The Klein bottle is a one-sided surface named after the great 19th-century mathematician Felix Klein, who was one of the first mathematicians to explore its unusual properties. The Klein bottle is formed from a square piece of paper by gluing opposite edges according to the following rules. The top and bottom edges are glued to each other by a reverse gluing; that is, the top edge is twisted 180 degrees before it is glued to the bottom edge. Performing only this gluing will turn the square into a surface called the Möbius strip. The left and right edges are glued in a very simple way: each point on the left edge gets glued to the opposite point on the right edge. Performing only this gluing will turn the sheet of paper into a cylinder.

Each of these gluings separately is easy to carry out. However...