  We study the S-matrix of planar $\mathcal{N}=4$ supersymmetric Yang-Mills
theory when external momenta are restricted to a two-dimensional subspace of
Minkowski space. We find significant simplifications and new, interesting
structures for tree and loop amplitudes in two-dimensional kinematics; in
particular, the higher-point amplitudes we consider can be obtained from those
with lowest-points by a collinear uplifting. Based on a compact formula for
one-loop N${}^2$MHV amplitudes, we use an equation proposed previously to
compute, for the first time, the complete two-loop NMHV and three-loop MHV
octagons, which we conjecture to uplift to give the full $n$-point amplitudes
up to simpler logarithmic terms or dilogarithmic terms.
