Archimedean solid
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.
"Identical vertices" are usually taken to mean that for any two vertices, there must be an isometry of the entire solid that takes one vertex to the other. Sometimes it is instead only required that the faces that meet at one vertex are related isometrically to the faces that meet at the other. This difference in definitions controls whether the pseudorhombicuboctahedron is considered an Archimedean solid or a Johnson solid.
Prisms and antiprisms, whose symmetry groups are the dihedral groups, are generally not considered to be Archimedean solids, despite meeting the above definition. With this restriction, there are only finitely many Archimedean solids. They can all be made via Wythoff constructions from the Platonic solids with tetrahedral, octahedral and icosahedral symmetry.
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,[1] 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).
| Name (Vertex configuration) |
Transparent | Solid | Net | Faces | Edges | Vertices | Point group | |
|---|---|---|---|---|---|---|---|---|
| truncated tetrahedron (3.6.6) |
 (Animation) |
|
|
8 | 4 triangles 4 hexagons |
18 | 12 | Td |
| cuboctahedron (3.4.3.4) |
 (Animation) |
|
|
14 | 8 triangles 6 squares |
24 | 12 | Oh |
| truncated cube or truncated hexahedron (3.8.8) |
 (Animation) |
|
|
14 | 8 triangles 6 octagons |
36 | 24 | Oh |
| truncated octahedron (4.6.6) |
 |
|
|
14 | 6 squares 8 hexagons |
36 | 24 | Oh |
| rhombicuboctahedron or small rhombicuboctahedron (3.4.4.4 ) |
 (Animation) |
|
|
26 | 8 triangles 18 squares |
48 | 24 | Oh |
| truncated cuboctahedron or great rhombicuboctahedron (4.6.8) |
 (Animation) |
|
|
26 | 12 squares 8 hexagons 6 octagons |
72 | 48 | Oh |
| 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 | Ih |
| truncated dodecahedron (3.10.10) |
 (Animation) |
|
|
32 | 20 triangles 12 decagons |
90 | 60 | Ih |
| Truncated icosahedron (5.6.6 ) |
 (Animation) |
|
|
32 | 12 pentagons 20 hexagons |
90 | 60 | Ih |
| rhombicosidodecahedron or small rhombicosidodecahedron (3.4.5.4) |
 (Animation) |
|
|
62 | 20 triangles 30 squares 12 pentagons |
120 | 60 | Ih |
| truncated icosidodecahedron or great rhombicosidodecahedron (4.6.10) |
 (Animation) |
|
|
62 | 30 squares 20 hexagons 12 decagons |
180 | 120 | Ih |
| snub dodecahedron or snub icosidodecahedron (2 chiral forms) (3.3.3.3.5) |
 (Animation)  (Animation) |
|
|
92 | 80 triangles 12 pentagons |
150 | 60 | I |
Some definitions of semiregular polyhedron include one more figure, the elongated square gyrobicupola or "pseudo-rhombicuboctahedron".[2]
Properties
The number of vertices is 720° divided by the vertex angle defect.
The cuboctahedron and icosidodecahedron are edge-uniform and are called quasi-regular.
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.
Chirality
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).
See also
- Aperiodic tiling
- Catalan solid
- List of uniform polyhedra
- Platonic solid
- Quasicrystal
- semiregular polyhedron
- regular polyhedron
- uniform polyhedron
Notes
- ^ Field J., Rediscovering the Archimedean Polyhedra: Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Dürer, Daniele Barbaro, and Johannes Kepler, Archive for History of Exact Sciences, 50, 1997, 227
- ^ Malkevitch (1988), p. 85
References
- Jayatilake, Udaya (2005). "Calculations on face and vertex regular polyhedra". Mathematical Gazette. 89 (514): 76–81.
{{cite journal}}: Unknown parameter|month=ignored (help) - 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)
- Malkevitch, Joseph (1988), "Milestones in the history of polyhedra", in Senechal, M.; Fleck, G. (eds.), Shaping Space: A Polyhedral Approach, Boston: Birkhäuser, pp. 80–92.
External links
- Weisstein, Eric W. "Archimedean solid". MathWorld.
- Archemedian Solids by Eric W. Weisstein, Wolfram Demonstrations Project.
- Paper models of Archimedean Solids and Catalan Solids
- Free 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
- Stella: Polyhedron Navigator: Software used to create many of the images on this page.
- Paper Models of Archimedean (and other) Polyhedra