Separation coordinates, moduli spaces and Stasheff polytopes

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 M¯0,n+2(R) of stable curves of genus zero with n + 2 marked points. We use the combinatorics of Stasheff polytopes tessellating M¯0,n+2(R) to classify the different canonical forms of separation coordinates and deduce an explicit construction of separation coordinates, as well as of Stäckel systems from the mosaic operad structure on M¯0,n+2(R)