An Algebraic Surface with Negative Curvature and Euler Characteristic

Classical examples of complete  surfaces of negative Gaussian curvature in Euclidean space R³, which are given by z=xy and z²=x²+y²-1, have Euler characteristic 1 and 0 respectively. The following equation

1 + 2 (x² + y²)² = 2z² + 3z⁴ + 8xy(1+z²)

generates a negatively curved surface with negative Euler characteristic, which is depicted above on the left. The other three pictures are some projective transformations of this surface, which have exactly four flat points each and are negatively curved elsewhere. For more on these surfaces, see this paper, which is joint work with Chris Connell, and the accompanying Mathematica notebook, or its html version.