Section 7.1 Elliptic Geometry
¶We introduced the Klein disk in Section 6.1, and described it as a model for elliptic geometry. But what are the postulates of elliptic geometry?
We'd like to use the neutral postulates, then add the elliptic parallel postulate, which asserts that there are no parallel lines. That is, any two distinct lines must intersect! But there's a problem with this program: it's not possible. Why not? Because we showed in Section 4.5 that parallel lines exist in neutral geometry!
In order to use the elliptic parallel postulate, we will have to modify the neutral postulates. A key step in the argument that parallel lines exist was the Exterior Angle Theorem, which asserts that the exterior angles of a triangle are larger than the nonadjacent interior angles, as discussed in Section 4.3. So far, we have two models of elliptic geometry in mind, namely the Klein disk and the sphere. So, what goes wrong with this theorem on a sphere?
Figure 7.1.1 shows the construction used in the Exterior Angle Theorem, drawn on a sphere. What happens if you make the triangle larger? That is, what happens if you move point \(C\) toward the south pole? If you move it far enough (which may require rotating the sphere), you should be able to convince yourself that the Exterior Angle Theorem does not hold on a sphere.
But we still don't know which postulates have to be changed...