4D-Klein-Sliced

Code for visualizing hyperplane sections of the embedded Klein bottle in 4D.

We embed the Klein bottle into $\mathbb{R}^4$ using the parametrization

$$\Phi(u,v)=\left(\sin v\sin\frac{u}{2},\sin v\cos\frac{u}{2},(2+\cos v)\sin u,(2+\cos v)\cos u\right)$$

for $(u,v)\in[0,2\pi)\times[0,2\pi)$.

demo.mp4

In the visualization above, the height being scanned corresponds to the coordinate along the fourth dimension of $\mathbb{R}^4$. Interpreted as a Morse function on the Klein bottle, this height function has two index-one critical points at height values $\pm1$.

For more details, please refer to my note on topology.