Boy’s surface

The surface in the picture

Non-orientable surfaces, like the Klein bottle, cannot be embedded into three-dimensional space. What if we allow self-intersections, i.e. immersions? Some non-orientable surfaces, like the Klein bottle, can be immersed into three-dimensoinal space; however, it was not known, before 1901, whether the projective plane can be immersed into three-dimensional space.

The (real) projective plane $ℝ{\text{P}}^2$ has several equivalent descriptions. The simpliest description is the quotient of the $2$-sphere under the antipodal map; that is, take a $2$-sphere and glue antipodal points together. It’s immediate that we can ignore one hemisphere of the sphere, so we get the second description: take a hemisphere, or equivalently a $2$-disk, and identify antipodal points on its boundary. Now if we focus on the boundary, we notice that the boundary is still a circle after the identification, but the way of attachment of disk is different now: the disk is attached to the circle by winding its boundary two times around the circle. Therefore we get a third description, which essentially gives it a CW complex structure.

David Hilbert asked his student, Werner Boy, to prove that $ℝ{\text{P}}^2$ cannot be immersed in ${ℝ}^3$. However, in 1901, he proved that it actually can be immersed; this immersion is now called Boy’s surface.

Method of production

The Boy’s surface has many parametrizations. I chose the Kusner–Bryant parametrization here for its round and beautiful shape. The parametrization is given by $\left(𝑥,𝑦,𝑧\right)=\frac{1}{{𝑔}_1^2+{𝑔}_2^2+{𝑔}_3^3}\left({𝑔}_1,{𝑔}_2,{𝑔}_3\right)$, where

$$\begin{array}{cc}{𝑔}_1=& -\frac{3}{2}\text{Im}\frac{{𝑤}^2\left(1-{𝑤}^4\right)}{{𝑤}^6+\sqrt{5}{𝑤}^3-1},\\ {𝑔}_2=& -\frac{3}{2}\text{Re}\frac{{𝑤}^2\left(1+{𝑤}^4\right)}{{𝑤}^6+\sqrt{5}{𝑤}^3-1},\\ {𝑔}_3=& \text{Im}\frac{1+{𝑤}^6}{{𝑤}^6+\sqrt{5}{𝑤}^3-1}-\frac{1}{2},\end{array}$$

and $𝑤$ inside the complex unit disk.

The image is produced in Blender. I thought it would be easy given the parametrization, but it turns out quite complicated, mainly because the surface is closed but the domain of $𝑤$ has a boundary, so the resulting surface had a seam. I needed to figure out the periodicity conditions of this parametrizations to close up the seam, and after some trial-and-error I finally got it right.