Klein bottle

In mathematics, the Klein bottle (/ˈklaɪn/) is an example of a non-orientable surface; that is, informally, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down. More formally, the surface is a two-dimensional manifold on which one cannot define a consistent direction perpendicular to the surface (normal vector) that varies continuously over the whole shape.

The Klein bottle is related to other non-orientable surfaces like the Möbius strip, which also has only one side but does have a boundary. In contrast, the Klein bottle is boundaryless, like a sphere or torus, though it cannot be embedded in ordinary three-dimensional space without intersecting itself.

The Klein bottle was first described in 1882 by the mathematician Felix Klein.^[1]