We consider the 2 -> 2 scattering amplitude of identical massive particles. We identify the Landau curves in the multi-particle region 16m(2) <= s, t < 36m(2). We systematically generate and select the relevant graphs and numerically solve the associated Landau equations for the leading singularity. We find an infinite sequence of Landau curves that accumulates at finite s and t on the physical sheet. We expect that such accumulations are generic for s, t > 16m(2). Our analysis sheds new light on the complicated analytic structure of nonperturbative relativistic scattering amplitudes.