A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used; a common one is that the faces are congruent regular polygons which are assembled in the same way around each vertex. A regular polyhedron is identified by its Schläfli symbol of the form {n, m}, where n is the number of sides of each face and m the number of faces meeting at each vertex. There are 5 finite convex regular polyhedra (the Platonic solids), and four regular star polyhedra (the Kepler–Poinsot polyhedra), making nine regular polyhedra in all. In addition, there are five regular compounds of the regular polyhedra. The regular polyhedra There are five convex regular polyhedra, known as the Platonic solids, four regular star polyhedra, the Kepler–Poinsot polyhedra, and five regular compounds of regular polyhedra: Platonic solids Kepler–Poinsot polyhedra Regular compounds Characteristics Equivalent properties The property of having a similar arrangement of faces around each vertex can be replaced by any of the following equivalent conditions in the definition: The vertices of a convex regular polyhedron all lie on a sphere. All the dihedral angles of the polyhedron are equal All the vertex figures of the polyhedron are regular polygons. All the solid angles of the polyhedron are congruent. Concentric spheres A convex regular polyhedron has all of three related spheres (other polyhedra lack at least one kind) which share its centre: An insphere, tangent to all faces. An intersphere or midsphere, tangent to all edges. A circumsphere, tangent to all vertices. Symmetry The regular polyhedra are the most symmetrical of all the polyhedra. They lie in just three symmetry groups, which are named after the Platonic solids: Tetrahedral Octahedral (or cubic) Icosahedral (or dodecahedral) Any shapes with icosahedral or octahedral symmetry will also contain tetrahedral symmetry. Euler characteristic The five Platonic solids have an Euler characteristic of 2. This simply reflects that the surface is a topological 2-sphere, and so is also true, for example, of any polyhedron which is star-shaped with respect to some interior point. Interior points The sum of the distances from any point in the interior of a regular polyhedron to the sides is independent of the location of the point (this is an extension of Viviani's theorem.) However, the converse does not hold, not even