Boy's surface

The Boy's surface is a geometric object. It is an immersion of the real projective plane in three-dimensional space. Werner Boy discovered the area was named after him in 1901. Like the Klein bottle but it has self-intersections, and also a triple point.

Background

The real projective plane is the simplest non- orientable closed surface. It arises from the Möbius strip by gluing a disc to its edge.

Like all non- orientable closed surfaces can not be embedded into the Projective plane.

The Boy's surface realized but after an immersion of the projective plane in the. Boy Werner designed this space as a counterexample to a conjecture of David Hilbert, that there would be no immersions of the projective plane in the. Boy was able to make several drawings of this area, and discovered their potential three-fold rotational symmetry, but could not find a parametric representation for it. Only in 1978 was Bernard Morin a parametrization with computer support.

The first analytical representation was given in 1981 by a semi- empirical method. This is to describe the meridians of the surface of ellipses, which are then parameterized.

There are now numerous parameterizations of Boyschen surfaces, for example by polynomials of degree 4.

Bryant - Kusner parametrization

The following parameterization is defined for complex numbers t with.

With

Opposing points on the boundary of the unit disk have the same pixel, so it is indeed a parameterization of a projective plane.

This parameterization minimizes the Willmore energy under all immersions of the projective plane in the.

Model in Oberwolfach

Before the Mathematical Research Institute Oberwolfach is a 1991 donated by Mercedes- Benz model of the Boy surface. This model has a symmetry group of order 3 and minimizes the Willmore energy. It consists of steel rods, representing the image of the polar coordinate system in the Bryant - Kusner parametrization. The radial rods ( images of the rays) are ordinary once twisted Möbius bands. The longitudinal rods ( images of the circles ) are ungetwistet with one exception, the exception (equivalent ) is a triple getwistetes Möbius strip, as it is also used in the emblem of the Mathematical Research Institute.

Swell