An Archimedean solid is a convex semiregular polyhedron – a solid made from regular polygonal sides of two or more types that meet in a uniform pattern around each corner. (A regular polyhedron, or Platonic solid, has only one type of polygonal side.)

 

There are 13 Archimedean solids. Although they are named after their discoverer, the first surviving record of them is in the fifth book of the Collection of Pappus of Alexandria. The duals of the Archimedean solids (made by replacing each face with a vertex, and each vertex with a face) are commonly known as Catalan solids. Apart from the Platonic and Archimedean solids, the only other convex uniform polyhedra with regular faces are prisms and antiprisms. This was shown by Johannes Kepler, who also gave the names generally used for the Archimedean solids. See also Johnson solids.

The Archimedean solids
name	vertices	faces	edges
truncated tetrahedron	12	8	18
truncated cube	24	14	36
truncated octahedron	24	14	36
truncated dodecahedron	60	32	90
truncated icosahedron	60	32	90
cuboctahedron	12	14	24
icosidodecahedron	30	32	60
snub dodecahedron	60	92	150
rhombicuboctahedron	24	26	48
great rhombicosidodecahedron	120	62	180
rhombicosidodecahedron	60	62	120
great rhombicuboctahedron	48	26	72
snub cube	24	38	60

 

A chemical Archimedean solid

In 2011, chemists announced that they had made a molecular sized version of the truncated octahedron. The tiny, hollow structure serves as a cage, capable of enclosing a wide variety of ions and molecules in a stable configuration. It also helps in the synthesis of substances that wouldn't otherwise form.

 

Michael Ward of New York University, and colleagues, built their cage, which has eight hexagonal and six square faces, by combining two types of carefully-designed molecular "tiles", one made of chemical groups known as guanidiniums, the other ringed by sulfonate groups. These were assembled into the truncated octahedron with the formation 72 hydrogen bonds.¹

 

Reference

1. Liu, Y., Hu, C., Comotti, A, and Ward, M. "Supramolecular Archimedean Cages Assembled with 72 Hydrogen Bonds", Science, 333 (no. 6041), 436–440 (2011).