The Archimedean solids are convex Polyhedra which have a similar arrangement of nonintersecting regular plane Convex Polygons of two or more different types about each Vertex with all sides the same length. The Archimedean solids are distinguished from the Prisms, Antiprisms, and Elongated Square Gyrobicupola by their symmetry group: the Archimedean solids have a spherical symmetry, while the others have ``dihedral'' symmetry. The Archimedean solids are sometimes also referred to as the Semiregular Polyhedra.
Pugh (1976, p. 25) points out the Archimedean solids are all capable of being circumscribed by a regular Tetrahedron so that four of their faces lie on the faces of that Tetrahedron. A method of constructing the Archimedean solids using a method known as ``expansion'' has been enumerated by Stott (Stott 1910; Ball and Coxeter 1987, pp. 139140).
Let the cyclic sequence represent the degrees of the faces surrounding a vertex (i.e., is a list of the number of sides of all polygons surrounding any vertex). Then the definition of an Archimedean solid requires that the sequence must be the same for each vertex to within Rotation and Reflection. Walsh (1972) demonstrates that represents the degrees of the faces surrounding each vertex of a semiregular convex polyhedron or Tessellation of the plane Iff
Condition (1) simply says that the figure consists of two or more polygons, each having at least three sides. Condition (2) requires that the sum of interior angles at a vertex must be equal to a full rotation for the figure to lie in the plane, and less than a full rotation for a solid figure to be convex.
The usual way of enumerating the semiregular polyhedra is to eliminate solutions of conditions (1) and (2) using several classes of arguments and then prove that the solutions left are, in fact, semiregular (Kepler 1864, pp. 116126; Catalan 1865, pp. 2532; Coxeter 1940, p. 394; Coxeter et al. 1954; Lines 1965, pp. 202203; Walsh 1972). The following table gives all possible regular and semiregular polyhedra and tessellations. In the table, `P' denotes Platonic Solid, `M' denotes a Prism or Antiprism, `A' denotes an Archimedean solid, and `T' a plane tessellation.
Figure  Solid  Schläfli Symbol  
(3, 3, 3)  P  Tetrahedron  
(3, 4, 4)  M  Triangular Prism  t 
(3, 6, 6)  A  Truncated Tetrahedron  t 
(3, 8, 8)  A  Truncated Cube  t 
(3, 10, 10)  A  Truncated Dodecahedron  t 
(3, 12, 12)  T  (Plane Tessellation)  t 
(4, 4, )  M  gonal Prism  t 
(4, 4, 4)  P  Cube  
(4, 6, 6)  A  Truncated Octahedron  t 
(4, 6, 8)  A  Great Rhombicuboctahedron  t 
(4, 6, 10)  A  Great Rhombicosidodecahedron  t 
(4, 6, 12)  T  (Plane Tessellation)  t 
(4, 8, 8)  T  (Plane Tessellation)  t 
(5, 5, 5)  P  Dodecahedron  
(5, 6, 6)  A  Truncated Icosahedron  t 
(6, 6, 6)  T  (Plane Tessellation)  
(3, 3, 3, )  M  gonal Antiprism  s 
(3, 3, 3, 3)  P  Octahedron  
(3, 4, 3, 4)  A  Cuboctahedron  
(3, 5, 3, 5)  A  Icosidodecahedron  
(3, 6, 3, 6)  T  (Plane Tessellation)  
(3, 4, 4, 4)  A  Small Rhombicuboctahedron  r 
(3, 4, 5, 4)  A  Small Rhombicosidodecahedron  r 
(3, 4, 6, 4)  T  (Plane Tessellation)  r 
(4, 4, 4, 4)  T  (Plane Tessellation)  
(3, 3, 3, 3, 3)  P  Icosahedron  
(3, 3, 3, 3, 4)  A  Snub Cube  s 
(3, 3, 3, 3, 5)  A  Snub Dodecahedron  s 
(3, 3, 3, 3, 6)  T  (Plane Tessellation)  s 
(3, 3, 3, 4, 4)  T  (Plane Tessellation)   
(3, 3, 4, 3, 4)  T  (Plane Tessellation)  s 
(3, 3, 3, 3, 3)  T  (Plane Tessellation) 
As shown in the above table, there are exactly 13 Archimedean solids (Walsh 1972, Ball and Coxeter 1987). They are called
the Cuboctahedron, Great Rhombicosidodecahedron,
Great Rhombicuboctahedron, Icosidodecahedron, Small
Rhombicosidodecahedron, Small Rhombicuboctahedron, Snub Cube, Snub Dodecahedron, Truncated
Cube, Truncated Dodecahedron, Truncated Icosahedron (soccer ball), Truncated Octahedron, and
Truncated Tetrahedron. The Archimedean solids satisfy
Here are the Archimedean solids shown in alphabetical order (left to right, then continuing to the next row).













The following table lists the symbol and number of faces of each type for the Archimedean solids (Wenninger 1989, p. 9).
Solid  Schläfli Symbol  Wythoff Symbol  C&R Symbol 
Cuboctahedron  2 3 4  
Great Rhombicosidodecahedron  t  2 3 5  
Great Rhombicuboctahedron  t  2 3 4  
Icosidodecahedron  2 3 5  
Small Rhombicosidodecahedron  t  3 5 2  3.4.5.4 
Small Rhombicuboctahedron  r  3 4 2  
Snub Cube  s  2 3 4  
Snub Dodecahedron  s  2 3 5  
Truncated Cube  t  2 3 4  
Truncated Dodecahedron  t  2 3 5  
Truncated Icosahedron  t  2 5 3  
Truncated Octahedron  t  2 4 3  
Truncated Tetrahedron  t  2 3 3 
Solid  
Cuboctahedron  12  24  8  6  
Great Rhombicosidodecahedron  120  180  30  20  12  
Great Rhombicuboctahedron  48  72  12  8  6  
Icosidodecahedron  30  60  20  12  
Small Rhombicosidodecahedron  60  120  20  30  12  
Small Rhombicuboctahedron  24  48  8  18  
Snub Cube  24  60  32  6  
Snub Dodecahedron  60  150  80  12  
Truncated Cube  24  36  8  6  
Truncated Dodecahedron  60  90  20  12  
Truncated Icosahedron  60  90  12  20  
Truncated Octahedron  24  36  6  8  
Truncated Tetrahedron  12  18  4  4 
Let be the Inradius, the Midradius, and the Circumradius. The following tables give the analytic and numerical values of , , and for the Archimedean solids with Edges of unit length.
Solid  
Cuboctahedron  0.75  0.86603  1 
Great Rhombicosidodecahedron  3.73665  3.76938  3.80239 
Great Rhombicuboctahedron  2.20974  2.26303  2.31761 
Icosidodecahedron  1.46353  1.53884  1.61803 
Small Rhombicosidodecahedron  2.12099  2.17625  2.23295 
Small Rhombicuboctahedron  1.22026  1.30656  1.39897 
Snub Cube  1.15763  1.24719  1.34371 
Snub Dodecahedron  2.03969  2.09688  2.15583 
Truncated Cube  1.63828  1.70711  1.77882 
Truncated Dodecahedron  2.88526  2.92705  2.96945 
Truncated Icosahedron  2.37713  2.42705  2.47802 
Truncated Octahedron  1.42302  1.5  1.58114 
Truncated Tetrahedron  0.95940  1.06066  1.17260 
The Duals of the Archimedean solids, sometimes called the Catalan Solids, are given in the following table.
Here are the Archimedean Duals (Holden 1971, Pearce 1978) displayed in alphabetical order (left to right, then continuing to the next row).
Here are the Archimedean solids paired with their Duals.
The Archimedean solids and their Duals are all Canonical Polyhedra.
See also Archimedean Solid Stellation, Catalan Solid, Deltahedron, Johnson Solid, KeplerPoinsot Solid, Platonic Solid, Semiregular Polyhedron, Uniform Polyhedron
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 136, 1987.
Behnke, H.; Bachman, F.; Fladt, K.; and Kunle, H. (Eds.). Fundamentals of Mathematics, Vol. 2. Cambridge, MA: MIT Press, pp. 269286, 1974.
Catalan, E. ``Mémoire sur la Théorie des Polyèdres.'' J. l'École Polytechnique (Paris) 41, 171, 1865.
Coxeter, H. S. M. ``The Pure Archimedean Polytopes in Six and Seven Dimensions.'' Proc. Cambridge Phil. Soc. 24, 19, 1928.
Coxeter, H. S. M. ``Regular and SemiRegular Polytopes I.'' Math. Z. 46, 380407, 1940.
Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973.
Coxeter, H. S. M.; LonguetHiggins, M. S.; and Miller, J. C. P. ``Uniform Polyhedra.'' Phil. Trans. Roy. Soc. London Ser. A 246, 401450, 1954.
Critchlow, K. Order in Space: A Design Source Book. New York: Viking Press, 1970.
Cromwell, P. R. Polyhedra. New York: Cambridge University Press, pp. 7986, 1997.
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., 1989.
Holden, A. Shapes, Space, and Symmetry. New York: Dover, p. 54, 1991.
Kepler, J. ``Harmonice Mundi.'' Opera Omnia, Vol. 5. Frankfurt, pp. 75334, 1864.
Kraitchik, M. Mathematical Recreations. New York: W. W. Norton, pp. 199207, 1942.
Pearce, P. Structure in Nature is a Strategy for Design. Cambridge, MA: MIT Press, pp. 3435, 1978.
Pugh, A. Polyhedra: A Visual Approach. Berkeley: University of California Press, p. 25, 1976.
Rawles, B. A. ``Platonic and Archimedean SolidsFaces, Edges, Areas, Vertices, Angles, Volumes, Sphere Ratios.'' http://www.intent.com/sg/polyhedra.html.
Rorres, C. ``Archimedean Solids: Pappus.'' http://www.mcs.drexel.edu/~crorres/Archimedes/Solids/Pappus.html.
Steinitz, E. and Rademacher, H. Vorlesungen über die Theorie der Polyheder. Berlin, p. 11, 1934.
Stott, A. B. Verhandelingen der Koninklijke Akad. Wetenschappen, Amsterdam 11, 1910.
Walsh, T. R. S. ``Characterizing the Vertex Neighbourhoods of SemiRegular Polyhedra.'' Geometriae Dedicata 1, 117123, 1972.
Wenninger, M. J. Polyhedron Models. New York: Cambridge University Press, 1989.
© 19969 Eric W. Weisstein