In nearly all (if not all) projective geometry texts I have bumped into the following theorem:

"Principle of duality: If in a theorem in $\mathfrak{P}$ one switches the word point for the word line and the corresponding incidence relations once again one obtains a theorem of $\mathfrak{P}$."

So far so good. Then I found this awesome list by G. Eric Moorhouse: http://www.uwyo.edu/moorhouse/pub/planes/ (Wayback Machine)

I noted the distinction between a Hall Plane and its dual. So looking a bit into the matter I kept running into the claim "Hall Planes are non-Desarguian and non-self-dual" and the modified version of the principle of duality, which claims the dualized theorem is true *on the dual plane* (this makes much more sense in my mind).

My question is twofold:

- How does one prove that Hall Planes are not self dual? I haven't managed to find a proof of this fact!
- What would be the true duality principle? If duality holds in the dual plane it should not hold in Hall Planes (as they're not self dual), yet all texts I've read claim that duality holds in any projective plane.

Thanks in advance!

P.S. Any good references on the concept of duality in projective geometry from a geometrical point of view would be much appreciated!