Boy’s surface is a nonorientable surface found by Werner Boy in 1901. Boy’s surface is an immersion of the real projective plane in 3-dimensional without infinities and singularities, but it meets itself in a triple point (self-intersect). The images below are generated using 3D-XplorMath, and with the “optimal” Bryant-Kusner parametrization when , , and :

Other viewpoints :

**Projective Plane
**

There are many ways to make a model of the Boy’s surface using the projective plane, one of them is to take a disc, and join together opposite points on the edge with self intersection (note : In fact, this can not be done in three dimensions without self intersections); so the disc must pass through itself somewhere. The Boy’s surface can be obtained by sewing a corresponding band (Möbius band) round the edge of a disc.

**The** **Parametrization of Boy’s surface**

Rob Kusner and Robert Bryant discovered the beautiful parametrization of the Boy’s surface on a given complex number , where , so that giving the Cartesian coordinates of a point on the surface.

In 1986 Apéry gave the analytic equations for the general method of nonorientable surfaces. Following this standard form, parametrization of the Boy’s surface can also be written as a smooth deformation given by the equations :

where

,

varies from 0 to 1.

Here are some links related to Boy’s surface :

- MathWorld, Boy Surface, Nonorientable Surface, Projective Plane.
- Wikipedia, Boy’s Surface, Real Projective Plane.
- Sanderson, B.
*Boy’s Will be Boy’s*, available here.