Elementary particle scattering is perhaps the most basic physical process in Nature. The data specifying the scattering process defines a "kinematic space", associated with the propagation of particles out to infinity. By contrast the textbook approach to computing scattering amplitudes using Feynman diagrams, invokes auxiliary structures beyond this kinematic space--local interactions in the interior of spacetime, and unitary evolution in Hilbert space. This description makes space-time locality and quantum-mechanical unitarity manifest, but hides extraordinary simplicity and infinite hidden symmetries of the amplitude that have been uncovered over the past thirty years. The past decade has seen the emergence of a new picture, where scattering amplitudes are seen as the answer to an entirely different sort of question involving the notion of "positive geometries" directly in the kinematic space, with surprising and deep connections to a number of contemporary areas of research in mathematics. In this talk I will describe these ideas in a number of examples of direct relevance to real-world physics, where we can see, quite concretely, how the usual rules of space-time and quantum mechanics can arise, joined at the hip, from fundamentally geometric and combinatorial origins.