In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the semi-regular convex polyhedra composed of regular polygons meeting in identical vertices, excluding the 5 Platonic solids (which are composed of only one type of polygon) and excluding the prisms and antiprisms. They differ from the Johnson solids, whose regular polygonal faces do not meet in identical vertices.
"Identical vertices" means that each two vertices are symmetric to each other: A global isometry of the entire solid takes one vertex to the other while laying the solid directly on its initial position. Branko Grünbaum (2009) pointed out a widespread error in the literature on Archimedean solids: Some authors give a weaker definition of an Archimedean solid, in which "identical vertices" means merely that the faces surrounding each vertex are of the same types (Each vertex looks the same from close uo.), so only a local isometry is required, but then they omit a 14th polyhedron that meets this weaker definition, the elongated square gyrobicupola (pseudo-rhombicuboctahedron), and only list the 13 Archimedean solids. There is a 14th solid that meets the weaker definition but not the stronger one, the (pseudo-rhombicuboctahedron). Because of this repeated mistake, Grünbaum suggested that the weaker definition should be used and that the elongated square gyrobicupola should be counted as an Archimedean solid, which would give 14 Archimedean solids. However, most authors, as well as Archimedes himself, do not include it in their lists of Archimedean solids, and to be consistent with this choice, they must use the stronger global isometry-based definition.
Prisms and antiprisms, whose symmetry groups are the dihedral groups, are generally not considered to be Archimedean solids, even though their faces are regular polygons and their symmetry groups act transitively on their vertices. Excluding these two infinite families, there are 13 Archimedean solids. All the Archimedean solids (but not the elongated square gyrobicupola) can be made via Wythoff constructions from the Platonic solids with tetrahedral, octahedral and icosahedral symmetry....LESS