I have some questions concerning the nine geometries of the plane and their physical significance. (Euclidean, Hyperbolic, Elliptical, Minkowski, anti-Minkowski, Galilean, For starters, what are some of the limitations or problems we encounter when using Euclidean geometry in physics [special relativity(?)]? And how do other geometries fix this? How do we derive other geometries from Projective Geometry? Like de Sitter, Minkowski, anti-euclidean geometry, etc. Lastly, I read that some of these geometries can described using complex numbers. How so? Please help!!