In mathematics, the Klein bottle is a certain non-orientable surface, i.e., a surface (a two-dimensional manifold) with no distinct "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.
The Möbius strip or Möbius band (pronounced /ˈmiːbiəs/ or /ˈmoʊbiəs/ in English, IPA: [ˈmøːbiʊs] in German) (alternatively written Mobius or Moebius in English) is a surface with only one side and only one boundary component. The Möbius strip has the mathematical property of being non-orientable. It is also a ruled surface. It was discovered independently by the German mathematicians August Ferdinand Möbius and Johann Benedict Listing in 1858.
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