The Klein Bottle

The Klein bottle is the non-orientable surface with Euler characteristic equal to 0.

A Klein bottle can be made from a rectangular piece of the plane by identifying the top and bottom edges using the same orientation, but identifying the left and right edges with opposite orientation (as in the formation of a Möbius band). The first step forms a tube, but the second step can not be carried out without causing self-intersection: the tube must pass through itself in order to attach the ends correctly.

Another way to make the Klein bottle is to take two Möbius bands and join them along their boundaries (each band has a single boundary curve). Finally, the Klein bottle is the connected sum of two real projective planes, since the projective plane minus a disk is just a Möbius band.

The Klein bottle can not be embedded in three-space, but it can be immersed there.

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Article Source: http://www.math.rochester.edu/misc/klein-bottle.html