ABSTRACT
The Klein bottle is the non-orientable surface of genus two. The double torus is the
orientable surface of genus two. We begin with the Klein bottle. It has several pos-
sible polygonal representations. The most common one seems to be a rectangle with
one pair of opposite sides glued with a twist, as in Figure 16.1, (also Figure 15.9),
giving the form a+b+a+b−, which is not the standard form of Theorem 15.2. Numerous interesting 3D models of the Klein bottle can be found on the internet. We
see that in this rectangular representation, if the two horizontal edges of the rectangle
are glued, the result is a cylinder, in which opposite ends have opposite orientation.
But if the vertical edges are glued, we obtain a Mo¨bius band, whose boundary must
be glued to itself.
