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Business News/ Opinion / Straight line fever
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Straight line fever

A look at the work of Maryam Mirzakhani, the first woman to win the prestigious Fields medal for mathematics

Understanding the basic difference between the characteristics of a flat surface and a curved one is a small key to understanding the work of Mirzakhani. Photo: The Seoul ICM via AFPPremium
Understanding the basic difference between the characteristics of a flat surface and a curved one is a small key to understanding the work of Mirzakhani. Photo: The Seoul ICM via AFP

Took someone I know to the airport to catch a flight directly to New York, one of those 15-hour things. Came home and fired up my latest hobby horse, flightradar24.com, to track the progress of the flight. Now, if you look at your typical world map, you’ll see that New York is generally west of Bombay, if also slightly north. If you draw a straight line between the cities on the map—the shortest distance between them, you think, and a reasonable presumption of the route the flight would take—it would pass through Saudi Arabia and most of North Africa. From somewhere in Morocco, it would strike out across the Atlantic to touch down in NYC.

So why then did flightradar24 show this flight heading almost directly north, soaring over Gujarat? And later Afghanistan, Uzbekistan, Russia, northern Scandinivia and Greenland, before entering North American airspace?

Simple: because the shortest distance on a map is not the same thing as the shortest distance on a globe. The shortest route between two cities is along what’s called the great circle between them. That is, imagine tracing a circle that passes through both cities, with its centre at the centre of the planet. (Alternatively, imagine slicing the planet into two halves, with your cut passing through the two cities and the centre). That’s a great circle, and that turns out to be shortest route between the two cities. On a globe, the great circle connecting Bombay and New York passes through some of the northernmost reaches of the planet. That’s the route the flight will follow.

Understanding this basic difference between the characteristics of a map and a globe—a flat surface and a curved one—is a small key to understanding the work of Maryam Mirzakhani, the first woman to win mathematics’ Fields Medal. The point is, assumptions we make about flat surfaces don’t hold for curved ones. Here’s another: the angles of a triangle add up to 180 degrees, remember? But consider the triangle made by the North Pole, the point where the 0 degree meridian crosses the equator (off the coast of Ghana) and the point where the 90 degree meridian crosses the equator (in the Indian Ocean about 1500km south of the Andamans). All three angles are 90 degrees. Their total: 270 degrees.

Curved surfaces for you.

Mirzakhani works especially with “hyperbolic" surfaces, which curve in other ways. Imagine a doughnut with two holes. Break off bits from either end, so you have something resembling a misshapen “H", and imagine those legs stretching away to infinity, up and down. What can we say about such a surface? Well, the notion of a shortest route gets another workover now. If both points are on the arm of the “H", then the shortest route between them is on the great circle around that arm. But if one is on each of the upper legs, then the shortest route is down one leg, across the arm and up the other leg. That is, one of those straight lines is part of a finite circle (a “closed" loop), but the other is part of a line that stretches to infinity in either direction. You wouldn’t be able to draw that first kind of straight line on a map (a straight line can’t turn into a circle?); you wouldn’t be able to draw the second kind on a globe. So we have a qualitatively different kind of surface here.

And on such a surface, some of these straight line closed loops can fold back on or intersect themselves. (Imagine the shortest route between a point on the front of the arm and a point on the back of the right lower leg). In fact, most closed loops do this.

Those that don’t are called “simple". Mirzakhani’s PhD research related the length of a simple loop to the number of loops of that length you’ll find on the hyperbolic surface. In fact, she found a formula that captures that relationship.

Why is all this important? Well, the universe we live in is perhaps best understood by considering it as a hyperbolic surface. And we want to understand how various heavenly bodies—planets, stars, asteroids and so on—move on such a surface in relation to each other. In that pursuit, these loops help us. Thus we try to comprehend their nature. Much like how a billiards ball won’t wander around when you strike it with your cue, but instead will travel in a straight line until it bounces off the edge of the table, stars and planets also seek to travel in straight lines. Except that, as we have seen, “straight line" takes on a new meaning altogether on a hyperbolic surface.

Naturally, Mirzakhani’s work is far more complex than this novice-level outline can achieve. But it might give you a glimpse into her field, dynamics: broadly, the study of bodies in motion.

It might tell you that even the most abstruse mathematics has roots in simple, easily understood ideas. Think of that when you next draw a straight line.

To read Dilip D’Souza’s column on the work of Indian-origin mathematician Manjul Bhargava, who also won the Fields Medal, go to mintne.ws/1uStGxb

Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners.

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Published: 18 Aug 2014, 05:27 PM IST
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