Views from Flatland

By Ivars Peterson

Muse, October 2000, p. 26.

What do you think the world would look like if you and everything in it were squished flatter than a pancake? Like shadows, you and your friends would freely flit about the surface. But if you couldn't rise above or below the surface, objects with any thickness would look very strange to you.

That's the idea behind a book called Flatland, written more than 100 years ago by Edwin A. Abbott. Head of a school for boys in London, Abbott imagined a world in which the inhabitants are geometric shapes: straight lines, triangles, squares, pentagons, circles, and other figures, all living on the surface of a perfectly flat world.

Curiously, though Flatlanders are two-dimensional, they appear to one another to be straight lines. To see why, place a coin on a table. From directly above, the coin looks circular. Seen at an angle, the coin looks more like an oval. If your eye is level with the table, the oval thins to little more than a straight line.

Unlike many people in Great Britain at the time he wrote Flatland, Abbott believed that girls deserve as good an education as boys. He also favored granting more rights to women, including the right to vote. To make fun of the way women were treated in British society, Abbott made Flatland males triangles and other polygons, but he made Flatland females straight lines. That made them dangerous because they can run you through like a needle. Unlike males, females can become invisible at will. Can you see how?

In one dramatic episode in Flatland, a three-dimensional sphere visits the Flatlanders' world. You can think of a sphere as made of a stack of circles. Only one of these circles would intersect Flatland at any given moment. To a Flatlander, each circle would look like a line. The length of the line would depend on the size of the circle, so to a Flatlander, a sphere rising through Flatland would look like a dot that grew longer and longer, then shorter and shorter, until it became a dot again and vanished.

A tiny water critter skimming along the surface of a still pond would get the same sort of view if you were wading nearby. It would see the nearly circular cross sections of your legs as mysteriously shifting lines.

The complete text of Flatland is available at www.geom.umn.edu/~banchoff/Flatland/.

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