Archimedean solid

The Archimedean solids are a set of thirteen convex polyhedra whose faces are regular polygons, but not all alike, and whose vertices are all symmetric to each other.

The elongated square gyrobicupola or pseudo­rhombi­cub­octa­hedron is an extra polyhedron with regular faces and congruent vertices, but it is not generally counted as an Archimedean solid because it is not vertex-transitive.

Expansion involves moving each face away from the center (by the same distance to preserve the symmetry of the Platonic solid) and taking the convex hull.

An example is the rhombicuboctahedron, constructed by separating the cube or octahedron's faces from the centroid and filling them with squares.

[14] The net of Archimedean solids appeared in Albrecht Dürer's Underweysung der Messung, copied from the Pacioli's work.

Kepler once stated that there were fourteen Archimedean solids, yet his published enumeration only includes the thirteen uniform polyhedra.

[16] The solid appeared when some mathematicians mistakenly constructed the rhombicuboctahedron: two square cupolas attached to the octagonal prism, with one of them rotated in forty-five degrees.

[17] The thirteen solids have the property of vertex-transitive, meaning any two vertices of those can be translated onto the other one, but the elongated square gyrobicupola does not.

Grünbaum pointed out a frequent error in which authors define Archimedean solids using some form of this local definition but omit the fourteenth polyhedron.