Four red vectors are given, one per quadrant, $[0,90^\circ)$,
$[90^\circ,180^\circ)$, etc.
A rigid *star* of six green vectors separated by $60^\circ$
can be positioned at
$(\theta,
\theta+60^\circ,
\theta+120^\circ,
\theta+180^\circ,
\theta+240^\circ,
\theta+300^\circ)$.
The goal is to spin the green star so that the red vectors are centralized
in green sectors as much as possible.
Define the *deviation* $\delta(r)$ of a red vector $r$ as the larger of the
(absolute value of the) two
angles from the red vector to the boundaries of the green sector in which it lies.
I want to minimize the largest red deviation.

For example, let the red vectors be at $(0^\circ, 90^\circ, 180^\circ, 270^\circ)$. Then choosing $\theta=15^\circ$ yields a deviation of $45^\circ$ for all red vectors. For example, $\delta(0^\circ) = \max \{ 15^\circ, 45^\circ \}$.

My question is: What is the largest deviation of any four red vectors? I thought it might approach $60^\circ$, but it seems that perhaps $52.5^\circ$ is the worst ($52.5^\circ = 7 \pi / 24$).

The problem generalizes to $k$ red vectors and $m > k$ green sectors. Likely the logic to establish the answer for $(k,m)=(4,6)$ will work for any $(k,m)$.