Cross-Cap Surface

cross-cap t5hFb
cross-cap 
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}, Boxed -> False, Axes -> False,
 BoundaryStyle -> {Thin, Gray}, PlotPoints -> 20, MaxRecursion -> 4,
 PlotStyle ->
  Directive[RandomColor[], Opacity[.8], Specularity[1, 20]]]

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]

Other surfaces immersed in 3D that are topologically equivalent to the projective plane are: