Boy's surface is an immersion of the real projective plane in 3-dimensional space found by Werner Boy. Unlike the Roman surface and the cross-cap, it has no singularities (pinch points), but it does self-intersect.
To make a Boy's surface:
Start with a sphere. Remove a cap.
Attach one end of each of three strips to alternate sixths of the edge left by removing the cap.
Bend each strip and attach the other end of each strip to the sixth opposite the first end, so that the inside of the sphere at one end is connected to the outside at the other. Make the strips skirt the middle rather than go through it.
Join the loose edges of the strips. The joins intersect the strips.
Boy's surface is discussed (and illustrated) in Jean-Pierre Petit's Le Topologicon.
