A key characteristic of the Archimedean solids is that each face is a regular polygon, and around every vertex, the same polygons appear in the same sequence, e.g., hexagon-hexagon-triangle in the truncated tetrahedron, shown above. Two or more different polygons appear in each of the Archimedean solids, unlike the Platonic solids which each contain only a single type of polygon. The polyhedron is required to be convex.
Here are the possibilities as to what can appear at a vertex. The notation (3, 4, 3, 4) means each vertex contains a triangle, a square, a triangle, and a square, in that cyclic order.
Although the accepted polyhedron names are less than ideal, there is a certain logic to the names above. (They are adapted from Kepler's Latin terminology.) The term snub refers to a process of surrounding each polygon with a border of triangles as a way of deriving for example the snub cube from the cube. The term truncated refers to the process of cutting off corners. Compare, for example, the cube and the truncated cube. Truncation adds a new face for each previously existing vertex, and replaces n-sided polygons with 2n-sided ones, e.g., octagons instead of squares.
There is also another class of polyhedra in which the same regular polygons appear at each vertex: the prisms and antiprisms, which have vertex types (4, 4, n) and (3, 3, 3, n) respectively. But as these form two infinite series, they can not be all listed, and so are described as a separate group.
Exercise: Just saying that the same regular polygons appear in the same sequence at each vertex is not a sufficient definition of these polyhedra. The Archimedean solid (shown at right) in which three squares and a triangle meet at each vertex is the rhombicuboctahedron. Look at it, and then imagine another, similar, convex solid with three squares and an equilateral triangle at each vertex. Figure out what it must be before looking at the answer.
Answer: Read about the pseudo-rhombicuboctahedron.
Exercise: The first two entries in the list above, the cuboctahedron and the icosidodecahedron, have certain special properties. What do you notice about their edges ?
Answer: They belong to a
special class of polyhedra.