There are many ways to create and classify polyhedra in terms of faces, edges, and vertices. This paper describes a method for generating the Platonic solids and most Archimedean solids using 3D simulated particles having attractive and repulsive forces. It could be described as a variation on the Thomson problem, referring to the distribution of points on a sphere. This method is meant to inspire creative approaches and insights on polyhedral symmetry. Particles are initialized with random positions; the forces cause the particles to converge on the vertices of polyhedra. Faces and edges are derived as secondary features as determined by the relative positions of particles. A categorization scheme is proposed, based on three polyhedral symmetry groups (tetrahedral, octahedral, and icosahedral). This could be used as the basis for an educational tool, with references to molecular modeling and dynamical systems. It also can be a way to algorithmically design a variety of 3D models.