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In geometry, Boy's surface is an immersion of the real projective plane in 3-dimensional space found by Werner Boy in 1901 (he discovered it on assignment from David Hilbert to prove that the projective plane could not be immersed in 3-space). Unlike the Roman surface and the cross-cap, it has no singularities (i.e., pinch-points), but it does self-intersect.
Boy's surface is discussed (and illustrated) in Jean-Pierre Petit's Le Topologicon.
Boy's surface was first parametrized explicitly by Bernard Morin in 1978. See below for another parametrization, discovered by Rob Kusner and Robert Bryant...