Surfaces

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# 2.3.2 Möbius band

Let us now see what happens if we try to identify two opposite edges of a rectangle in opposing directions. (Here, we are identifying the two ends of the rectangle rather than the top and bottom.) We start with a rectangle, as before, but this time the edges to be identified have their arrowheads pointing in opposite directions. This means that we cannot glue the edges directly, but have to twist one of them through radians before gluing. More formally, we take a closed rectangle ABA'B' in the plane and identify the opposite edges AB and A'B', as shown in Figure 28. This means that:

• we imagine A and A' to be the same point;
• we imagine B and B' to be the same point;
• we pair up all corresponding points in between (such as P and P'), taking note of the directions of the arrowheads.

We obtain a Möbius band.

Figure 28 Making a Möbius band from a rectangle

Again, the labelling AB implies a direction from A to B, and similarly A'B' implies a direction from A' to B'.

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