# Sudanese Surface

This surface gets its name not from the country of Sudan, but from the two topologists who named it – Sue Goodman and Daniel Asimov (it was discovered by Blaine Lawson in his Stanford Ph.D. thesis).

Starting from an immersion of the Klein bottle into the 3-sphere $S^3 = \{(z,w)\in \mathbb{C}^2|\ |z|^2 + |w|^2 = 1\}$ due to Blaine Lawson, we take a half-strip of the bottle to obtain a Möbius band in $S^3$. We then pick a reference point from which we will project the band into 3-dimensional space, like ($\frac{1}{\sqrt{2}},\frac{i}{\sqrt{2}}$). Living inside of $\mathbb{R}^4$, this point has an orthogonal complement that is homeomorphic to $\mathbb{R}^3$. We can then just pick an orthonormal basis of this subspace, and we have a copy of $\mathbb{R}^3$

Our Sudanese surface is thus obtained by stereographically projecting the embedded band into the orthogonal complement of ($\frac{1}{\sqrt{2}}, \frac{i}{\sqrt{2}}$) equipped with orthonormal coordinates, so that we get a surface in $\mathbb{R}^3$.