Cayley-Klein Geometries and physics

[GcSanchez05]

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!

 

[Tinyboss]

My knowledge of geometry is nowhere near broad enough to answer everything you asked, but I can recommend a really good book, which was the textbook in a course I graded for last semester: John Stillwell - The Four Pillars of Geometry. It doesn't cover everything you asked about, but it's a really nice overview of four different approaches to geometry (via axioms, linear algebra, projective geometry, or transformation groups) and relationships between different geometries.

 

[arkajad]

Lastly, I read that some of these geometries can described using complex numbers. How so?

Please help!

For instance Hermitian 2x2 complex matrices can be naturally identified with Minkowski space. Determinant of the matrix defines in this case the quadratic form of Minkowski space geometry.