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Feb 1 at 3:09 comment added ziggurism here is a link to whitehead's book: download.tuxfamily.org/openmathdep/geometry_projective/…
Feb 1 at 3:08 comment added ziggurism for the record, a quaternary relation is the first thing the Whitehead defines in his 1906 axiomatization of projective geometry. He calls them "harmonic points", not "harmonic throw".
Nov 1, 2016 at 21:56 comment added John Baez Speaking of characteristic 2, I read that Coxeter had some axioms for a projective plane that excluded characteristic 2.
Nov 1, 2016 at 21:54 comment added John Baez I'd be happy to do the case of projective lines coming with fields, for starters. With projective planes we know how to generalize, but that's only because we first understood projective planes over fields. For projective lines the first challenge is to find what structure is involved, then what axiom it obeys for kP1 when k is a field, then when k is a division ring, etc. Harmonic throws sound interesting: instead of a 4-ary operation as I was proposing (in my fatally flawed suggestion), these give a 4-ary incidence relation.
Nov 1, 2016 at 15:59 comment added MvG @DavidRoberts: Well, OP asked about lines coming from fields. Of course one might as well use this approach to formulate the axioms of skew fields or division algebras instead of those of a field, but I would assume that some of my harmonic throw conditions would not work there the way I stated them, which may or may not be recoverable using a more general statement. I haven't investigated this so far, though.
Nov 1, 2016 at 13:11 comment added David Roberts What about things like the quaternionic or octonionic projective lines? Perfectly good characteristic zero, but no field in sight.
Nov 1, 2016 at 8:43 history answered MvG CC BY-SA 3.0