The 1D unit inverval $[0,1)$ can be mapped to the 2D unit square $\left[0, 1\right)^2$ in a way that preserves locality via the Hilbert Curve.
Is there any similarly simple locality-preserving mapping from the 1D unit interval to the 2D surface of a sphere $\{(x,y,z) | x^2 + y^2 + z^2 = 1\}$?
It would be great if $0$ and $1$ mapped to the exact same point on the sphere under this mapping (as opposed to e.g. being mapped to opposite ends of the sphere).
P.S.: I am a dilettante in this area, so I have no idea which tags or areas are appropriate for this post. If someone could help editing in relevant tags, it would be great.