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Self intersecting, highly symmetrical mapping of the real projective plane into 3D space

In mathematics, the Roman surface or Steiner surface is a self intersecting mapping of the real projective plane into three dimensional space, with an unusually high degree of symmetry. This mapping is not an immersion of the projective plane; however, the figure resulting from removing six singular points is one. Its name arises because it was discovered by Jakob Steiner when he was in Rome in 1844. The simplest construction is as the image of a sphere centered at the origin under the map This gives an implicit formula of

Also, taking a parametrization of the sphere in terms of longitude (θ) and latitude (φ), gives parametric equations for the Roman surface as follows:

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 coordinate axis which terminate in six pinch points. The entire surface has tetrahedral symmetry. It is a particular type (called type 1) of Steiner surface, that is, a 3 dimensional linear projection of the Veronese surface.

The Roman surface is non orientable, i. e. one sided. This is not quite obvious. To see this, look again at Figure 3. Imagine an ant on top of the "third" hyperbolic paraboloid, z = x y. Let this ant move North. As it moves, it will pass through the other two paraboloids, like a ghost passing through a wall. These other paraboloids only seem like obstacles due to the self intersecting nature of the immersion. Let the ant ignore all double and triple points and pass right through them. So the ant moves to the North and falls off the edge of the world, so to speak. It now finds itself on the northern lobe, hidden underneath the third paraboloid of Figure 3. The ant is standing upside down, on the "outside" of the Roman surface. Let the ant move towards the Southwest. It will climb a slope (upside down) until it finds itself "inside" the Western lobe. Now let the ant move in a Southeastern direction along the inside of the Western lobe towards the z = 0 axis, always above the x y plane. As soon as it passes through the z = 0 axis the ant will be on the "outside" of the Eastern lobe, standing rightside up. Then let it move Northwards, over "the hill", then towards the Northwest so that it starts sliding down towards the x = 0 axis. As soon as the ant crosses this axis it will find itself "inside" the Northern lobe, standing right side up. Now let the ant walk towards the North. It will climb up the wall, then along the "roof" of the Northern lobe. The ant is back on the third hyperbolic paraboloid, but this time under it and standing upside down. (Compare with Klein bottle.)

The Roman surface has four "lobes". The boundaries of each lobe are a set of three lines of double points. Between each pair of lobes there is a line of double points. The surface has a total of three lines of double points, which lie (in the parametrization given earlier) on the coordinate axes. The three lines of double points intersect at a triple point which lies on the origin. The triple point cuts the lines of double points into a pair of half lines, and each half line lies between a pair of lobes. One might expect from the preceding statements that there could be up to eight lobes, one in each octant of space which has been divided by the coordinate planes. But the lobes occupy alternating octants: four octants are empty and four are occupied by lobes. If the Roman surface were to be inscribed inside the tetrahedron with least possible volume, one would find that each edge of the tetrahedron is tangent to the Roman surface at a point, and that each of these six points happens to be a Whitney singularity. These singularities, or pinching points, all lie at the edges of the three lines of double points, and they are defined by this property: that there is no plane tangent to any surface at the singularity.

A. Coffman, A. Schwartz, and C. Stanton: The Algebra and Geometry of Steiner and other Quadratically Parametrizable Surfaces. In Computer Aided Geometric Design (3) 13 (April 1996), p. 257 286
Bert Jüttler, Ragni Piene: Geometric Modeling and Algebraic Geometry. Springer 2008, , p. 30