Question: Let $S^2$ denote the 2-sphere embedded in $\mathbb{R}^3$. Consider the group $SO(3)$ of rotations acting on $S^2$. Is there a strict total order $<$ on the point set $P\subseteq S^2, \vert P\vert \ge 3$ such that for any $p,q\in P$ and any rotation $R\in SO(3)$, $p<q$ if and only if $R(p) < R(q)$?
In other words, I want to sort the points on the sphere $S^2$ strictly, and the order resulting from sorting is not affected by any rotation. Any insights, references, or suggestions for further reading would be greatly appreciated.
No. There's a rotation that swaps any two points.
One can show more generally that the orbits of $SO(3)$ acting on pairs $S^2 \times S^2$ of points consists exactly of pairs of points at a fixed distance from each other. All of these are symmetric relations, and any $SO(3)$-invariant binary relation must be a union of orbits, so every $SO(3)$-invariant binary relation on $S^2$ is symmetric (and has the form "$p$ and $q$ are at a distance $d$ from each other, where $d$ lies in some set").