Boy's surface

Boy's surface was first parametrized explicitly by Bernard Morin in 1978. Another parametrization was 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.

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

Parametrization

Boy's surface can be parametrized in several ways. One parametrization, discovered by Rob Kusner and Robert Bryant, is the following: given a complex number w whose magnitude is less than or equal to one (${\displaystyle \Vert w\Vert \le 1}$), let $${\displaystyle \begin{array}{cc}{g}_1& =-\frac{3}{2}\mathrm{Im}\left[\frac{w\left(1-w^4\right)}{w^6+\sqrt{5}{w}^3-1}\right]\\ g_2& =-\frac{3}{2}\mathrm{Re}\left[\frac{w\left(1+w^4\right)}{w^6+\sqrt{5}{w}^3-1}\right]\\ g_3& =\mathrm{Im}\left[\frac{1+w^6}{w^6+\sqrt{5}{w}^3-1}\right]-\frac{1}{2}\end{array}}$$

and then set $${\displaystyle \left(\begin{array}{c}x\\ y\\ z\end{array}\right)=\frac{1}{g_1^2+g_2^2+g_3^2}\left(\begin{array}{c}{g}_1\\ g_2\\ g_3\end{array}\right)}$$

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

If one performs an inversion of this parametrization centered on the triple point, one obtains a complete minimal surface with three ends (that's how this parametrization was discovered naturally). 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.

Property of Bryant–Kusner parametrization

If w is replaced by the negative reciprocal of its complex conjugate, $-\frac{1}{w^{\star }},$ then the functions g₁, g₂, and g₃ of w are left unchanged.

By replacing w in terms of its real and imaginary parts w = s + it, and expanding resulting parameterization, one may obtain a parameterization of Boy's surface in terms of rational functions of s and t. This shows that Boy's surface is not only an algebraic surface, but even a rational surface. 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).

Relation to the real projective plane

Let ${\displaystyle P(w)=(x(w),y(w),z(w))}$ be the Bryant–Kusner parametrization of Boy's surface. Then $${\displaystyle P(w)=P\left(-\frac{1}{w^{\star }}\right).}$$

This explains the condition ${\displaystyle \Vert w\Vert \le 1}$ on the parameter: if ${\displaystyle \Vert w\Vert <1,}$ then $\Vert -\frac{1}{w^{\star }}\Vert >1.$ However, things are slightly more complicated for ${\displaystyle \Vert w\Vert =1.}$ In this case, one has $-\frac{1}{w^{\star }}=-w.$ This means that, if ${\displaystyle \Vert w\Vert =1,}$ the point of the Boy's surface is obtained from two parameter values: ${\displaystyle P(w)=P(-w).}$ In other words, the Boy's surface has been parametrized by a disk such that pairs of diametrically opposite points on the perimeter of the disk are equivalent. 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 surface also appears in a corresponding construction for the Klein bottle: the Ida surface, an immersion of the Klein bottle in three-dimensional space with no pinch points, is topologically the connected sum of two copies of Boy's surface, just as the Klein bottle is the connected sum of two copies of the real projective plane. It is obtained by applying to the Etruscan Venus, a singular Klein bottle analogous to Steiner's Roman surface, the same pinch-point-cancelling deformation that carries the Roman surface to Boy's surface.