The 13 Archimedean solids are the convex polyhedra that have a similar arrangement of nonintersecting regular convex
polygons of two or more different types arranged in the same way about each vertex with all sides the same
length (Cromwell 1997, pp. 91-92).
The Archimedean solids are distinguished by having very high symmetry, thus excluding solids belonging to a dihedral group
of symmetries (e.g., the two infinite families of regular prisms and antiprisms),
as well as the elongated
square gyrobicupola (because that surface's symmetry-breaking twist allows vertices
"near the equator" and those "in the polar regions" to be distinguished;
Cromwell 1997, p. 92). The Archimedean solids are sometimes also referred to
as the semiregular polyhedra.
The Archimedean solids are illustrated below in alphabetical order (left to right, then continuing to the next row).
The following table lists the uniform, Schläfli, Wythoff, and Cundy and Rollett symbols for the Archimedean solids (Wenninger 1989, p. 9).
The following table gives the number of vertices  , edges  , and faces  , together with the number of  -gonal faces  for the Archimedean solids. The sorted numbers
of edges are 18, 24, 36, 36, 48, 60, 60, 72, 90, 90, 120, 150, 180 (Sloane's A092536), numbers of faces are 8, 14, 14, 14, 26, 26, 32, 32,
32, 38, 62, 62, 92 (Sloane's A092537), and numbers of vertices are 12, 12, 24, 24, 24, 24,
30, 48, 60, 60, 60, 60, 120 (Sloane's A092538).
Seven of the 13 Archimedean solids (the cuboctahedron, icosidodecahedron, truncated cube, truncated
dodecahedron, truncated
octahedron, truncated
icosahedron, and truncated
tetrahedron) can be obtained by truncation
of a Platonic solid. The three
truncation series producing these seven Archimedean solids are illustrated above.
Two additional solids (the small rhombicosidodecahedron and small
rhombicuboctahedron) can be obtained by expansion
of a Platonic solid, and two
further solids (the great
rhombicosidodecahedron and great
rhombicuboctahedron) can be obtained by expansion
of one of the previous 9 Archimedean solids (Stott 1910; Ball and Coxeter 1987, pp. 139-140).
It is sometimes stated (e.g., Wells 1991, p. 8) that these four solids can be
obtained by truncation of other solids. The confusion originated with Kepler himself,
who used the terms "truncated icosidodecahedron" and "truncated cuboctahedron"
for the great rhombicosidodecahedron
and great rhombicuboctahedron,
respectively. However, truncation alone is not capable of producing these solids,
but must be combined with distorting to turn the resulting rectangles into squares
(Ball and Coxeter 1987, pp. 137-138; Cromwell 1997, p. 81).
The remaining two solids, the snub cube and snub dodecahedron, can
be obtained by moving the faces of a cube
and dodecahedron outward while
giving each face a twist. The resulting spaces are then filled with ribbons of equilateral triangles (Wells
1991, p. 8).
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.
The Archimedean solids satisfy
 |
(1)
|
where  is the sum of face-angles at a vertex
and  is the number of vertices (Steinitz
and Rademacher 1934, Ball and Coxeter 1987).
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.
 | fg. | 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 | 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 | tessellation | t | | (4,
8, 8) | T | tessellation | t | | (5,
5, 5) | P | dodecahedron |  | | (5,
6, 6) | A | truncated icosahedron | t | | (6, 6, 6) | T | 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 | tessellation |  | | (3, 4, 4, 4) | A | small rhombicuboctahedron | r | | (3,
4, 5, 4) | A | small rhombicosidodecahedron | r | | (3, 4, 6, 4) | T | tessellation | r | | (4,
4, 4, 4) | T | 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 | tessellation | s | | (3, 3, 3, 4, 4) | T | tessellation | -- | | (3, 3, 4, 3, 4) | T | tessellation | s | | (3,
3, 3, 3, 3) | T | 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.
Let  be the inradius
of the dual polyhedron (corresponding to the insphere,
which touches the faces of the dual solid),  be the
midradius of both the polyhedron
and its dual (corresponding to the midsphere,
which touches the edges of both the polyhedron and its duals),  the circumradius (corresponding to the circumsphere of the solid which touches the vertices of the
solid) of the Archimedean solid, and  the edge length
of the solid Since the circumsphere
and insphere are dual to each other,
they obey the relationship
 |
(2)
|
(Cundy and Rollett 1989, Table II following p. 144). In addition,
The following tables give the analytic and numerical values of  ,  , and  for the Archimedean
solids with polyhedron edges
of unit length (Coxeter et al. 1954; Cundy and Rollett 1989, Table II following
p. 144). Hume (1986) gives approximate expressions for the dihedral angles of the Archimedean solid (and exact expressions
for their duals).
*The complicated analytic expressions for the circumradii of these solids are given in the entries for the snub
cube and snub dodecahedron.
The Archimedean solids and their duals are all canonical polyhedra.
Since the Archimedean solids are convex, the convex
hull of each Archimedean solid is the solid itself.
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