The real projective plane RP² cannot be embedded in 3D without self-intersection.
The crosscap is an immersion showing its non-orientable nature - a path along
the surface can reverse orientation (like a Möbius strip).
x = ½·cos(u)·sin(2v) y = ½·sin(u)·sin(2v) z = ½·(cos²v - cos²u·sin²v)