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:
“She liked the blue drapes. They made a star
At the angle. A boy in leather moved in.”
—John Ashbery (b. 1927)
“I have passed down the river before sunrise on a summer morning, between fields of lilies still shut in sleep; and when, at length, the flakes of sunlight from over the bank fell on the surface of the water, whole fields of white blossoms seemed to flash open before me, as I floated along, like the unfolding of a banner, so sensible is this flower to the influence of the sun’s rays.”
—Henry David Thoreau (1817–1862)