We study the cluster algebra of the kinematic configuration space $${{\mathrm{Conf}}}_n(\mathbb {P}^3)$$ Conf n ( P 3 ) of an $$n$$ n -particle scattering amplitude restricted to the special 2D kinematics. We found that the $$n$$ n -point two-loop MHV remainder function in special 2D kinematics depends on a selection of the $${\mathcal {X}}$$ X -coordinates that are part of a special structure of the cluster algebra related to snake triangulations of polygons. This structure forms a necklace of hypercube beads in the corresponding Stasheff polytope. Furthermore at $$n = 12$$ n = 12 , the cluster algebra and the selection of the $${\mathcal {X}}$$ X -coordinates in special 2D kinematics replicates the cluster algebra and the selection of $${\mathcal {X}}$$ X -coordinates of the $$n=6$$ n = 6 two-loop MHV amplitude in 4D kinematics.