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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 $S=(p_1, p_2, \ldots, p_q)$ represent the degrees of the faces surrounding a vertex (i.e., $S$ 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 $S$ represents the degrees of the faces surrounding each vertex of a semiregular convex polyhedron or Tessellation of the plane Iff

1. $q\geq 3$ and every member of $S$ is at least 3,

2. $\sum_{i=1}^q {1\over p_i}\geq {\textstyle{1\over 2}}q-1$, with equality in the case of a plane Tessellation, and

3. for every Odd Number $p\in S$, $S$ contains a subsequence ($b$, $p$, $b$).


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.


$S$ Figure Solid Schläfli Symbol
(3, 3, 3) P Tetrahedron $\{3,3\}$
(3, 4, 4) M Triangular Prism t$\{2,3\}$
(3, 6, 6) A Truncated Tetrahedron t$\{3,3\}$
(3, 8, 8) A Truncated Cube t$\{4,3\}$
(3, 10, 10) A Truncated Dodecahedron t$\{5,3\}$
(3, 12, 12) T (Plane Tessellation) t$\{6,3\}$
(4, 4, $n$) M $n$-gonal Prism t$\{2,n\}$
(4, 4, 4) P Cube $\{4,3\}$
(4, 6, 6) A Truncated Octahedron t$\{3,4\}$
(4, 6, 8) A Great Rhombicuboctahedron t $\left\{{3\atop 4}\right\}$
(4, 6, 10) A Great Rhombicosidodecahedron t $\left\{{3\atop 5}\right\}$
(4, 6, 12) T (Plane Tessellation) t $\left\{{3\atop 6}\right\}$
(4, 8, 8) T (Plane Tessellation) t$\{4,4\}$
(5, 5, 5) P Dodecahedron $\{5,3\}$
(5, 6, 6) A Truncated Icosahedron t$\{3,5\}$
(6, 6, 6) T (Plane Tessellation) $\{6,3\}$
(3, 3, 3, $n$) M $n$-gonal Antiprism s $\left\{{2\atop n}\right\}$
(3, 3, 3, 3) P Octahedron $\{3,4\}$
(3, 4, 3, 4) A Cuboctahedron $\left\{{3\atop 4}\right\}$
(3, 5, 3, 5) A Icosidodecahedron $\left\{{3\atop 5}\right\}$
(3, 6, 3, 6) T (Plane Tessellation) $\left\{{3\atop 6}\right\}$
(3, 4, 4, 4) A Small Rhombicuboctahedron r $\left\{{3\atop 4}\right\}$
(3, 4, 5, 4) A Small Rhombicosidodecahedron r $\left\{{3\atop 5}\right\}$
(3, 4, 6, 4) T (Plane Tessellation) r $\left\{{3\atop 6}\right\}$
(4, 4, 4, 4) T (Plane Tessellation) $\{4,4\}$
(3, 3, 3, 3, 3) P Icosahedron $\{3,5\}$
(3, 3, 3, 3, 4) A Snub Cube s $\left\{{3\atop 4}\right\}$
(3, 3, 3, 3, 5) A Snub Dodecahedron s $\left\{{3\atop 5}\right\}$
(3, 3, 3, 3, 6) T (Plane Tessellation) s $\left\{{3\atop 6}\right\}$
(3, 3, 3, 4, 4) T (Plane Tessellation) --
(3, 3, 4, 3, 4) T (Plane Tessellation) s $\left\{{4\atop 4}\right\}$
(3, 3, 3, 3, 3) T (Plane Tessellation) $\{3,6\}$


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

\begin{displaymath} (2\pi-\sigma)V=4\pi, \end{displaymath}

where $\sigma$ is the sum of face-angles at a vertex and $V$ 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).

     

\begin{figure}\BoxedEPSF{U07_net.epsf scaled 225}\end{figure}

\begin{figure}\BoxedEPSF{U28_net.epsf scaled 225}\end{figure}

\begin{figure}\BoxedEPSF{U11_net.epsf scaled 350}\end{figure}

\begin{figure}\BoxedEPSF{U24_net.epsf scaled 300}\end{figure}

\begin{figure}\BoxedEPSF{U27_net.epsf scaled 225}\end{figure}

\begin{figure}\BoxedEPSF{U10_net.epsf scaled 275}\end{figure}

\begin{figure}\BoxedEPSF{U12_net.epsf scaled 225}\end{figure}

\begin{figure}\BoxedEPSF{U29_net.epsf scaled 300}\end{figure}

\begin{figure}\BoxedEPSF{U09_net.epsf scaled 300}\end{figure}

\begin{figure}\BoxedEPSF{U26_net.epsf scaled 225}\end{figure}

\begin{figure}\BoxedEPSF{U25_net.epsf scaled 225}\end{figure}

\begin{figure}\BoxedEPSF{U08_net.epsf scaled 225}\end{figure}

\begin{figure}\BoxedEPSF{U02_net.epsf scaled 400}\end{figure}

     


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 $\left\{{3\atop 4}\right\}$ 2 $\vert$ 3 4 $(3.4)^2$
Great Rhombicosidodecahedron t $\left\{{3\atop 5}\right\}$ 2 3 5 $\vert$  
Great Rhombicuboctahedron t $\left\{{3\atop 4}\right\}$ 2 3 4 $\vert$  
Icosidodecahedron $\left\{{3\atop 5}\right\}$ 2 $\vert$ 3 5 $(3.5)^2$
Small Rhombicosidodecahedron t $\left\{{3\atop 5}\right\}$ 3 5 $\vert$ 2 3.4.5.4
Small Rhombicuboctahedron r $\left\{{3\atop 4}\right\}$ 3 4 $\vert$ 2 $3.4^3$
Snub Cube s $\left\{{3\atop 4}\right\}$ $\vert$ 2 3 4 $3^4.4$
Snub Dodecahedron s $\left\{{3\atop 5}\right\}$ $\vert$ 2 3 5 $3^4.5$
Truncated Cube t$\{4,3\}$ 2 3 $\vert$ 4 $3.8^2$
Truncated Dodecahedron t$\{5,3\}$ 2 3 $\vert$ 5 $3.10^2$
Truncated Icosahedron t$\{3,5\}$ 2 5 $\vert$ 3 $5.6^2$
Truncated Octahedron t$\{3,4\}$ 2 4 $\vert$ 3 $4.6^2$
Truncated Tetrahedron t$\{3,3\}$ 2 3 $\vert$ 3 $3.6^2$

Solid $v$ $e$ $f_3$ $f_4$ $f_5$ $f_6$ $f_8$ $f_{10}$
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 $r$ be the Inradius, $\rho$ the Midradius, and $R$ the Circumradius. The following tables give the analytic and numerical values of $r$, $\rho$, and $R$ for the Archimedean solids with Edges of unit length.

Solid $r$ $\rho$ $R$
Cuboctahedron ${\textstyle{3\over 4}}$ ${\textstyle{1\over 2}}\sqrt{3}$ 1
Great Rhombicosidodecahedron ${\textstyle{1\over 241}}(105+6\sqrt{5}\,)\sqrt{31+12\sqrt{5}}$ ${\textstyle{1\over 2}}\sqrt{30+12\sqrt{5}}$ ${\textstyle{1\over 2}}\sqrt{31+12\sqrt{5}}$
Great Rhombicuboctahedron ${\textstyle{3\over 97}}(14+\sqrt{2}\,)\sqrt{13+6\sqrt{2}}$ ${\textstyle{1\over 2}}\sqrt{12+6\sqrt{2}}$ ${\textstyle{1\over 2}}\sqrt{13+6\sqrt{2}}$
Icosidodecahedron ${\textstyle{1\over 8}}(5+3\sqrt{5}\,)$ ${\textstyle{1\over 2}}\sqrt{5+2\sqrt{5}}$ ${\textstyle{1\over 2}}(1+\sqrt{5}\,)$
Small Rhombicosidodecahedron ${\textstyle{1\over 41}}(15+2\sqrt{5}\,)\sqrt{11+4\sqrt{5}}$ ${\textstyle{1\over 2}}\sqrt{10+4\sqrt{5}}$ ${\textstyle{1\over 2}}\sqrt{11+4\sqrt{5}}$
Small Rhombicuboctahedron ${\textstyle{1\over 17}}(6+\sqrt{2}\,)\sqrt{5+2\sqrt{2}}$ ${\textstyle{1\over 2}}\sqrt{4+2\sqrt{2}}$ ${\textstyle{1\over 2}}\sqrt{5+2\sqrt{2}}$
Snub Cube * * *
Snub Dodecahedron * * *
Truncated Cube ${\textstyle{1\over 17}}(5+2\sqrt{2}\,)\sqrt{7+4\sqrt{2}}$ ${\textstyle{1\over 2}}(2+\sqrt{2}\,)$ ${\textstyle{1\over 2}}\sqrt{7+4\sqrt{2}}$
Truncated Dodecahedron ${\textstyle{5\over 488}}(17\sqrt{2}+3\sqrt{10}\,)\sqrt{37+15\sqrt{5}}$ ${\textstyle{1\over 4}}(5+3\sqrt{5}\,)$ ${\textstyle{1\over 4}}\sqrt{74+30\sqrt{5}}$
Truncated Icosahedron ${\textstyle{9\over 872}}(21+\sqrt{5}\,)\sqrt{58+18\sqrt{5}}$ ${\textstyle{3\over 4}}(1+\sqrt{5}\,)$ ${\textstyle{1\over 4}}\sqrt{58+18\sqrt{5}}$
Truncated Octahedron ${\textstyle{9\over 20}}\sqrt{10}$ ${\textstyle{3\over 2}}$ ${\textstyle{1\over 2}}\sqrt{10}$
Truncated Tetrahedron ${\textstyle{9\over 44}}\sqrt{22}$ ${\textstyle{3\over 4}}\sqrt{2}$ ${\textstyle{1\over 4}}\sqrt{22}$
*The complicated analytic expressions for the Circumradii of these solids are given in the entries for the Snub Cube and Snub Dodecahedron.


Solid $r$ $\rho$ $R$
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.


Archimedean Solid Dual
Cuboctahedron Rhombic Dodecahedron
Great Rhombicosidodecahedron Disdyakis Triacontahedron
Great Rhombicuboctahedron Disdyakis Dodecahedron
Icosidodecahedron Rhombic Triacontahedron
Small Rhombicosidodecahedron Deltoidal Hexecontahedron
Small Rhombicuboctahedron Deltoidal Icositetrahedron
Snub Dodecahedron (laevo) Pentagonal Hexecontahedron (dextro)
Snub Cube (laevo) Pentagonal Icositetrahedron (dextro)
Truncated Cube Small Triakis Octahedron
Truncated Dodecahedron Triakis Icosahedron
Truncated Icosahedron Pentakis Dodecahedron
Truncated Octahedron Tetrakis Hexahedron
Truncated Tetrahedron Triakis Tetrahedron

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.

\begin{figure}\begin{center}\BoxedEPSF{DualsArchimedeanSolids1.epsf scaled 1250}\end{center}\end{figure}

\begin{figure}\begin{center}\BoxedEPSF{DualsArchimedeanSolids2.epsf scaled 1250}\end{center}\end{figure}


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

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.


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© 1996-9 Eric W. Weisstein
1999-05-25