The Steiner Roman surface

This is the projection of the Veronese surface in projective 5-space to projective 3-space, using the 2-plane as center. Alternatively one can first project to P^4 from a point in P^5 which is not on any secant line of the Veronese surface, which gives an an isomorphism of P^2 with a subvariety of P^4, then project this to P^3 from a point not on it. The projection of the real locus does not hit the plane x_0=0 in any real points, so we draw the projection of the real locus of the Veronese surface in affine 3-space. Steiner discovered this surface in 1844. It can be given by the agreeable equation


which makes clear its symmetry under the tetrahedral group permuting x,y,z or changing two of the signs of x,y,z. The real surface is an immersion of the real projective plane in 3 space. It is nonorientable and has a pinch point at each end of the 3 lines of self intersection. All Steiner surfaces are rational surfaces since the Veronese surface is.

Note - If you pinch a ball of clay between the thumb and index finger of each hand held at right angles the surface of the clay will look like this