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.*