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.