Index

9. No, the converse of this theorem is not true. Here is a counter-example:

Notice that each point is incident with 3 lines, but there are three lines incident with 2 points, and three lines incident with 3 points. This diagram is often referred to as the punctured Fano plane. It is created by removing any point from the Fano plane, and shortening the corresponding lines. [Note: So this geometry is the derived geometry at the point, but different from the near-linear restricted geometry at the point, which only consists of the point itself.] This procedure does not change the incidence of each point, as no lines are removed and each point remains on three lines. It clearly follows however, that the three lines in the Fano plane that are incident with the removed point will no longer be incident with 3 points, but rather will now be incident with 2. Note that the removal of any one of the seven points will create an isomorphic space, as the Fano plane is a homogenous geometry. This means that there is an automorphism that takes any point to any other. Essentially, each point is indistinguishable from any other.