Editing Archimedean solid (section)

== Background of discovery ==
The names of Archimedean solids were taken from 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 that Archimedes listed thirteen polyhedra and briefly described them in terms of how many faces of each kind these polyhedra have.<ref>{{multiref
|{{harvp|Cromwell|1997|p=[https://books.google.com/books?id=OJowej1QWpoC&pg=PA156 156]}}
|{{harvp|Grünbaum|2009}}
|{{harvp|Field|1997|p=248}}
}}</ref>

{{multiple image
 | image1 = Piero della Francesca - Libellus de quinque corporibus regularibus - p52b (cropped).jpg
 | caption1 = Truncated icosahedron in ''[[De quinque corporibus regularibus]]''
 | image2 = De divina proportione - Vigintisex Basium Planum Vacuum.jpg
 | caption2 = Rhombicuboctahedron drawn by [[Leonardo da Vinci]]
 | image3 = Perspectiva Corporum Regularium 17b.jpg
 | caption3 = Cuboctahedron in ''[[Perspectiva Corporum Regularium]]''
 | total_width = 450
}}
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.{{sfnp|Banker|2005}} Yet, he did not credit those shapes to Archimedes and know of Archimedes' work but rather appeared to be an independent rediscovery.{{sfnp|Field|1997|p=248}} 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]].<ref>{{multiref
|{{harvp|Cromwell|1997|p=[https://books.google.com/books?id=OJowej1QWpoC&pg=PA156 156]}}
|{{harvp|Field|1997|p=253&ndash;254}}
}}</ref> The [[Net (polyhedron)|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 [[prism (geometry)|prisms]], [[antiprisms]], and the non-convex solids known as [[Kepler&ndash;Poinsot polyhedra]].{{sfnp|Schreiber|Fischer|Sternath|2008}}

{{multiple image
| align             = right
| total_width       = 300
| image1            = Polyhedron small rhombi 6-8, davinci.png
| image2            = Elongated square gyrobicupola, davinci.png
| footer            = [[Rhombicuboctahedron]] and [[elongated square gyrobicupola]]. The latter is not vertex-transitive, and thus not Archimedean.
}}

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.{{sfnp|Grünbaum|2009}} The solid appeared when some mathematicians mistakenly constructed the [[rhombicuboctahedron]]: two [[square cupola]]s attached to the [[octagonal prism]], with one of them rotated in forty-five degrees.<ref>{{multiref
 |{{harvp|Cromwell|1997|p=[https://books.google.com/books?id=OJowej1QWpoC&pg=PA91 91]}}
 |{{harvp|Berman|1971}}
}}</ref> 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. {{harvtxt|Grünbaum|2009}} 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 into the [[Johnson solid|Johnson solids]] instead, a convex polyhedron in which all of the faces are regular polygons.{{sfnp|Grünbaum|2009}}