Construction of sphere-filling ropes

The left movie shows the construction of $\beta^{6,i}, i=0,1,2,3,4,5,6$. Each of the curves has a thickness $\Theta_6=\sin\frac{\pi}{2\cdot6}$. The second and fifth curve are the only two one-component, closed, length maximizing curves of that thickness on the two-sphere. The second curve is blown up, to show, that the curve indeed fills the whole sphere. The other curves have more than one component but are as well sphere filling.

Read the whole story in: Download the movie (License cc-by-sa-3.0): beta_6_i.mp4
Also availabe as 3D-model.
Rendered with POV-Ray.
Valid xhmtl.