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The Klein Bottle

The Klein Bottle is the closed non-orientable surface with Euler characteristic = 0. It may be obtained by joining opposite ends of a cylinder with a twist, or by attatching two Möbius bands along their boundary circles. The Klein Bottle can not be embedded in R^3. There exist two immersed (self-intersecting) images of the Klein Bottle in R^3 which are not homotopic.

Description and pictures of the standard immersion of the Klein Bottle.

Description and pictures of the figure-8 immersion of the Klein Bottle.

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