While traditional methods for calculating scattering amplitudes prove too computationally intensive to be useful at higher loop orders, a great deal is now known about the analytic and kinematic properties of amplitudes to all orders in planar N=4. This information can be leveraged to construct these amplitudes directly, by putting together an ansatz of the relevant class of functions and requiring that it share the distinctive properties of a given amplitude. In this talk, I will describe how this bootstrap-type approach can be used to uniquely determine all six-particle amplitudes in this theory through (at least) five loops, focusing on how these methods make transparent the Steinmann relations and bear out the predictions of cosmic Galois theory. I will also discuss how these methods can be generalized to quantities of direct relevance to particle physics experiments.