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2.3.3 Torus

Let us now return to the cylinder we obtained in Figure 27. What happens if we bend it around and glue the two ends? Now, continuing with our idea of identifying edges of our original rectangle, we want to bend the cylinder round and glue its ends in such a way that the points A and B are identified. Furthermore, bending as shown in Figure 29 corresponds to identifying the edges AA' and BB' of the original rectangle in the same direction, as indicated in the figure. We obtain a torus.

Figure 29
Figure 29 Making a torus from a cylinder

In terms of our original rectangle, the whole construction process corresponds to identifying opposite edges in pairs in the directions indicated by the arrowheads in Figure 30. For ease of reference, we label each pair of edges that is to be identified with the same letter, a for the first pair to be identified and b for the second pair.

We often omit the vertex labels from polygons used to construct surfaces by identification and simply use these edge labels.

Figure 30
Figure 30 Making a torus from a rectangle

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