Archimedean Solid

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 above.

Nets of the Archimedean solids are illustrated above.

The following table lists the uniform, Schläfli, Wythoff, and Cundy and Rollett symbols for the Archimedean solids (Wenninger 1989, p. 9).

	solid	uniform polyhedron	Schläfli symbol	Wythoff symbol	Cundy and Rollett symbol
1	cuboctahedron
2	great rhombicosidodecahedron		t		4.6.10
3	great rhombicuboctahedron		t		4.6.8
4	icosidodecahedron
5	small rhombicosidodecahedron		r		3.4.5.4
6	small rhombicuboctahedron		r
7	snub cube		s
8	snub dodecahedron		s
9	truncated cube		t
10	truncated dodecahedron		t
11	truncated icosahedron		t
12	truncated octahedron		t
13	truncated tetrahedron		t

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 (OEIS A092536), numbers of faces are 8, 14, 14, 14, 26, 26, 32, 32, 32, 38, 62, 62, 92 (OEIS A092537), and numbers of vertices are 12, 12, 24, 24, 24, 24, 30, 48, 60, 60, 60, 60, 120 (OEIS A092538).

	solid
1	cuboctahedron	12	24	14	8	6
2	great rhombicosidodecahedron	120	180	62		30		20		12
3	great rhombicuboctahedron	48	72	26		12		8	6
4	icosidodecahedron	30	60	32	20		12
5	small rhombicosidodecahedron	60	120	62	20	30	12
6	small rhombicuboctahedron	24	48	26	8	18
7	snub cube	24	60	38	32	6
8	snub dodecahedron	60	150	92	80		12
9	truncated cube	24	36	14	8				6
10	truncated dodecahedron	60	90	32	20					12
11	truncated icosahedron	60	90	32			12	20
12	truncated octahedron	24	36	14		6		8
13	truncated tetrahedron	12	18	8	4			4

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, 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,

			(3)
			(4)
			(5)
			(6)
			(7)
			(8)

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).

	solid
1	cuboctahedron			1
2	great rhombicosidodecahedron
3	great rhombicuboctahedron
4	icosidodecahedron
5	small rhombicosidodecahedron
6	small rhombicuboctahedron
7	snub cube	*	*	*
8	snub dodecahedron	*	*	*
9	truncated cube
10	truncated dodecahedron
11	truncated icosahedron
12	truncated octahedron
13	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
1	cuboctahedron	0.75	0.86603	1
2	great rhombicosidodecahedron	3.73665	3.76938	3.80239
3	great rhombicuboctahedron	2.20974	2.26303	2.31761
4	icosidodecahedron	1.46353	1.53884	1.61803
5	small rhombicosidodecahedron	2.12099	2.17625	2.23295
6	small rhombicuboctahedron	1.22026	1.30656	1.39897
7	snub cube	1.15763	1.24719	1.34371
8	snub dodecahedron	2.03969	2.09688	2.15583
9	truncated cube	1.63828	1.70711	1.77882
10	truncated dodecahedron	2.88526	2.92705	2.96945
11	truncated icosahedron	2.37713	2.42705	2.47802
12	truncated octahedron	1.42302	1.5	1.58114
13	truncated tetrahedron	0.95940	1.06066	1.17260

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.