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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