Archimedean Solid

The Archimedean solids are convex Polyhedra which have a similar arrangement of nonintersectingregular plane Convex Polygons of two or more different types about each Vertex with all sides the same length. The Archimedean solids are distinguished from thePrisms, 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 aresometimes 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 sothat four of their faces lie on the faces of that Tetrahedron. A method of constructing the Archimedean solids using amethod 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 thedefinition of an Archimedean solid requires that the sequence must be the same for each vertex to within Rotationand Reflection. Walsh (1972) demonstrates that represents the degrees of the faces surrounding each vertex of asemiregular 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 severalclasses of arguments and then prove that the solutions left are, in fact, semiregular (Kepler 1864, pp. 116-126; Catalan1865, 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 calledthe Cuboctahedron, Great Rhombicosidodecahedron,Great Rhombicuboctahedron, Icosidodecahedron, SmallRhombicosidodecahedron, Small Rhombicuboctahedron, Snub Cube, Snub Dodecahedron, TruncatedCube, Truncated Dodecahedron, Truncated Icosahedron (soccer ball), Truncated Octahedron, andTruncated 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 tablesgive 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

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 toright, 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.

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. 269-286, 1974.

Catalan, E. ``Mémoire sur la Théorie des Polyèdres.'' J. l'École Polytechnique (Paris) 41, 1-71, 1865.

Coxeter, H. S. M. ``The Pure Archimedean Polytopes in Six and Seven Dimensions.'' Proc. Cambridge Phil. Soc. 24, 1-9, 1928.

Coxeter, H. S. M. ``Regular and Semi-Regular Polytopes I.'' Math. Z. 46, 380-407, 1940.

Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973.

Coxeter, H. S. M.; Longuet-Higgins, M. S.; and Miller, J. C. P. ``Uniform Polyhedra.'' Phil. Trans. Roy. Soc. London Ser. A 246, 401-450, 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. 79-86, 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. 75-334, 1864.

Kraitchik, M. Mathematical Recreations. New York: W. W. Norton, pp. 199-207, 1942.

Pearce, P. Structure in Nature is a Strategy for Design. Cambridge, MA: MIT Press, pp. 34-35, 1978.

Pugh, A. Polyhedra: A Visual Approach. Berkeley: University of California Press, p. 25, 1976.

Rawles, B. A. ``Platonic and Archimedean Solids--Faces, 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 Semi-Regular Polyhedra.'' Geometriae Dedicata 1, 117-123, 1972.

Wenninger, M. J. Polyhedron Models. New York: Cambridge University Press, 1989.

• Numeric Function
• Numerology
• NURBS Curve
• NURBS Surface
• Nyquist Frequency
• Nyquist Sampling
• Néron-Severi Group
• Nöther
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• Oblateness
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• Oblate Spheroidal Coordinates
• Oblate Spheroidal Wave Function
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