Surfaces can be constructed in a similar way from plane figures other than polygons. For example, starting with a disc, we can fold the left-hand half over onto the right-hand half, and identify the edges labelled a, as shown in Figure 36; this is rather like zipping up a purse, or ‘crimping’ a Cornish pastie. We can then stretch and inflate the object so obtained until we obtain a sphere.
We can even start with more than one plane figure. For example, a sphere can be formed from two discs by identifying edges as shown in Figure 37 and then inflating.
Which surfaces are obtained by identifying edges (in the given directions) in each of the triangles in Figure 38?
Hint: In (b), you might consider cutting the triangle before identifying the edges, and then repairing the cut.
(a) Identifying the edges we obtain a cone which can then be flattened out. The result is a disc.
(b) The three vertices of the triangle are all identified to a single point which lies on the boundary of the resulting surface; the two edges that are identified become a single closed curve through this point. The result is a Möbius band. One way of seeing this is to bisect the triangle, glue the two parts together by identifying the edges labelled a, to create a rectangle, then repair the bisection (which needs a twist).
Throughout this section, our emphasis has been on describing the edge identifications geometrically. A more rigorous treatment, involving the so-called ‘identification topology’, is given in Section 5.