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.