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Polyhedra in which all vertices are the same

In geometry, an Archimedean solid is one of 13 convex polyhedra whose faces are regular polygons and whose vertices are all symmetric to each other. They were first enumerated by Archimedes. They belong to the class of convex uniform polyhedra, the convex polyhedra with regular faces and symmetric vertices, which is divided into the Archimedean solids, the five Platonic solids (each with only one type of polygon face), and the two infinite families of prisms and antiprisms. The 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. An even larger class than the convex uniform polyhedra is the Johnson solids, whose regular polygonal faces do not need to meet in identical vertices. In these polyhedra, the vertices are identical, in the sense that a global isometry of the entire solid takes any one vertex to any other. observed that a 14th polyhedron, the elongated square gyrobicupola (or pseudo rhombicuboctahedron), meets a weaker definition of an Archimedean solid, in which "identical vertices" means
merely that the parts of the polyhedron near any two vertices look the same (they have the same shapes of faces meeting around each vertex in the same order and forming the same angles). Grünbaum pointed out a frequent error in which authors define Archimedean solids using some form of this local definition but omit the 14th polyhedron. If only 13 polyhedra are to be listed, the definition must use global symmetries of the polyhedron rather than local neighborhoods. 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.

The Archimedean solids take their name from Archimedes, who discussed them in a now lost work. Pappus refers to it, stating that Archimedes listed 13 polyhedra. During the Renaissance, artists and mathematicians valued pure forms with high symmetry, and by around 1620 Johannes Kepler had completed the rediscovery of the 13 polyhedra, as well as defining the prisms, antiprisms, and the non convex solids known as Kepler Poinsot polyhedra. (See for more information about the rediscovery of the Archimedean solids during the renaissance.) Kepler may have also found the elongated square gyrobicupola (pseudorhombicuboctahedron): at least, he once stated that there were 14 Archimedean solids. However, his published enumeration only includes the 13 uniform polyhedra, and the first clear statement of the pseudorhombicuboctahedron's existence was made in 1905, by Duncan Sommerville.

The number of vertices is 720° divided by the vertex angle defect. The cuboctahedron and icosidodecahedron are edge uniform and are called quasi regular. The duals of the Archimedean solids are called the Catalan solids. Together with the bipyramids and trapezohedra, these are the face uniform solids with regular vertices.

The snub cube and snub dodecahedron are known as chiral, as they come in a left handed form (Latin: levomorph or laevomorph) and right handed form (Latin: dextromorph). When something comes in multiple forms which are each other's three dimensional mirror image, these forms may be called enantiomorphs. (This nomenclature is also used for the forms of certain chemical compounds.)

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