# Roman surface

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

Also, taking a parametrization of the sphere in terms of

longitude (θ) and

latitude (φ), we get parametric equations for the roman surface as follows:

*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

The origin is a triple point, and each of the xy-, yz-, and xz-planes are tangential to the surface there. The other places of self-intersection are double points, defining segments along each axis which terminate in pinch points. The entire surface has

tetrahedral symmetry. It is a particular type (called type 1) of

Steiner surface.