## ::Klein bottle

### ::concepts

Klein::bottle    Theta::bottle    Space::right    Image::torus    Surface::plane    Thumb::strip

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

In mathematics, the Klein bottle {{#invoke:IPAc-en|main}} 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. Whereas 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).

The Klein bottle was first described in 1882 by the German mathematician Felix Klein. It may have been originally named the Kleinsche Fläche ("Klein surface") and then misinterpreted as Kleinsche Flasche ("Klein bottle"), which ultimately led to the adoption of this term in the German language as well.<ref>{{#invoke:citation/CS1|citation |CitationClass=book }}, Extract of page 95</ref>

Klein bottle sections