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

The Klein bottle is a two-dimensional non-orientable surface. It was first described in 1882 by the German mathematician Felix Klein.

The Klein bottle canot be embedded in three-dimensional space. That's why images of the Klein bottle always display self-intersections, which do no exist in the Klein bottle itself. However, it *can* be embedded in four-dimensional space.

Note that if you slice a Klein bottle in half along its plane of symmetry, we get the mirror image of two Möbias strips. One will have a left twist, the other will have a right twist.