The Cross-cap is a representation of the projective plane. It is like a shrinked Torus where there's no middle hole, and the side has been clipped so that they cross. The significance of this surface is only its topological property.
This cross-cap image is done with the following parametric formula: x = (aa * aa) * (sin(u) * sin(2 * v) / 2) y = (aa * aa) * (sin(2 * u) * cos(v) * cos(v)) z = (aa * aa) * (cos(2 * u) * cos(v) * cos(v)) where aa is a constant.
See also: boys Apery, Boys (Bryant-Kusner) surface, Cross-cap, Steiner Surface