An isotoxal polyhedron, also called an edge-transitive polyhedron, is a polyhedron whose geometric symmetry group maps any polyhedron edge to any other (Cromwell 1997, p. 371). Every isotoxal polyhedron is an equilateral polyhedron, although the converse does not hold.
There are exactly nine convex polyhedra that are isotoxal. These are the five Platonic solids, the cuboctahedron, the regular icosidodecahedron, the rhombic dodecahedron, and the rhombic triacontahedron (Göring and Winter 2023, p. 2).
When nonconvex polyhedra with intersecting faces are admitted, the symmetry group need not act transitively on either the polyhedron vertices or the faces. For example, the object designated O32b_1 in Collins' catalog has 20 polyhedron vertices, 48 polyhedron edges, and 18 faces. Some of its polyhedron vertices meet four polyhedron edges and others meet six, while its faces have either four or eight sides (Collins).
