## ::Klein bottle

### ::concepts

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**

Intro Construction Properties Dissection Simple-closed curves Parameterization Generalizations Klein surface See also Notes References External links

PREVIOUS: Intro | NEXT: Construction |

<< | >> |