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 (x2,y2, z2,yz,xz,xy). This map sends opposite points on the unit sphere to the same point. This is called the Veronese embedding of the projective plane. We have taken the data set for the sphere and created a data set for a projective plane using the Veronese embedding. Coordinate projection onto the the last three coordinates produces a surface in 3-space known as Steiner's Roman surface. This is shown on the left above. This was the used as the initial conditions for the Sammon map. The Sammon map transformed Steiner's Roman surface to a figure known as the crosscap which is one of the easiest ways to visualize a projective plane in 3-space. The crosscap is shown on the right above.