Regular polyhedron

A regular polyhedron is a polyhedron with regular and congruent polygons as faces. Its 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.

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:

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:

A convex regular polyhedron has all of three related spheres (other polyhedra lack at least one kind) which share its centre:

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:

Any shapes with icosahedral or octahedral symmetry will also contain tetrahedral symmetry.

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.