In this paper, we demonstrate that the boundary element method can be used effectively to simulate the sedimentation of a pair of nonspherical particles. Results are reported for tetrahedra, octahedra, cubes, and icosahedra. It was found that the rate of sedimentation of these particles is hardly affected by their shape and orientation. For octahedra, accurate results can be obtained with 24 elements on the surface of each particle. This allows more particles to be modeled and trajectories to be calculated; examples for eight and nine octahedra sedimenting in an unbounded fluid are given.