Boy's surface ( Trivia # 22 )

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 .


To make a Boy's surface:
1. Start with a sphere. Remove a cap.
2. Attach one end of each of three strips to alternate sixths of the edge left by removing the cap.
3. 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.
4. Join the loose edges of the strips. The joins intersect the strips.

Boy's surface can be used in sphere eversion, as a half-way model. A half-way model is an immersion of the sphere with the property that a rotation interchanges inside and outside, and so can be employed to evert (turn inside-out) a sphere. Boy's surface is the beginning of a sequence of half-way models with higher symmetry first proposed by George Francis.