The

**Roman surface**(so called because Jakob Steiner was in Rome when he thought of it) is a self-intersecting immersion of the real projective plane into three-dimensional space, with an unusually high degree of symmetry.

The simplest construction is as the image of a sphere centered at the origin under the map *f*(*x*,*y*,*z*) = (*yz*,*xz*,*xy*). This gives us an implicit formula of

*x*^{2}*y*^{2}+*y*^{2}*z*^{2}+*x*^{2}*z*^{2}−*r*^{2}*xyz*= 0

*x*=*r*^{2}cos θ cos φ sin φ*y*=*r*^{2}sin θ cos φ sin φ*z*=*r*^{2}cos θ sin θ cos^{2}φ