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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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III.28 Fuchsian Groups

Jeremy Gray

One of the most basic objects in geometry is the torus: a surface that has the shape of the surface of a bagel. If you want to construct one, you can do so by taking a square and gluing opposite edges together. When you glue the top and bottom edges together you have a cylinder, and when you glue the other two edges together, which have now become circles, you obtain your torus.

A more mathematical way of making a torus is as follows. We start with the usual (x,y) coordinate plane and the square in it with vertices at (0, 0), (1, 0), (1, 1), and (0, 1), which consists of the points whose coordinates satisfy 0 ≤ x ≤ 1, 0 ≤ y ≤ 1. This square can be moved around horizontally and vertically. If we shift it

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