A convex polyhedron is regular if all its faces are alike and all its vertices are alike. More precisely, this means that

1. (i) all the faces are regular polygons having the same number p of edges, and

2. (ii) the same number q of edges meet at each vertex.

Notice that the polyhedron shown here, with 6 triangular faces, satisfies (i) but is not regular because it does not satisfy (ii).

This solid is face-transitive but not regular

There are five regular convex polyhedra, known as the Platonic solids:

1. (i) the regular tetrahedron, with 4 triangular faces (p = 3, q = 3),

2. (ii) the cube, with 6 square faces (p = 4, q = 3),

3. (iii) the regular octahedron, with 8 triangular faces (p = 3, q = 4),

4. (iv) the regular dodecahedron, with 12 pentagonal faces (p = 5, q = 3),

5. (v) the regular icosahedron, with 20 triangular faces (p = 3, q = 5).

See also edge-transitive, face-transitive, Ludwig, Schläfli, Schläfli symbol, vertex-transitive.