a sextant to observe the sun and calculate their position as they sailed for South Georgia. These calculations assume that the sun is so far away that its rays hit the earth effectively in parallel. And so that nice bit of geometry my son is fiddling with comes into play: when a line cuts across parallel lines, the corresponding angles are equal. So if you know how high the sun is in the sky, meaning what angle it makes with the horizon—which is what Worsley’s sextant was telling him—calculating the latitude is a trivial bit of arithmetic. Longitude needs a different technique, but it’s not much harder.

Now of course there were complications, given overcast skies that made it hard to spot the sun, and seas so rough that staying steady enough to take accurate readings was a constant battle. But even so, this was essentially how Worsley determined exactly where in that stretch of sea their lifeboat was each day. It was how he managed to keep them en route to South Georgia.

The men sighted the island just over two weeks later. A few months after that, all the men from the expedition were safe.

It’s a thoroughly soul-stirring tale. And at the heart of it is what Worsley, without modern crutches like cellphones and global positioning systems, accomplished on that journey from Elephant Island to South Georgia. It must rank as one of the most magnificent feats of navigation in history.

And perhaps most remarkable of all is that at the heart of Worsley’s feat, in turn, was some relatively elementary geometry.

Talk about bringing schoolwork alive.

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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