I will review for a general audience some recent developments in our
understanding of the mathematical structure of scattering amplitudes
in quantum field theory. Many of these developments involve properties
that have been discovered ”experimentally”: not in actual experiments,
but by carrying out a tedious calculation and then observing
that the result has some remarkable hidden simplicity. I will give
examples of this phenomenon, and in particular I will discuss some
aspects of the geometry of the ”amplituhedron”, a geometric object
that is believed to completely encode certain scattering amplitudes.