Klein bottle

In mathematics, the Klein bottle ([klaɪ̯n]) is a non-orientable surface, informally, a surface (a two-dimensional manifold) with no identifiable "inner" and "outer" sides. Other related non-orientable objects include the Möbius strip and the real projective plane. Whereas a Möbius strip is a two-dimensional surface with boundary, a Klein bottle has no boundary. (For comparison, a sphere is an orientable surface with no boundary.)

The Klein bottle was first described in 1882 by the German mathematician Felix Klein. It is sometimes claimed that it was originally named the Kleinsche Fläche "Klein surface" and that this was incorrectly interpreted as Kleinsche Flasche "Klein bottle," which ultimately led to the adoption of this term in the German language as well.

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