What is a Flat Torus?

If you take a sheet of paper and glue the left side to the right side, you get a cylinder. If you take a different sheet of paper and glue the top to the bottom, you again get a cylinder. If you take a third sheet and try to glue the left side to the right side AND the top to the bottom, the paper will crumple up and you'll get a big mess (try it!). It's much easier to let the computer keep track of how the paper's sides are glued.

Grab the grapes below and drag them off the bottom of the square. What happens? What if you drag them off the left side? Can the grapes ever crash into a wall?

This 2-dimensional space is called a flat torus.

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