# Archimedean solid

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**In geometry an Archimedean solid is a highly symmetric, semi-regular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices. They are distinct from the Platonic solids, which are composed of only one type of polygon meeting in identical vertices, and from the Johnson solids, whose regular polygonal faces do not meet in identical vertices. The symmetry of the Archimedean solids excludes the members of the dihedral group, the prisms and antiprisms. **
The Archimedean solids can all be made via Wythoff constructions from the Platonic solids with tetrahedral, octahedral and icosahedral symmetry. See Convex uniform polyhedron.

## Origin of name

The Archimedean solids take their name from Archimedes, who discussed them in a now-lost work. During the Renaissance, artists and mathematicians valued *pure forms* and rediscovered all of these forms. This search was completed around 1620 by Johannes Kepler, who defined prisms, antiprisms, and the non-convex solids known as the Kepler-Poinsot polyhedra.

## Classification

There are 13 Archimedean solids (15 if the mirror images of two enantiomorphs, see below, are counted separately). Here the *vertex configuration* refers to the type of regular polygons that meet at any given vertex. For example, a vertex configuration of (4,6,8) means that a square, hexagon, and octagon meet at a vertex (with the order taken to be clockwise around the vertex).

The number of vertices is 720° divided by the vertex angle defect.

Name (Vertex configuration) | Transparent | Solid | Net | Faces | Faces (By type) | Edges | Vertices | Symmetry group |
---|---|---|---|---|---|---|---|---|

truncated tetrahedron (3.6.6) | (Animation) | 8 | 4 triangles 4 hexagons | 18 | 12 | T_{d}
| ||

cuboctahedron (3.4.3.4) | (Animation) | 14 | 8 triangles 6 squares | 24 | 12 | O_{h}
| ||

truncated cube or truncated hexahedron (3.8.8) | (Animation) | 14 | 8 triangles 6 octagons | 36 | 24 | O_{h}
| ||

truncated octahedron (4.6.6) | (Animation) | 14 | 6 squares 8 hexagons | 36 | 24 | O_{h}
| ||

rhombicuboctahedron or small rhombicuboctahedron (3.4.4.4 ) | (Animation) | 26 | 8 triangles 18 squares | 48 | 24 | O_{h}
| ||

truncated cuboctahedron or great rhombicuboctahedron (4.6.8) | (Animation) | 26 | 12 8 hexagons 6 octagons | 72 | 48 | O_{h}
| ||

snub cube or snub hexahedron or snub cuboctahedron (2 chiral forms) (3.3.3.3.4) | (Animation) (Animation) | 38 | 32 triangles 6 squares | 60 | 24 | O | ||

icosidodecahedron (3.5.3.5) | (Animation) | 32 | 20 triangles 12 pentagons | 60 | 30 | I_{h}
| ||

truncated dodecahedron (3.10.10) | (Animation) | 32 | 20 triangles 12 decagons | 90 | 60 | I_{h}
| ||

truncated icosahedron or buckyball or football/soccer ball (5.6.6 ) | (Animation) | 32 | 12 pentagons 20 hexagons | 90 | 60 | I_{h}
| ||

rhombicosidodecahedron or small rhombicosidodecahedron (3.4.5.4) | (Animation) | 62 | 20 triangles 30 squares 12 pentagons | 120 | 60 | I_{h}
| ||

truncated icosidodecahedron or great rhombicosidodecahedron (4.6.10) | (Animation) | 62 | 30 squares 20 hexagons 12 decagons | 180 | 120 | I_{h}
| ||

snub dodecahedron or snub icosidodecahedron (2 chiral forms) (3.3.3.3.5) | (Animation) (Animation) | 92 | 80 triangles 12 pentagons | 150 | 60 | I |

The cuboctahedron and icosidodecahedron are edge-uniform and are called quasi-regular.

The snub cube and snub dodecahedron are known as *chiral*, as they come in a left-handed (Latin: levomorph or laevomorph) form and right-handed (Latin: dextromorph) form. When something comes in multiple forms which are each other's three-dimensional mirror image, these forms may be called enantiomorphs. (This nomenclature is also used for the forms of certain chemical compounds).

The duals of the Archimedean solids are called the Catalan solids. Together with the bipyramids and trapezohedra, these are the face-uniform solids with regular vertices.

## See also

## References

- Williams, Robert (1979).
*The Geometrical Foundation of Natural Structure: A Source Book of Design*. Dover Publications, Inc. ISBN 0-486-23729-X. (Section 3-9)

## External links

- Weisstein, Eric W., "Archimedean solid" from MathWorld.
- Archemedian Solids by Eric W. Weisstein, Wolfram Demonstrations Project.
- Paper models of Archimedean Solids and Catalan Solids
- Paper models(nets) of Archimedean solids
- The Uniform Polyhedra by Dr. R. Mäder
- Virtual Reality Polyhedra,
*The Encyclopedia of Polyhedra*by George W. Hart - Penultimate Modular Origami by James S. Plank
- Interactive 3D polyhedra in Java
- Contemporary Archimedean Solid Surfaces Designed by Tom Barber
- Stella: Polyhedron Navigator: Software used to create many of the images on this page.

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