Boy’s Surfaces

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 $a= 0.5$ , $-1.45 < u < 0$ , and $0 < v < 2\pi$ :

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 $\mathbb{R}^{3}$ Parametrization of Boy’s surface

Rob Kusner and Robert Bryant discovered the beautiful parametrization of the Boy’s surface on a given complex number $z$ , where $\left | z \right |\leq 1$ , so that giving the Cartesian coordinates $\left ( X,Y,Z \right )$ of a point on the surface.

In 1986 Apéry gave the analytic equations for the general method of nonorientable surfaces. Following this standard form, $\mathbb{R}^{3}$ parametrization of the Boy’s surface can also be written as a smooth deformation given by the equations :

$x\left ( u,v \right )= \frac {\sqrt{2}\cos\left ( 2u \right ) \cos^{2}\left ( v \right )+\cos \left ( u \right )\sin\left ( 2v \right ) }{D},$