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) |
Error creating thumbnail: (Animation) |
File:Truncated tetrahedron.png | Error creating thumbnail: | 8 | 4 triangles 4 hexagons |
18 | 12 | Td |
| cuboctahedron (3.4.3.4) |
Cuboctahedron (Animation) |
File:Cuboctahedron.png | Error creating thumbnail: | 14 | 8 triangles 6 squares |
24 | 12 | Oh |
| truncated cube or truncated hexahedron (3.8.8) |
Truncated hexahedron (Animation) |
File:Truncated hexahedron.png | File:Truncated hexahedron flat.svg | 14 | 8 triangles 6 octagons |
36 | 24 | Oh |
| truncated octahedron (4.6.6) |
Truncated octahedron (Animation) |
File:Truncated octahedron.png | File:Truncated octahedron flat.png | 14 | 6 squares 8 hexagons |
36 | 24 | Oh |
| rhombicuboctahedron or small rhombicuboctahedron (3.4.4.4 ) |
Rhombicuboctahedron (Animation) |
File:Small rhombicuboctahedron.png | File:Rhombicuboctahedron flat.png | 26 | 8 triangles 18 squares |
48 | 24 | Oh |
| truncated cuboctahedron or great rhombicuboctahedron (4.6.8) |
Truncated cuboctahedron (Animation) |
Error creating thumbnail: | Error creating thumbnail: | 26 | 8 hexagons 6 octagons |
72 | 48 | Oh |
| snub cube or snub hexahedron or snub cuboctahedron (2 chiral forms) (3.3.3.3.4) |
 (Animation) Error creating thumbnail: (Animation) |
Error creating thumbnail: | File:Snub cube flat.png | 38 | 32 triangles 6 squares |
60 | 24 | O |
| icosidodecahedron (3.5.3.5) |
Error creating thumbnail: (Animation) |
File:Icosidodecahedron.png | File:Icosidodecahedron flat.png | 32 | 20 triangles 12 pentagons |
60 | 30 | Ih |
| truncated dodecahedron (3.10.10) |
Truncated dodecahedron (Animation) |
|
File:Truncated dodecahedron flat.png | 32 | 20 triangles 12 decagons |
90 | 60 | Ih |
| truncated icosahedron or buckyball or football/soccer ball (5.6.6 ) |
Truncated icosahedron (Animation) |
Error creating thumbnail: | File:Truncated icosahedron flat.png | 32 | 12 pentagons 20 hexagons |
90 | 60 | Ih |
| rhombicosidodecahedron or small rhombicosidodecahedron (3.4.5.4) |
Error creating thumbnail: (Animation) |
|
File:Rhombicosidodecahedron flat.png | 62 | 20 triangles 30 squares 12 pentagons |
120 | 60 | Ih |
| truncated icosidodecahedron or great rhombicosidodecahedron (4.6.10) |
Truncated icosidodecahedron (Animation) |
File:Great rhombicosidodecahedron.png | File:Truncated icosidodecahedron flat.png | 62 | 30 squares 20 hexagons 12 decagons |
180 | 120 | Ih |
| snub dodecahedron or snub icosidodecahedron (2 chiral forms) (3.3.3.3.5) |
Snub dodecahedron (Ccw) (Animation)  (Animation) |
File:Snub dodecahedron ccw.png | File:Snub dodecahedron flat.svg | 92 | 80 triangles 12 pentagons |
150 | 60 | I |
A graph of the Archimedean solids (click on picture for better quality)
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[]
- semiregular polyhedron
- uniform polyhedron
- List of uniform polyhedra
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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