KLEIN

BOTTLE

In mathematics, the Klein bottle /ˈklaɪn/ is an example of a non-orientable surface; informally, it is a surface (a two-dimensional manifold) in which notions of left and right cannot be consistently defined. Other related non-orientable objects include the Mobius strip and the real projective plane. Whereas a Mobius strip is a 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 may have been originally named the Kleinsche Flache ("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.

for 0 ≤ u < π and 0 ≤ v < 2π.

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