Making peace with the projective plane

In my newsletter I give an introduction to the latest few posts and occasionally give some additional commentary. In my latest newsletter I wrote Projective space and homogeneous coordinates seemed like arbitrary abstractions when I was a student because they were presented without any motivation. Much later I came to realize that they’re natural, almost inevitable. Exceptional cases and hand-wavy references to objects “at infinity” go away when moving to projective space. My mathematics education was very top-down: abstractions first, specifics later (if ever). I was OK with that for a while, until I hit my abstraction tolerance sometime in grad school. Now I would advocate more of a bottom-up approach, motivating abstractions first rather than presenting them as a fait acompli. The first time I saw the projective plane was probably in a topology class where someone said you could construct the plane by starting with a square and gluing opposite edges in a physically impossible way.…