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Today I was working on a problem in euclidean geometry, and I found it immensely useful to compare with Fano geometry, for contrast. Are there any other toy geometries like Fano geometry? I think it could be useful to have a repertoire of those.

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Felix Klein considered non-Euclidean geometries, like the hyperbolic and spherical geometry. These are the spaces of constant curvature: Euclidean with curvature $0$, hyperbolic with curvature $-1$ and spherical with curvature $1$. The hyperbolic plane is a good toy example, compared to the Euclidean plane.

Edit: If you wanted finite geometries, here are some more toy examples:http://www.beva.org/math323/asgn5/nov5.htm.

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  • $\begingroup$ Is there a set of axioms that govern the hyperbolic geometry? $\endgroup$ – Adam Nov 27 '13 at 21:06
  • $\begingroup$ Yes, something is written here: en.wikipedia.org/wiki/Hyperbolic_geometry. $\endgroup$ – Dietrich Burde Nov 27 '13 at 21:08
  • $\begingroup$ This hyperbolic geometry seems very complicated. Can I do hyperbolic geometry proofs using just paper? $\endgroup$ – Adam Nov 27 '13 at 21:12

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