Scattering amplitudes are the arena where quantum field theory directly meets collider experiments. An excellent model for scattering in QCD is provided by N=4 super-Yang-Mills theory, particularly in the planar limit of a large number of colors, where the theory becomes integrable, and amplitudes become dual to light-like polygonal Wilson-loop expectation values. The first nontrivial case is the 6-gluon amplitude (hexagonal Wilson loop), which can be computed to 7 loops using a bootstrap which is based on the rigidity of the function space of multiple polylogarithms, together with a few other conditions. It is also possible to bootstrap a particular form factor for the chiral stress-tensor operator to produce 3 gluons, through 8 loops. Remarkably, the two sets of results are related by a mysterious “antipodal” duality, which exchanges the role of branch cuts and derivatives. I will describe how the bootstrapping works and what we know about this new duality.