On Monday, September 2, 2013 2:46:22 AM UTC-5, lite.o...@gmail.com wrote: > [Yet another geometry question that involves no geometry exercises] So after reading various sources on the internet, it seems like Analytic Geometry is an extension of Synthetic Geometry
An instance or model of synthetic geometry that is arrived at by refining it with the additional of extra structures (most notably: the inclusion of the real number or complex number system as the underlying field).
Anything proven in the *synthetic* geometry is true for all of its instances. Whatever is proven in analytic geometry may or may not be true in the more exotic instances of a synthetic geometry (e.g. in a finite projective or affine geometry).
A geometry doesn't have to be a geometry at all. For instance, a tournament schedule can a finite geometry: each team is a point, each meeting or game is a line. If two teams meet only once, then the "two points lie on a unique line" axiom is satisfied. This also applies to other tourneys where 3 or more players or teams compete in a round (like a board game tourney). A finite projective geometry, for instance, may also double-over as the schedule for a multi-player tourney.