Cross-Cap Surface
The Cross-cap is a representation of the projective plane. (A “projective plane” is a plane with a line added at infinity. It developed out of study of projective geometry.)
Cross-cap is like a shrinked Torus where there's no middle hole and the side has been pinched together in such way that the top cross to the bottom.
One way to think of cross-cap is as a hemi-sphere (or a bowl), and imagine each point on the rim is connected to points on opposite side.
One way to think of cross-cap is, draw a filled circle on paper, and imagine each point on the rim is connected to the point on the opposite side.
The significance of this surface is that it gives us a visual guide of the topology of the real projective plane. 〔➤see Intro to Real Projective Plane〕
ParametricPlot3D[
{(Sin[u]*Sin[2*v]/2),
(Sin[2*u]*Cos[v]*Cos[v]),
(Cos[2*u]*Cos[v]*Cos[v])
},
{u, 0, Pi}, {v, 0, Pi}]
Other surfaces immersed in 3D that are topologically equivalent to the projective plane are: