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 with its rims sewed together (points on opposite sides connect.).
The significance of this surface is its topological property representing the projective plane.
On a piece of paper, draw a circle. Imagine the opposite points on the circle are connected. That is a projective plane.
Formula:
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: Boys surface (of Apery), Boys surface (of Bryant-Kusner), Steiner Surface
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