# Archimedean Duals

When one forms the dual of a given polyhedron, one creates a new polyhedron in which the faces and vertices of the dual correspond to the vertices and faces, respectively, of the original polyhedron.  When one takes the duals of the Archimedean solids, one gets an interesting set of new polyhedra that are called the Archimedean duals or the Catalan polyhedra, after the French mathematician Eugene Catalan, who first described them in 1865.

In the Archimedean duals, every face is identical but there are two or more different types of vertices. This is because in the original Archimedean solids every vertex is identical but there are two or more different types of faces, and when taking the dual, faces transform into vertices and vice versa. In these Archimedean duals, the faces are not regular polygons; there are different lengths and angles within each face, but each face is identical and the vertex figures are regular. Furthermore, from the construction it follows that all the dihedral angles of an Archimedean dual are equal.

The relationships between the Archimedean solids and their respective duals is nicely brought out by studying a compound of a solid and its dual. For example, the snub cube and its dual, the pentagonal icositetrahedron, combine into this beautiful compound. The five-sided faces of the dual correspond to the fact that five polygons (four triangles and a square) meet at each vertex of the snub cube. (By the way, be sure and take a peek into the inside of this object.)

Check out this list of all thirteen Archimedean duals and compounds. In each compound note how the vertices of each solid are over the centers of the faces of the other. Also observe that a polyhedron and its dual have the same number of edges. The edges cross at right angles at a point which is the midpoint of the Archimedean solid's edge, but not always the midpoint of the dual solid's edge.

The above list also includes an image of the face of each archimedean dual. This is handy if you want to make a paper model of an archimedean dual. Simply print out the face and use it as a template by copying it or poking pins through its corners into heavy paper. Note however that some printers (especially dot matrix printers) stretch an image horizontally or vertically when printing it. If you have this problem, the printed template will not assemble properly, and you should find another printer. You can test by printing out this image of a square and measuring to see if your printer prints it as square.

Exercise: Which Archimedean duals are chiral ?

Answer: Only this one and this one.