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Archimedean solid
Polyhedra in which all vertices are the same
Polyhedra in which all vertices are the same
The Archimedean solids are a set of thirteen convex polyhedra whose faces are regular polygons and are vertex-transitive, although they are not face-transitive. The solids were named after Archimedes, although he did not claim credit for them. They belong to the class of uniform polyhedra, the polyhedra with regular faces and symmetric vertices. Some Archimedean solids were portrayed in the works of artists and mathematicians during the Renaissance.
The elongated square gyrobicupola or ** is an extra polyhedron with regular faces and congruent vertices. Still, it is not generally counted as an Archimedean solid because it is not vertex-transitive.
The solids
The Archimedean solids have a single vertex configuration and highly symmetric properties. A vertex configuration indicates which regular polygons meet at each vertex. For instance, the configuration 3 \cdot 5 \cdot 3 \cdot 5 indicates a polyhedron in which each vertex is met by alternating two triangles and two pentagons. Highly symmetric properties in this case mean the symmetry group of each solid was derived from the Platonic solids, resulting from their construction. Some sources say the Archimedean solids are synonymous with the semiregular polyhedron. Yet, the definition of a semiregular polyhedron may also include the infinite prisms and antiprisms, including the elongated square gyrobicupola.{{multiref | |
The skeleton of Archimedean solids can be drawn in a graph, named Archimedean graph. Such graphs are regular, polyhedral (and therefore by necessity also 3-vertex-connected planar graphs), and also Hamiltonian graphs.
| Name | Solids | Vertex configurations | Faces | Edges | Vertices | Point |
|---|---|---|---|---|---|---|
| group | ||||||
| Truncated tetrahedron | [[Image:Truncatedtetrahedron.svg | 70px | Truncated tetrahedron]] | 3.6.6 | ||
| [[Image:Polyhedron truncated 4a vertfig.png | 50px]] | 4 triangles | ||||
| 4 hexagons | 18 | 12 | Td | |||
| Cuboctahedron | [[Image:Cuboctahedron.svg | 70px | Cuboctahedron]] | 3.4.3.4 | ||
| [[Image:Polyhedron 6-8 vertfig.png | 50px]] | 8 triangles | ||||
| 6 squares | 24 | 12 | Oh | |||
| Truncated cube | [[Image:Truncatedhexahedron.svg | 70px | Truncated hexahedron]] | 3.8.8 | ||
| [[Image:Polyhedron truncated 6 vertfig.png | 50px]] | 8 triangles | ||||
| 6 octagons | 36 | 24 | Oh | |||
| Truncated octahedron | [[Image:truncatedoctahedron.svg | 70px | Truncated octahedron]] | 4.6.6 | ||
| [[Image:Polyhedron truncated 8 vertfig.png | 50px]] | 6 squares | ||||
| 8 hexagons | 36 | 24 | Oh | |||
| Rhombicuboctahedron | [[Image:rhombicuboctahedron.jpg | 70px | Rhombicuboctahedron]] | 3.4.4.4 | ||
| [[Image:Polyhedron small rhombi 6-8 vertfig.png | 50px]] | 8 triangles | ||||
| 18 squares | 48 | 24 | Oh | |||
| Truncated cuboctahedron | [[Image:truncatedcuboctahedron.jpg | 70px | Truncated cuboctahedron]] | 4.6.8 | ||
| [[Image:Polyhedron great rhombi 6-8 vertfig light.png | 50px]] | 12 squares | ||||
| 8 hexagons | ||||||
| 6 octagons | 72 | 48 | Oh | |||
| Snub cube | [[Image:snubhexahedronccw.jpg | 70px | Snub hexahedron (Ccw)]] | 3.3.3.3.4 | ||
| [[Image:Polyhedron snub 6-8 left vertfig.png | 50px]] | 32 triangles | ||||
| 6 squares | 60 | 24 | O | |||
| Icosidodecahedron | [[Image:icosidodecahedron.jpg | 70px | Icosidodecahedron]] | 3.5.3.5 | ||
| [[Image:Polyhedron 12-20 vertfig.png | 50px]] | 20 triangles | ||||
| 12 pentagons | 60 | 30 | Ih | |||
| Truncated dodecahedron | [[Image:truncateddodecahedron.jpg | 70px | Truncated dodecahedron]] | 3.10.10 | ||
| [[Image:Polyhedron truncated 12 vertfig.png | 50px]] | 20 triangles | ||||
| 12 decagons | 90 | 60 | Ih | |||
| Truncated icosahedron | [[Image:truncatedicosahedron.svg | 70px | Truncated icosahedron]] | 5.6.6 | ||
| [[Image:Polyhedron truncated 20 vertfig.png | 50px]] | 12 pentagons | ||||
| 20 hexagons | 90 | 60 | Ih | |||
| Rhombicosidodecahedron | [[Image:rhombicosidodecahedron.jpg | 70px | Rhombicosidodecahedron]] | 3.4.5.4 | ||
| [[Image:Polyhedron small rhombi 12-20 vertfig.png | 50px]] | 20 triangles | ||||
| 30 squares | ||||||
| 12 pentagons | 120 | 60 | Ih | |||
| Truncated icosidodecahedron | [[Image:truncatedicosidodecahedron.jpg | 70px | Truncated icosidodecahedron]] | 4.6.10 | ||
| [[Image:Polyhedron great rhombi 12-20 vertfig light.png | 50px]] | 30 squares | ||||
| 20 hexagons | ||||||
| 12 decagons | 180 | 120 | Ih | |||
| Snub dodecahedron | [[Image:snubdodecahedroncw.jpg | 70px | Snub dodecahedron (Cw)]] | 3.3.3.3.5 | ||
| [[Image:Polyhedron snub 12-20 left vertfig.png | 50px]] | 80 triangles | ||||
| 12 pentagons | 150 | 60 | I |
The construction of some Archimedean solids begins from the Platonic solids. The truncation involves cutting away corners; to preserve symmetry, the cut is in a plane perpendicular to the line joining a corner to the center of the polyhedron and is the same for all corners, and an example can be found in truncated icosahedron constructed by cutting off all the icosahedron's vertices, having the same symmetry as the icosahedron, the icosahedral symmetry.{{multiref | |
At least ten of the Archimedean solids have the Rupert property: each can pass through a copy of itself, of the same size. They are the cuboctahedron, truncated octahedron, truncated cube, rhombicuboctahedron, icosidodecahedron, truncated cuboctahedron, truncated icosahedron, truncated dodecahedron, and the truncated tetrahedron.{{multiref | | |
The dual polyhedron of an Archimedean solid is a Catalan solid.
Background of discovery
The names of Archimedean solids were taken from the Ancient Greek mathematician Archimedes, who discussed them in a now-lost work. Although they were not credited to Archimedes originally, Pappus of Alexandria in the fifth section of his titled compendium Synagoge, referring to Archimedes, listed thirteen polyhedra and briefly described them in terms of how many faces of each kind these polyhedra have.{{multiref | | |
During the Renaissance, artists and mathematicians valued pure forms with high symmetry. Some Archimedean solids appeared in Piero della Francesca's De quinque corporibus regularibus, in attempting to study and copy the works of Archimedes, as well as include citations to Archimedes. Yet, he did not credit those shapes to Archimedes and knew of Archimedes' work, but rather appeared to be an independent rediscovery. Other appearance of the solids appeared in the works of Wenzel Jamnitzer's Perspectiva Corporum Regularium, and both Summa de arithmetica and Divina proportione by Luca Pacioli, drawn by Leonardo da Vinci.{{multiref | |
Kepler may have also found another solid known as elongated square gyrobicupola or pseudorhombicuboctahedron. Kepler once stated that there were fourteen Archimedean solids, yet his published enumeration only includes the thirteen uniform polyhedra. The first clear statement of such solid existence was made by Duncan Sommerville in 1905. The solid appeared when some mathematicians mistakenly constructed the rhombicuboctahedron: two square cupolas attached to the octagonal prism, with one of them rotated forty-five degrees.{{multiref | |
References
Footnotes
Works cited
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- {{citation | editor1-last = Senechal | editor1-first = M. | editor1-link = Marjorie Senechal | editor2-last = Fleck | editor2-first = G.
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- {{citation | editor-last = Cocchiarella | editor-first = Luigi
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References
- An Atlas of Graphs, p. 267-270
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