A Klein bottle can be made from a rectangular piece of the plane by identifying the top and bottom edges using the same orientation, but identifying the left and right edges with opposite orientation (as in the formation of a Möbius band). The first step forms a tube, but the second step can not be carried out without causing self-intersection: the tube must pass through itself in order to attach the ends correctly.

The Klein bottle can not be embedded in three-space, but it can be immersed there.

* 8/12/94 dpvc@geom.umn.edu -- *

*The Geometry Center*