We present a new formulation of string and particle amplitudes that emerges from simple one-dimensional models. The key is a new way to parametrize the positive part of Teichmüller space. The formulation works at all orders in the perturbation series, including non-planar contributions to the amplitudes. The relationship between string and particle amplitudes is made manifest as a "tropical limit". The results are well adapted to studying the scattering of large numbers of particles or amplitudes at high loop order. The talk will in part cover results from arXiv:2309.15913, 2311.09284.