Steiner's Roman SurfaceA realization of the real projective plane 

Introduction:Steiner's Roman surface is one realization of a mathematical object known as the real projective plane. The real projective plane is defined as "the set that results when antipodal points of S^{2}(a) are identified; thus RP^{2}(a) = {{p,p}  p = a}." (Gray 330) Two points are said to be antipodal if p_{1} = p_{2}. In order to realize the real projective plane (i.e., make it into a surface in R^{3}), we need a map with the antipodal property. That is, a map which satisfies F(p) = F(p). The socalled Roman map, defined by romanmap(x, y, z) = (xy, yz, zx), has this property. When we apply it to a sphere, we obtain the parameterization for Steiner's Roman surface given below. Note that, like the Möbius strip and the Klein bottle, Steiner's Roman surface is nonorientable. This is clear from an attempted frontback coloring of the polygons that approximate the surface: For another realization of the real projective plane, see the cross cap. Definition:, where a is the radius of the sphere to which the Roman map is applied. Properties:
