## Archimedean Solid

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. 139-140).

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

1. and every member of is at least 3,

2. , with equality in the case of a plane Tessellation, and

3. for every Odd Number , contains a subsequence (, , ).

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. 116-126; Catalan 1865, pp. 25-32; Coxeter 1940, p. 394; Coxeter et al. 1954; Lines 1965, pp. 202-203; 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

where is the sum of face-angles at a vertex and is the number of vertices (Steinitz and Rademacher 1934, Ball and Coxeter 1987).

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 1 Great Rhombicosidodecahedron Great Rhombicuboctahedron Icosidodecahedron Small Rhombicosidodecahedron Small Rhombicuboctahedron Snub Cube * * * Snub Dodecahedron * * * Truncated Cube Truncated Dodecahedron Truncated Icosahedron Truncated Octahedron Truncated Tetrahedron
*The complicated analytic expressions for the Circumradii of these solids are given in the entries for the Snub Cube and Snub Dodecahedron.

 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, Kepler-Poinsot Solid, Platonic Solid, Semiregular Polyhedron, Uniform Polyhedron

References

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