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