So far our efforts have been focused on the general projective plane. While this has done us well so far, there are reasons we may want to change our focus, if only for a while.
Firstly, recall we based our axioms on observations of the Euclidean plane. Furthermore, the real projective plane is basically the Euclidean plane with a line at infinity. However, the Euclidean plane is just the two dimensional version of Euclidean geometry. So far, we haven't found a way to generalize projective planes in the same way, and it would be interesting to find out how, if possible.
Secondly, while we have built up a good collection of simple theorems from the plane axioms, we can't get much further with the plane axioms alone. The problem is that while the plane axioms ensure the presence of projectivities, they cannot really control their behaviour, and as such there is not much that can be proven about projectivities or projective planes from the plane axioms alone. We thus seek a natural restriction on projective planes that will allow us to 'tame' the projectivities. This control will payoff in the existence of complex structure and sophisticated tools that otherwise may be absent.
To get an idea of how to construct 'n-dimensional' analogues of projective space, we shall look at the Axioms 1-6 one by one.
Axioms 1 and 2, not only are both true in Euclidean and projective planes, but also in every Euclidean space, so that so it seems that these two axioms should remain unchanged. Note in particular this means that Theorems 1.1 and 1.2 also apply.
Axiom 3 is tricky, so we leave that for now.
As for Axioms 4-6, there is nothing wrong with Axiom 4, but for full generality, we also want projective analogs of Euclidean spaces of dimension lower than two. Axioms 5 and 6 are incompatible with this, so they must go.
That leaves us with Axioms 1, 2, and 4, but we still need to deal with Axiom 3. That will be done in the next post.