Boy's surface

In geometry, Boy's surface is an immersion of the real projective plane in three-dimensional space.

It was discovered in 1901 by the German mathematician Werner Boy, who had been tasked by his doctoral thesis advisor David Hilbert to prove that the projective plane could not be immersed in three-dimensional space.

[2] Boy's surface is one of the two possible immersions of the real projective plane which have only a single triple point.

[3] Unlike the Roman surface and the cross-cap, it has no other singularities than self-intersections (that is, it has no pinch-points).

One parametrization, discovered by Rob Kusner and Robert Bryant,[4] is the following: given a complex number w whose magnitude is less than or equal to one (

), let and then set we then obtain the Cartesian coordinates x, y, and z of a point on the Boy's surface.

This implies that the Bryant–Kusner parametrization of Boy's surfaces is "optimal" in the sense that it is the "least bent" immersion of a projective plane into three-space.

The remark of the preceding paragraph shows that the generic fiber of this parameterization consists of two points (that is that almost every point of Boy's surface may be obtained by two parameters values).

This shows that the Boy's surface is the image of the real projective plane, RP2 by a smooth map.

That is, the parametrization of the Boy's surface is an immersion of the real projective plane into the Euclidean space.

Boy's (the case p = 3) and Morin's (the case p = 2) surfaces begin a sequence of half-way models with higher symmetry first proposed by George Francis, indexed by the even integers 2p (for p odd, these immersions can be factored through a projective plane).

The Oberwolfach Research Institute for Mathematics has a large model of a Boy's surface outside the entrance, constructed and donated by Mercedes-Benz in January 1991.

This model has 3-fold rotational symmetry and minimizes the Willmore energy of the surface.

It consists of steel strips representing the image of a polar coordinate grid under a parameterization given by Robert Bryant and Rob Kusner.

The meridians (rays) become ordinary Möbius strips, i.e. twisted by 180 degrees.

A model was made in glass by glassblower Lucas Clarke, with the cooperation of Adam Savage, for presentation to Clifford Stoll.