### 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
x^2*y^2+y^2*z^2+x^2*z^2+x*y*z=0

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