examples of toy non-Euclidean geometries

Question

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.

Answer 1

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.