Tessellating cushions

We consider a class of planar tree-level four-point functions in N = 4 SYM in a special kinematic regime: one BMN operator with two scalar excitations and three half-BPS operators are put onto a line in configuration space; addtionally, for the half-BPS operators a co-moving frame is chosen in flavour space.  In configuration space, the four-punctured sphere is naturally triangulated by tree-level planar diagrams. We demonstrate on a number of examples that each tile can be associated with a modified hexagon form-factor in such a?  way as to efficiently reproduce the tree-level four-point function.  Our tessellation is not of the OPE type, fostering the hope of finding an independent, integrability-based approach to the computation of planar four-point functions.