Klein bottle

In topology, a branch of mathematics, the Klein bottle (/ˈkln/) is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down. Other related non-orientable objects include the Möbius strip and the real projective plane. While a Möbius strip is a surface with boundary, a Klein bottle has no boundary. For comparison, a sphere is an orientable surface with no boundary.

A two-dimensional representation of the Klein bottle immersed in three-dimensional space
Structure of a three-dimensional Klein bottle

The Klein bottle was first described in 1882 by the German mathematician Felix Klein.


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