Sexy math: Drawing parallels in geometry.

This year's Abel Prize honors some of the most revolutionary contributions to geometry since those of Euclid.

From: The Time by Marcus du Sautoy. April 8, 2009.

The winner of the Abel Prize this year was announced recently by the Norwegian Academy of Science and Letters in Oslo. Although a relative newcomer to the accolades that mathematicians can receive, the Abel Prize is quickly becoming the Nobel prize that Alfred Nobel never endowed. The recipient of the million-dollar prize this year is the Russian-born mathematician Mikhail Gromov, who is being rewarded for his revolutionary contributions to geometry.

For 2,000 years the world of geometry was dominated by one name: Euclid. In the greatest textbook of all time, Elements, he set down basic axioms about the way points and lines behave. For example, it seems self-evident that between any two points you can draw a line. From these principles, Euclid began to draw conclusions about the shapes that you could draw with these points and lines. For example, he discovered that the angles in a triangle always add up to 180 degrees, something that all schoolchildren are taught.

But as time went on suspicion began to arise over one of Euclid's “self-evident truths”, the so-called parallel postulate. Euclid believed that if you draw a line and a point to one side of that line, you can draw a second line through the point that is parallel to the first line, ie, it will never meet the first line. He also believed that there was only one parallel line through that point and that any other line you drew would eventually intersect the first line.

It took a long time for mathematicians to realize that there are geometries in which the parallel postulate is false. Eventually, in the 19th century, Riemann and Gauss realized that Euclid's geometry wasn't the only one.

The word geometry is Greek in origin and means measuring the Earth. If you don't travel far, the Earth appears flat, which gives rise to the geometry that Euclid captured. His geometry is the one that works on a flat piece of paper in an exercise book. But as soon as you start to navigate the surface of the Earth, another geometry appears.

Anyone who has flown from London to San Francisco may have been shocked halfway through the flight to look out and see ice below. The map in the in-flight magazine doesn't show any ice on the line joining London and San Francisco and you worry that you've got on the wrong flight. But, because the Earth is curved, the shortest path between London and San Francisco is not the line you'd draw on a map but a great circle, like a line of longitude, with London being one of the poles. In the geometry of the surface of the Earth, lines are actually these great circles because they are the shortest paths from one point to another. This results in some surprising properties of triangles drawn in this geometry.

Mark a point at the North Pole and two points on the Equator. Now draw three straight lines between the three points to make a triangle. A straight line between the two points on the Equator is just a line running along the Equator. The two lines joining these points to the pole are the two lines of longitude running from the pole to the points. These lines of longitude meet the Equator line at an angle of 90 degrees. So already the two angles at the Equator add up to 180 degrees. Add on to this the angle between the lines of longitude and the angles in a triangle add up to more than 180 degrees. Does that mean that Euclid got it wrong? Will the whole of mathematics collapse because of this apparent contradiction?

Not at all. This is a geometry that doesn't satisfy the parallel postulate. In fact, in this geometry there are no parallel lines. Take a great circle and a point off that circle: any great circle through this point must meet the first great circle at two places, polar opposite to one another.

Even stranger geometries were discovered at the beginning of the 19th century by a Transylvanian mathematician called Janos Bolyai and a Russian, Nikolai Lobachevsky. In these geometries, angles in triangles add up to less than 180 degrees, and there are many parallel lines through each point.

Mikhail Gromov has been awarded the Abel Prize for his work making sense of the range of geometries that appeared in the 19th century and he is a worthy successor to the likes of Euclid, Riemann and Gauss, Bolyai and Lobachevsky.

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