Ants over a floating sphere A floating stationary sphere of a certain size has 5 ants on its surface.  All ants must move at the same fixed speed and never change their initial direction vector. During the move, they must not be closer than 72.5 degrees (or surface distance of approximately sphere radius x 1.26536) one to another.  The ants are identified from 'A' to 'E'.  Is it possible to create an algorithm to plan the ants movement, position and direction, and view the solution in 3D?
 A: A few considerations (not a solution):
Consider an ant passing through the north pole at time $0$ and walking on the "Greenwich meridian"
$$ r_A(t)=\begin{bmatrix}0\\\sin t\\\cos t\end{bmatrix}$$
and another ant on the meridian $\alpha$ and with a phase difference $\phi$ (where $-180^\circ\le\alpha\le180^\circ$, ):
$$ r_B(t)=\begin{bmatrix}\sin \alpha\sin( t-\phi)\\\cos \alpha \sin (t-\phi)\\\cos (t-\phi)\end{bmatrix}.$$
Then 
$$\begin{align}(r_a(t)-r_B(t))^2&=
\sin^2 \alpha\sin^2( t-\phi)+(\cos \alpha \sin (t-\phi)-\sin t)^2+(\cos (t-\phi)-\cos t)^2\\
&=2-2\cos \alpha \sin (t-\phi)\sin t-2\cos(t-\phi)\cos t\\
&=2-\cos \alpha\, (\cos \phi -\cos(2t-\phi))-(\cos\phi+\cos(2t-\phi))\\
&=2-\cos \alpha \cos \phi-\cos\phi +(\cos\alpha-1)\cos(2t-\phi)).
\end{align} $$
Then clearly
$$ \min_{t\in\Bbb R}(r_a(t)-r_B(t))^2=1-\cos \alpha \cos \phi-\cos\phi +\cos\alpha=(1+\cos\alpha)(1-\cos\phi)$$
We must pick $\alpha, \phi$ so that this expression is $\ge 2(1-\cos74^\circ)$, the value obtained for $\alpha=0$ and $\phi=74^\circ$.
Thus for given $\alpha$, we need
$$ \cos\phi\le 1-\frac{2(1-\cos74^\circ)}{1+\cos\alpha},$$
and this is possible only if $\frac{2(1-\cos74^\circ)}{1+\cos\alpha}\le 2$, or
$$|\alpha|\le 106^\circ .$$
In particular, for $\alpha=90^\circ$ we obtain
$ \cos\phi\le -1+2\cos74^\circ$, or $|\phi|>116.6619\ldots ^\circ$.
While all these restrictions do give some leeway for two of the ants, it seems hard to find a configuration suitable for five ants. For example, the last result shows that an ant travelling on a meridian allows at most two ants travelling on the equator; any somewhat regular pattern seems to be prohibited (e.g., three meridians at $0°, 120°, 240°$ with one ant each and two ants on the equator? Prohibited by $|\alpha|<106°$). This alone does not yet make a solution impossible, but I am not very confoident ...
