2.3.4 Klein bottle

There are two other surfaces that can be obtained by identifying both pairs of opposite edges of a rectangle. In one of these, shown in Figure 31, we first identify the edges AB and A'B', labelled a, in the direction shown by the arrowheads. This gives us a cylinder, as before. We then try to identify the cylinder ends, labelled b, in opposite directions, as indicated by the arrowheads – this is equivalent to identifying the edges A'A and BB', labelled b, of the original rectangle. Unfortunately, this cannot be done in three dimensions, and so is hard to visualise: if we try to link the ends, the cylinder has to ‘pass through itself’. So, although we have a surface, it is not a surface in space. It is called a Klein bottle.

Figure 31 Making a Klein bottle from a rectangle