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

x²y² + y²z² + x²z² − r²xyz = 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 = r² cos θ cos φ sin φ

y = r² sin θ cos φ sin φ

z = r² cos θ sin θ cos² 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.