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2.3.6 Torus with 1 hole

We need not restrict ourselves to rectangles: we can also build surfaces by identifying edges of other polygons. For example, if we start with a pentagon and identify two pairs of its edges as shown in Figure 33, what do we get? Identifying the edges labelled a and c in the directions indicated, we obtain a cylinder with a hole; the boundary of the hole is the edge labelled c. If we now identify the edges labelled b (now at each end of the cylinder) in the directions indicated, after some bending and stretching we obtain a torus with a hole.

Figure 33
Figure 33 Making a torus with 1 hole from a pentagon

From now on, we shall usually put a direction on every edge (such as edge c), even if it is not to be identified. For such an edge, we can choose an arbitrary direction.


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