Archimedean Solids |
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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).
| Name | Solid | Transparent | Faces | Edges | Vertices | Vertex configuration | Symmetry group | |
|---|---|---|---|---|---|---|---|---|
| truncated tetrahedron |
 |
 (Video) |
8 | 4 triangles 4 hexagons |
18 | 12 | 3.6.6 | Td |
| cuboctahedron |
 |
 (Video) |
14 | 8
triangles 6 squares |
24 | 12 | 3.4.3.4 | Oh |
|
truncated cube or truncated hexahedron |
 |
 (Video) |
14 | 8 triangles 6 octagons |
36 | 24 | 3.8.8 | Oh |
| truncated octahedron |
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 (Video) |
14 | 6 squares 8 hexagons |
36 | 24 | 4.6.6 | Oh |
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rhombicuboctahedron or small rhombicuboctahedron |
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 (Video) |
26 | 8 triangles 18 squares |
48 | 24 | 3.4.4.4 | Oh |
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truncated cuboctahedron or great rhombicuboctahedron |
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 (Video) |
26 | 12 squares 8 hexagons 6 octagons |
72 | 48 | 4.6.8 | Oh |
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snub cube or snub cuboctahedron (2 chiral forms) |
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 (Video)  (Video) |
38 | 32 triangles 6 squares |
60 | 24 | 3.3.3.3.4 | O |
| icosidodecahedron |
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 (Video) |
32 | 20 triangles 12 pentagons |
60 | 30 | 3.5.3.5 | Ih |
| truncated dodecahedron |
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 (Video) |
32 | 20 triangles 12 decagons |
90 | 60 | 3.10.10 | Ih |
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truncated icosahedron or buckyball or football/soccer ball |
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 (Video) |
32 | 12 pentagons 20 hexagons |
90 | 60 | 5.6.6 | Ih |
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rhombicosidodecahedron or small rhombicosidodecahedron |
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 (Video) |
62 | 20 triangles 30 squares 12 pentagons |
120 | 60 | 3.4.5.4 | Ih |
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truncated icosidodecahedron or great rhombicosidodecahedron |
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 (Video) |
62 | 30 squares 20 hexagons 12 decagons |
180 | 120 | 4.6.10 | Ih |
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snub dodecahedron or snub icosidodecahedron (2 chiral forms) |
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 (Video)  (Video) |
92 | 80 triangles 12 pentagons |
150 | 60 | 3.3.3.3.5 | I |
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The following table lists the symbols for the Archimedean solids (Wenninger 1989, p. 9).
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solid | uniform polyhedron | Schläfli symbol | Wythoff symbol | Cundy and Rollett symbol |
| 1 | cuboctahedron |
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| 2 | great rhombicosidodecahedron |
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t |
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4.6.10 |
| 3 | great rhombicuboctahedron |
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t |
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4.6.8 |
| 4 | icosidodecahedron |
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| 5 | small rhombicosidodecahedron |
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r |
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3.4.5.4 |
| 6 | small rhombicuboctahedron |
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r |
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| 7 | snub cube |
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s |
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| 8 | snub dodecahedron |
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s |
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| 9 | truncated cube |
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t |
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| 10 | truncated dodecahedron |
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t |
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| 11 | truncated icosahedron |
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t |
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| 12 | truncated octahedron |
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t |
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| 13 | truncated tetrahedron |
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t |
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The following table gives the number of vertices
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edges
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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).
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Solid |
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| 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 |