Regular polyhedron

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 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, known as the Platonic solids. These are the: tetrahedron {3, 3}, cube {4, 3}, octahedron {3, 4}, dodecahedron {5, 3} and icosahedron {3, 5}. There are also four regular star polyhedra, making nine regular polyhedra in all.

There are five convex regular polyhedra, known as the Platonic solids, and four regular star polyhedra, the Kepler-Poinsot 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 regular polyhedron has all of three related spheres (other polyhedra lack at least one kind) which share its centre:

...
Wikipedia