File:Immersed-Moebius-4.png

• The only singularities are self-intersections. In particular, the surface has no pinch-points.
• The surface has 3-fold rotational symmetry, and it has a 2-fold symmetry when flipping it: The surface looks the same when viewed from above or from below. The symmetry group is that of an equilateral triangle, namely the dihedral group of order 6.
• The locus of self-intersections has the form of the border of a 3-leaved clover with the triple point in the center.
• The surface is Boy's surface in Bryant-Kusner-parametrization with the top dome removed and adjusted in such a way that it attains the symmetries mentioned above.
• Conversely, poking a hole into a sphere and gluing this version of the Möbius strip into the hole results in the immersion of the real projective plane in 3-space. The gluing can be performed in such a way that no new self-intersections are needed.

For a complex number w let

${\displaystyle {\begin{aligned}g_{1}&=-{\frac {3}{2}}\operatorname {Im} \left({w\left(1-w^{4}\right) \over w^{6}+{\sqrt {5}}w^{3}-1}\right)\\g_{2}&=-{\frac {3}{2}}\operatorname {Re} \left({w\left(1+w^{4}\right) \over w^{6}+{\sqrt {5}}w^{3}-1}\right)\\g_{3}&=\operatorname {Im} \left({1+w^{6} \over w^{6}+{\sqrt {5}}w^{3}-1}\right)+\delta \\\end{aligned}}}$

and let

${\displaystyle {\begin{pmatrix}x\\y\\z\end{pmatrix}}={\frac {1}{g_{1}^{2}+g_{2}^{2}+g_{3}^{2}}}{\begin{pmatrix}g_{1}\\g_{2}\\g_{3}\end{pmatrix}}}$

Then:

• With ${\displaystyle |w|\leqslant 1}$ and ${\displaystyle \delta =-1/2}$: x, y, and z are the Cartesian coordinates of a point on Boy's surface in Bryant-Kusner parametrization, which is an immersion of the real projective plane in 3-space. ${\displaystyle \delta =1/2}$ results in an upside-down version of Boy's surface.
• With ${\displaystyle 0.35\leqslant |w|\leqslant 1}$ and ${\displaystyle \delta =0}$: x, y, and z are the Cartesian coordinates of the shown object. The extra condition ${\displaystyle |w|\geqslant 0.35}$ "pokes" a hole in the surface and ${\displaystyle \delta =0}$ stretches the hole open, leading to the additional flip-symmetry.