Projective Plane

The projective plane is one of the simplest non-orientable surfaces. Like the Möbius strip the projective plane only has one side. However unlike the Möbius strip which has a single circular edge the projective plane plane has no edges. One way to think of the projective plane is to imagine attaching a disk to a Möbius strip along its circular boundaries. This is difficult to imagine because in three dimensions the resulting closed surface must intersect itself. There are several ways that the self-intersections can occur in three dimensions but some self-intersection must occur. Each choice for the self-intersection produces a different three dimensional figure. The projective plane can be embedded in four or more dimensions so that it does not intersect itself |

Another way to
imagine the projective plane is to take a sphere and attach each pair
of points on opposite sides to one another. This is still hard to imagine
but it provides a nice way to embed the projective plane in a six dimensional
space. Let (x,y,z) be a point on the unit sphere in 3-space. Map this
point to (x |