Аннотация:We show that the orthogonal separation coordinates on the sphere $S^n$ are naturally
parametrised by the real version of the Deligne-Mumford-Knudsen moduli
space $\bar M_{0,n+2}(\mathbb R)$ of stable curves of genus zero with
$n+2$ marked points. We use the combinatorics of Stasheff polytopes
tessellating $\bar M_{0,n+2}(\mathbb R)$ to classify the different
canonical forms of separation coordinates and deduce an explicit
construction of separation coordinates as well as of St\"ackel systems from the
mosaic operad structure on $\bar M_{0,n+2}(\mathbb R)$.