Surfaces
Surfaces

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Surfaces

2.3.5 Projective plane

The final surface that can be obtained by identifying edges of a rectangle is even more complicated. Again it cannot be constructed in three dimensions, so is not a surface in space and is hard to visualise. This time we identify both pairs of opposite edges of the rectangle in opposite directions, as indicated by the arrowheads in Figure 32: we identify the edges AB and B'A' (labelled a) and the edges A'A and BB' (labelled b). The resulting surface is called the projective plane.

Figure 32
Figure 32 Projective plane rectangle

The existence of the Klein bottle and the projective plane, which are not surfaces in space, is one reason why mathematicians cannot restrict their study of surfaces to surfaces in space.

We discuss the projective plane in Section 3.

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