I've been working hard to get a good grasp on projective geometry - reading as much as I can on books.google.com as well as purchasing what appears to be a key exposition for beginners - The Geometry of Incidence by Harold L. Dorwart.
However, it is quite a bit of work to get the brain to accept projective space. Understanding linear projection of vectors via the dot product was hard... projective geometry is a new level of hard.
There seems to be hope, though... a drug called Salvia which is becoming quite the buzz (no pun intended).
"Generally, smoked salvia effects come on quickly, peak for 5-20 minutes, and then begin to subside," the FAQ notes. Users report visions; feelings of fright; loss of physical coordination; uncontrollable laughter; confusion; feelings of being underground, or underwater, or flying, or floating; experiences of "non-Euclidean" spaces; and more, according to Erowid.
Experiences of non-Euclidian spaces? Like, perhaps "projective spaces"? Just what I've been searching for! Need to check if it is legal in my state...