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.
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.
Boy's surface was first parametrized explicitly by Bernard Morin in 1978. See below for another parametrization, discovered by Rob Kusner and Robert Bryant.
Boy's surface is one of the two possible immersions of the real projective plane which have only a single triple point.
Read more about Boy's Surface: Symmetry of The Boy's Surface, Model At Oberwolfach, Applications, Parametrization of Boy's Surface
Famous quotes containing the words boy and/or surface:
“The good, the admirable reader identifies himself not with the boy or the girl in the book, but with the mind that conceived and composed that book.”
—Vladimir Nabokov (1899–1977)
“A lifeless planet. And yet, yet still serving a useful purpose, I hope. Yes, a sun. Warming the surface of some other world. Giving light to those who may need it.”
—Franklin Coen, and Joseph Newman. Exeter (Jeff Morrow)