Traditional methods for computing quantities such as scattering amplitudes and form factors in quantum field theory become intractable at high perturbative orders. However, a great deal is now known about the mathematical properties of these quantities, especially in supersymmetric gauge theories. In this talk, I will describe how this knowledge can be leveraged to 'bootstrap' amplitudes and form factors directly, by constructing an ansatz with the appropriate mathematical structure and requiring it to have certain expected behavior in special kinematic limits. I will focus on the example of three- and four-point form factors in maximally supersymmetric gauge theory, which have recently been found to be amenable to bootstrap techniques. I will then describe a remarkable new 'antipodal' self-duality that is respected by the four-point form factor, which descends to a duality between three-point form factors and six-particle amplitudes in collinear limits.