The non-Euclidean geometries are both based on conic sections: Elliptic and Hyperbolic. Conic sections are described with 2nd-degree polynomials.
Are there any geometries based on higher degree polynomials? If not, why is the number 2 so special? e.g. It might be because there is no 3rd-degree analog of the Pythagorean theorem, but I have never found a good explanation about why this is the case.