Klein Surface

Klein bottle, a closed one-sided surface, the concept for which was introduced by F. Klein in 1874.

A Klein surface can be made from a tube that is open at both ends by bending the tube, passing its narrower end through the

Figure 1. Klein surface

wall, and “identifying” the two end circles (see Figure 1,a). A surface is obtained that has a self-intersecting line (see Figure l,b). A Klein surface without self-intersection can only be realized in four-dimensional space.