Archimedean solid

File:Archimedian Solids 15.jpg

The Archimedean solids. Two of them are chiral, with both forms shown, making 15 models in all.

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 Template:Shy 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.Template:Sfnp Some sources say the Archimedean solids are synonymous with the semiregular polyhedron.Template:Sfnp Yet, the definition of a semiregular polyhedron may also include the infinite prisms and antiprisms, including the elongated square gyrobicupola.

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.

The thirteen Archimedean solids

Name	Solids	Vertex configurationsTemplate:Sfnp	FacesTemplate:Sfnp	EdgesTemplate:Sfnp	VerticesTemplate:Sfnp	Point groupTemplate:Sfnp
Truncated tetrahedron	Truncated tetrahedron	3.6.6 File:Polyhedron truncated 4a vertfig.png	4 triangles 4 hexagons	18	12	Td
Cuboctahedron	Cuboctahedron	3.4.3.4 File:Polyhedron 6-8 vertfig.png	8 triangles 6 squares	24	12	Oh
Truncated cube	Truncated hexahedron	3.8.8 File:Polyhedron truncated 6 vertfig.png	8 triangles 6 octagons	36	24	Oh
Truncated octahedron	Truncated octahedron	4.6.6 File:Polyhedron truncated 8 vertfig.png	6 squares 8 hexagons	36	24	Oh
Rhombicuboctahedron	Rhombicuboctahedron	3.4.4.4 File:Polyhedron small rhombi 6-8 vertfig.png	8 triangles 18 squares	48	24	Oh
Truncated cuboctahedron	Truncated cuboctahedron	4.6.8 File:Polyhedron great rhombi 6-8 vertfig light.png	12 squares 8 hexagons 6 octagons	72	48	Oh
Snub cube	Snub hexahedron (Ccw)	3.3.3.3.4 File:Polyhedron snub 6-8 left vertfig.png	32 triangles 6 squares	60	24	O
Icosidodecahedron	Icosidodecahedron	3.5.3.5 File:Polyhedron 12-20 vertfig.png	20 triangles 12 pentagons	60	30	Ih
Truncated dodecahedron	Truncated dodecahedron	3.10.10 File:Polyhedron truncated 12 vertfig.png	20 triangles 12 decagons	90	60	Ih
Truncated icosahedron	Truncated icosahedron	5.6.6 File:Polyhedron truncated 20 vertfig.png	12 pentagons 20 hexagons	90	60	Ih
Rhombicosidodecahedron	Rhombicosidodecahedron	3.4.5.4 File:Polyhedron small rhombi 12-20 vertfig.png	20 triangles 30 squares 12 pentagons	120	60	Ih
Truncated icosidodecahedron	Truncated icosidodecahedron	4.6.10 File:Polyhedron great rhombi 12-20 vertfig light.png	30 squares 20 hexagons 12 decagons	180	120	Ih
Snub dodecahedron	Snub dodecahedron (Cw)	3.3.3.3.5 File:Polyhedron snub 12-20 left vertfig.png	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. If the truncation is exactly deep enough such that each pair of faces from adjacent vertices shares exactly one point, it is known as a rectification. Expansion involves moving each face away from the center (by the same distance to preserve the symmetry of the Platonic solid) and taking the convex hull. An example is the rhombicuboctahedron, which is constructed by separating the cube or octahedron's faces from their centroids and filling them with squares.Template:Sfnp Snub is a construction process of polyhedra by separating the polyhedron faces, twisting their faces in certain angles, and filling them up with equilateral triangles. Examples can be found in snub cube and snub dodecahedron. The resulting construction of these solids gives the property of chirality, meaning they are not identical when reflected in a mirror.Template:Sfnp However, not all of them can be constructed in such a way, or they could be constructed alternatively. For example, the icosidodecahedron can be constructed by attaching two pentagonal rotunda bases-to-base, or a rhombicuboctahedron that can be constructed alternatively by attaching two square cupolas on the bases of an octagonal prism.Template:Sfnp

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.

The dual polyhedron of an Archimedean solid is a Catalan solid.Template:Sfnp

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.

Template:Multiple image 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.Template:Sfnp Yet, he did not credit those shapes to Archimedes and knew of Archimedes' work, but rather appeared to be an independent rediscovery.Template:Sfnp 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. The net of Archimedean solids appeared in Albrecht Dürer's Underweysung der Messung, copied from the Pacioli's work. By around 1620, Johannes Kepler in his Harmonices Mundi had completed the rediscovery of the thirteen polyhedra, as well as defining the prisms, antiprisms, and the non-convex solids known as Kepler–Poinsot polyhedra.Template:Sfnp

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.Template:Sfnp 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. The thirteen solids have the property of vertex-transitive, meaning any two vertices of those can be translated onto the other one, but the elongated square gyrobicupola does not. Template:Harvtxt observed that it meets a weaker definition of an Archimedean solid, in which "identical vertices" means merely that the parts of the polyhedron near any two vertices look the same (they have the same shapes of faces meeting around each vertex in the same order and forming the same angles). Grünbaum pointed out a frequent error in which authors define Archimedean solids using some form of this local definition but omit the fourteenth polyhedron. If only thirteen polyhedra are to be listed, the definition must use global symmetries of the polyhedron rather than local neighborhoods. In the aftermath, the elongated square gyrobicupola was withdrawn from the Archimedean solids and included in the Johnson solids instead, a convex polyhedron in which all of the faces are regular polygons.Template:Sfnp

See also

• Conway polyhedron notation
• Johnson solid

References

Works cited

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Further reading

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External links

• Archimedean 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
• Archimedean Solids at Visual Polyhedra by David I. McCooey
• Virtual Reality Polyhedra, The Encyclopedia of Polyhedra by George W. Hart
• Penultimate Modular Origami by James S. Plank
• Interactive 3D polyhedra in Java
• Solid Body Viewer is an interactive 3D polyhedron viewer that allows you to save the model in SVG, STL, or OBJ format.
• Stella: Polyhedron Navigator: Software used to create many of the images on this page.
• Paper Models of Archimedean (and other) Polyhedra