Index

One of the most exciting things about projective planes is that for any statement that is true for a given projective plane, the dual of that statement must also be true. We define a dual statement as being created by interchanging the words "point" and "line" in a given statement.

To illustrate the principle of duality, we construct an interesting model of an infinite projective plane (recall that up to this point, we have mostly only dealt with finite geometries) which we will call the real projective plane. We will create our real projective plane using the unit sphere. In the first picture below it is evident that we have 2 points A and B as well as a line a through these two points. To construct the real projective plane we will identify each such pair of antipodal points, and so we will describe the point set of the real projective plane to consist of pairs of antipodal points. The line set is the set of great circles passing through these points, with antipodal pairs of points on the line identified of course. It may be easiest to just imagine the real projective plane as the unit sphere with lines as great circles, remembering that any two antipodal points really represent one point.

Now we look at the dual of the real projective plane, which will be the real projective plane itself. Any point (that is, pair of antipodal points on the unit sphere) becomes a line, or great circle, created by finding the "equator" between the two antipodal points. Similarly, any line (great circle) has as its dual a point (whose antipodal pair is the north and south pole of the great circle). We see that the dual of points A and B are now great circles and that they intersect at two antipodal points. This intersection of the circles corresponds to our lines in the original figure. Thus, our points become lines and our lines become points!

The real projective plane, with the point and line set described above, satifies the axioms for a projective plane. If we can show that the dual of each of our axioms is true for a given projective plane, then it follows that the dual of any true statement made about that projective plane plane must also be true.

Please enable Java for an interactive construction (with Cinderella). Please enable Java for an interactive construction (with Cinderella). Please enable Java for an interactive construction (with Cinderella).