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Cayley-Klein Geometries and physics!

  1. Feb 12, 2012 #1
    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!!
  2. jcsd
  3. Feb 12, 2012 #2
    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.
  4. Feb 14, 2012 #3
    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.
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