![]() ![]() ![]() The ant cannot tell whether there are infinitely many copies of the letter “P”, or if there is only one letter “P” that it keeps coming back around to. Or perhaps it lives on a torus? (Or an infinitely long cylinder?) The point is that there is no way for the ant to tell the difference. After exploring for quite a while, the ant thinks its world looks something like this: Sometimes it finds places where the ground is black, and after making some maps the ant realizes that these places are shaped like a giant letter “P”. Sometimes it finds places where two lines cross at right angles. Occasionally there is a straight line drawn on the ground, extending off into the distance. ![]() The ant starts walking around, exploring its world. (You may know some tiny creatures in a similar situation who live on a sphere.) Unlike those tiny creatures on the sphere, however, the ant has nothing it can use to draw with, no objects to leave behind, etc., so it has no way to tell whether it has ever been to a particular location before. The ant is so small that it can’t tell that the surface it lives on is curved. Now imagine a very tiny ant who lives by itself on the surface of the torus. First, we glue the top and bottom edges together, resulting in a cylinder then bend the ends of the cylinder around to match up the left and right edges, resulting in a torus (a donut shape). Now, instead of matching up the edges of a bunch of copies of the same square, we can think about taking one square and gluing matching edges together. ![]()
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