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An Archimedean solid is a convex uniform polyhedron, excluding the Platonic solids and the infinite families of uniform prisms and antiprisms. Here uniform means that every face is a regular polygon and that the symmetry group acts transitively on the vertices of a polyhedron: a symmetry of the entire solid carries any chosen vertex of a polyhedron to any other. Merely having equal numbers of faces of each type at every vertex of a polyhedron is a weaker local condition.^[ref]

There are thirteen types under this convention, counting a mirror-image pair as one type. The infinite prism and antiprism families are excluded by convention even though their uniform members satisfy the same regular-face and global vertex-equivalence conditions.^[ref]

For example, the truncated octahedron has six square faces and eight regular hexagon faces. Counting face-edge incidences gives $(6\cdot 4+8\cdot 6)/2=36$ edge of a polyhedron; Euler characteristic then gives $36-14+2=24$ vertices of a polyhedron. At each vertex of a polyhedron, one square and two regular hexagons meet. The icosidodecahedron, with twenty equilateral triangles and twelve regular pentagons, gives a different regular-face combination.

A Platonic solid with six square faces, twelve edges and eight vertices. Its rotational symmetry group of a cube has order 24; including orientation-reversing isometries gives the order-48 symmetry group of a cube.

The regular dodecahedron is the Platonic solid with 12 regular pentagonal faces, 20 vertices, and 30 edges.

A regular icosahedron is a Platonic solid with twenty equilateral triangular faces, twelve vertices and thirty edges. Five faces meet at each vertex. Its orientation-preserving rotational symmetry has order sixty: the orbit-stabilizer theorem gives twenty faces times the three rotations stabilizing a chosen face.