The Roman surface (so called because Jakob Steiner was in Rome when he thought of it) is a self-intersecting mapping of the real projective plane into three-dimensional space, with an unusually high degree of symmetry. The mapping is not an immersion; however the complement of six points of its domain (ie without the singularities described below) is one. The projective plane is non-orientable with an Euler characteristic of 1 and non-orientable genus of 1.

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 + z²x² − 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. It contains four pinch-points or cross-caps.