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@futurebird @dymaxion @whknott

🧵> In three dimensions it is more complicated: besides the obvious Euclid-like axioms, it seems one need also an axiom that says "for any eight lines A1,A2,A2,A4 and B1,B2,B3,B4, if 15 of the pairs Ai,Bj intersect, then the 16th will intersect too".

And it only got worse in higher dimensions.

So in the end I gave up on the axiomatic approach. I defined a model with real coordinates, and said "n-dim projective geometry is is any geometry isomorphic to this"