The Battle of Grand Gulf was fought on April 29, 1863, during th: The symmetry groups of the Platonic solids are a special class of three-dimensional point groups known as polyhedral groups. The high degree of symmetry of the Platonic solids can be interpreted in a number of ways. Most importantly, the vertices of each solid are all equivalent under the action of the symmetry group, as are the edges and faces. One says the action of the symmetry group is transitive on the vertices, edges, and faces. In fact, this is another way of defining regularity of a polyhedron: a polyhedron is regular if and only if it is vertex-uniform, edge-uniform, and face-uniform.