Finite Projective Planes

The idea of finite geometries sounds like an oxymoron the first time you hear of it, something like dry water or cold fire, but the axioms of some kinds of geometry can indeed be satisfied by finite models. The axioms of projective plane geometry are as follows. Any two distinct points lie on exactly one line. Any two distinct lines meet in exactly one point. There exist four points, no three of which are collinear. The most famous example of a finite projective plane is the Fano plane. It helps to switch back and forth between thinking of finite geometries geometrically and in terms of set theory. The fact that the Fano plane has an easy-to-understand illustration is fortunate, but not to be taken for granted. Also, it could be misleading to lean to heavily on the fact that the lines and points of finite geometry can, in this case, be represented by points and line segments (and a circle) in the Euclidean plane. The set-theoretic perspective is that the points are the one-element…

Commenting on Bubbles has been removed. Why? If you want to talk about this post, write to the author, use the comments on their blog if it has them, or reply with a "RE:" post on your own blog. Suggestions? feedback@bubbles.town