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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.

Mark B. Villarino wrote an elementary classification argument for Archimedean solids using Euler formula for a sphere. His paper proves the thirteen exceptional convex cases, treating prisms and antiprisms as separate infinite families.^[ref]

The counting argument relates the faces meeting at a vertex to a global constraint on vertices, edges and faces. This makes a short topological identity constrain possible three-dimensional shapes. The classification must specify its convexity and vertex-uniformity assumptions to avoid including different notions of semiregularity.

A uniform polyhedron has regular polygonal faces and a symmetry group acting transitively on its vertices. Thus every vertex has the same surrounding face pattern, but that pattern may use more than one regular polygon type. The definition includes both convex examples and, under the broader convention, star-faced or self-intersecting examples.

A cube is uniform: each vertex meets three congruent squares, and its rotational symmetries carry any vertex to any other. A cuboctahedron has triangles and squares, alternating around every vertex, and is also uniform. Uniformity does not require every pair of faces to be related by a symmetry; that stronger face condition distinguishes regular solids.

The requirement is geometric. Merely drawing a polyhedron with the same number of incident edges at every vertex does not ensure regular faces or a vertex-transitive symmetry group. Regular prisms and regular antiprisms supply infinite families of convex uniform polyhedra.