# Archimedean Polyhedra

These solids were described by Archimedes, although his original writings on the topic are  lost and only known of second-hand. All but one of these polyhedra were gradually rediscovered during the Renaissance by various artists, and Kepler finally reconstructed the entire set in 1619.

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.

Thus there are 13 solids classified as Archimedean, (not counting two mirror images twice). The snub cube and snub dodecahedron are chiral; they each come in two handednesses (two enantiomorphs, referred to as left-handed and right-handed forms or laevo and dextro forms). To see the enantiomorph of either of the snub figures, view the reflection of your computer screen in a mirror. Or, look at this side-by-side model of a snub cube and its enantiomorph.

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.