View source for Archimedean solid

{{short description|Polyhedra in which all vertices are the same}}
[[File:Archimedian Solids 15.jpg|thumb|The Archimedean solids.  Two of them are [[Chirality (mathematics)|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, but not all alike, and whose vertices are all symmetric to each other. 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 ''{{shy|pseudo|rhombi|cub|octa|hedron}}'' is an extra polyhedron with regular faces and congruent vertices, but it is not generally counted as an Archimedean solid because it is not [[vertex-transitive]].
== The solids ==
The Archimedean solids have the [[vertex configuration]] and highly symmetric properties. Vertex configuration means a polyhedron whose two or more polygonal faces meet at the vertex. For instance, the <math> 3 \cdot 5 \cdot 3 \cdot 5 </math> means 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 were derived from the [[Platonic solids]], resulting from their construction.{{sfnp|Diudea|2018|p=[https://books.google.com/books?id=p_06DwAAQBAJ&pg=PA39 39]}} Some sources say the Archimedean solids are synonymous with the [[semiregular polyhedron]].{{sfnp|Kinsey|Moore|Prassidis|2011|p=[https://books.google.com/books?id=fFpuDwAAQBAJ&pg=PA380 380]}} Yet, the definition of a semiregular polyhedron may also include the infinite [[Prism (geometry)|prism]]s and [[antiprism]]s, including the [[elongated square gyrobicupola]].<ref>{{multiref
|{{harvp|Rovenski|2010|p=[https://books.google.com/books?id=BhVCYqqP69kC&pg=PA116 116]}}
|{{harvp|Malkevitch|1988|p=85}}
}}</ref>
{| class="wikitable sortable" style="text-align:center"
|+ The thirteen Archimedean solids
|-
! Name
! class="unsortable" | Solids
! [[Vertex configuration|Vertex configurations]]{{sfnp|Williams|1979}}
!Faces{{sfnp|Berman|1971}}
! Edges{{sfnp|Berman|1971}}
! Vertices{{sfnp|Berman|1971}}
! [[List of spherical symmetry groups#Polyhedral symmetry|Point<BR>group]]{{sfnp|Koca|Koca|2013|p=[https://books.google.com/books?id=ILnBkuSxXGEC&pg=PA48 47&ndash;50]}}
|-
| [[Truncated tetrahedron]]
| [[Image:truncatedtetrahedron.jpg|70px|Truncated tetrahedron]]
| 3.6.6<BR>[[Image:Polyhedron truncated 4a vertfig.png|50px]]
| 4 triangles<br>4 [[hexagon]]s
| 18
| 12
| T<sub>d</sub>
|-
| [[Cuboctahedron]]
| [[Image:cuboctahedron.svg|70px|Cuboctahedron]]
| 3.4.3.4<BR>[[Image:Polyhedron 6-8 vertfig.png|50px]]
| 8 [[triangle]]s<br>6 [[square]]s
| 24
| 12
| O<sub>h</sub>
|-
| [[Truncated cube]]
| [[Image:truncatedhexahedron.svg|70px|Truncated hexahedron]]
| 3.8.8<BR>[[Image:Polyhedron truncated 6 vertfig.png|50px]]
| 8 triangles<br>6 [[octagon]]s
| 36
| 24
| O<sub>h</sub>
|-
| [[Truncated octahedron]]
| [[Image:truncatedoctahedron.jpg|70px|Truncated octahedron]]
| 4.6.6<BR>[[Image:Polyhedron truncated 8 vertfig.png|50px]]
| 6 squares<br>8 hexagons
| 36
| 24
| O<sub>h</sub>
|-
| [[Rhombicuboctahedron]]
| [[Image:rhombicuboctahedron.jpg|70px|Rhombicuboctahedron]]
| 3.4.4.4<BR>[[Image:Polyhedron small rhombi 6-8 vertfig.png|50px]]
|8 triangles<br>18 squares
| 48
| 24
| O<sub>h</sub>
|-
| [[Truncated cuboctahedron]]
| [[Image:truncatedcuboctahedron.jpg|70px|Truncated cuboctahedron]]
| 4.6.8<BR>[[Image:Polyhedron great rhombi 6-8 vertfig light.png|50px]]
| 12 squares<br>8 hexagons<br>6 octagons
| 72
| 48
| O<sub>h</sub>
|-
| [[Snub cube]]
| [[Image:snubhexahedronccw.jpg|70px|Snub hexahedron (Ccw)]]
| 3.3.3.3.4<BR>[[Image:Polyhedron snub 6-8 left vertfig.png|50px]]
|32 triangles<br>6 squares
| 60
| 24
| O
|-
| [[Icosidodecahedron]]
| [[Image:icosidodecahedron.svg|70px|Icosidodecahedron]]
| 3.5.3.5<BR>[[Image:Polyhedron 12-20 vertfig.png|50px]]
| 20 triangles<br>12 [[pentagon]]s
| 60
| 30
| I<sub>h</sub>
|-
| [[Truncated dodecahedron]]
| [[Image:truncateddodecahedron.jpg|70px|Truncated dodecahedron]]
| 3.10.10<BR>[[Image:Polyhedron truncated 12 vertfig.png|50px]]
|20 triangles<br>12 [[decagon]]s
| 90
| 60
| I<sub>h</sub>
|-
| [[Truncated icosahedron]]
| [[Image:truncatedicosahedron.jpg|70px|Truncated icosahedron]]
| 5.6.6<BR>[[Image:Polyhedron truncated 20 vertfig.png|50px]]
| 12 pentagons<br>20 hexagons
| 90
| 60
| I<sub>h</sub>
|-
| [[Rhombicosidodecahedron]]
| [[Image:rhombicosidodecahedron.jpg|70px|Rhombicosidodecahedron]]
| 3.4.5.4<BR>[[Image:Polyhedron small rhombi 12-20 vertfig.png|50px]]
| 20 triangles<br>30 squares<br>12 pentagons
| 120
| 60
| I<sub>h</sub>
|-
| [[Truncated icosidodecahedron]]
| [[Image:truncatedicosidodecahedron.jpg|70px|Truncated icosidodecahedron]]
| 4.6.10<BR>[[Image:Polyhedron great rhombi 12-20 vertfig light.png|50px]]
|30 squares<br>20 hexagons<br>12 decagons
| 180
| 120
| I<sub>h</sub>
|-
| [[Snub dodecahedron]]
| [[Image:snubdodecahedroncw.jpg|70px|Snub dodecahedron (Cw)]]
| 3.3.3.3.5<BR>[[Image:Polyhedron snub 12-20 left vertfig.png|50px]]
| 80 triangles<br>12 pentagons
| 150
| 60
| I
|}
The construction of some Archimedean solids begins from the Platonic solids. The [[truncation (geometry)|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 [[regular icosahedron|icosahedron]]'s vertices, having the same symmetry as the icosahedron, the [[icosahedral symmetry]].<ref>{{multiref
|{{harvp|Chancey|O'Brien|1997|p=[https://books.google.com/books?id=wcQIEAAAQBAJ&pg=PA13 13]}}
|{{harvp|Koca|Koca|2013|p=[https://books.google.com/books?id=ILnBkuSxXGEC&pg=PA48 48]}}
}}</ref> 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 (geometry)|rectification]]. [[Expansion (geometry)|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, constructed by separating the cube or octahedron's faces from the centroid and filling them with squares.{{sfnp|Viana|Xavier|Aires|Campos|2019|p=1123|loc=See Fig. 6}} [[Snub (geometry)|Snub]] is a construction process of polyhedra by separating the polyhedron faces, twisting their faces in certain angles, and filling them up with [[equilateral triangle]]s. Examples can be found in [[snub cube]] and [[snub dodecahedron]]. The resulting construction of these solids gives the property of [[Chirality (mathematics)|chiral]], meaning they are not identical when reflected in a mirror.{{sfnp|Koca|Koca|2013|p=[https://books.google.com/books?id=ILnBkuSxXGEC&pg=PA49 49]}} 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]] base-to-base, or rhombicuboctahedron that can be constructed alternatively by attaching two [[square cupola]]s on the bases of octagonal prism.{{sfnp|Berman|1971}}
There are at least for known ten solids that have the [[Rupert property]], a polyhedron that can pass through a copy of itself with the same or similar size. They are the cuboctahedron, truncated octahedron, truncated cube, rhombicuboctahedron, icosidodecahedron, truncated cuboctahedron, truncated icosahedron, truncated dodecahedron, and the truncated tetrahedron.<ref>{{multiref
|{{harvp|Chai|Yuan|Zamfirescu|2018}}
|{{harvp|Hoffmann|2019}}
|{{harvp|Lavau|2019}}
}}</ref>
The [[Catalan solids]] are the [[dual polyhedron]] of Archimedean solids.{{sfnp|Diudea|2018|p=[https://books.google.com/books?id=p_06DwAAQBAJ&pg=PA39 39]}}
== 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 = Polyhedron small rhombi 6-8, davinci.png
| caption2 = Rhombicuboctahedron drawn by [[Leonardo da Vinci]] (colorized)
| 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}}
[[File:Elongated square gyrobicupola.png|thumb|The [[elongated square gyrobicupola]], a polyhedron where mathematicians mistakenly constructed the rhombicuboctahedron.]]
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]] instead, a convex polyhedron in which all of the faces are regular polygons.{{sfnp|Grünbaum|2009}}
== See also ==
* [[Archimedean graph]], planar graphs resembling the thirteen Archimedean solids.
* [[Conway polyhedron notation]]
== References ==
=== Footnotes ===
{{reflist|20em}}
=== Works cited ===
{{refbegin|25em}}
*{{citation
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| pages = 165–169
| s2cid = 190211171
| title = A manuscript of the works of Archimedes in the hand of Piero della Francesca
| volume = 147
}}.
* {{citation
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| year = 1971
| title = Regular-faced convex polyhedra
| journal = Journal of the Franklin Institute
| volume = 291
| issue = 5
| pages = 329–352
| doi = 10.1016/0016-0032(71)90071-8
| mr = 290245
}}.
* {{citation
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| last2 = Yuan | first2 = Liping
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| title = Rupert Property of Archimedean Solids
| journal = [[The American Mathematical Monthly]]
| volume = 125 | issue = 6 | pages = 497–504
| year = 2018
| doi = 10.1080/00029890.2018.1449505
| s2cid = 125508192
}}.
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| last2 = O'Brien | first2 = M. C. M.
| year = 1997
| title = The Jahn-Teller Effect in C<sub>60</sub> and Other Icosahedral Complexes
| url =https://books.google.com/books?id=wcQIEAAAQBAJ
| publisher = [[Princeton University Press]]
| isbn = 978-0-691-22534-0
}}.
* {{citation
| last = Cromwell | first = Peter R.
| title = Polyhedra
| year = 1997
| url = https://archive.org/details/polyhedra0000crom
| publisher = [[Cambridge University Press]]
| isbn = 978-0-521-55432-9
}}.
* {{citation
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| year = 2018
| title = Multi-shell Polyhedral Clusters
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| url = https://books.google.com/books?id=p_06DwAAQBAJ
}}.
* {{citation
| last = Field | first = J. V. | author-link = Judith V. Field
| doi = 10.1007/BF00374595
| issue = 3–4
| journal = [[Archive for History of Exact Sciences]]
| jstor = 41134110
| mr = 1457069
| pages = 241–289
| s2cid = 118516740
| title = Rediscovering the Archimedean polyhedra: Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Dürer, Daniele Barbaro, and Johannes Kepler
| volume = 50
| year = 1997
}}.
* {{citation
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| mr = 2520469
| pages = 89–101
| title = An enduring error
| url = https://digital.lib.washington.edu/dspace/bitstream/handle/1773/4592/An_enduring_error.pdf
| volume = 64
| year = 2009| doi-access = free
}}. Reprinted in {{citation|title=The Best Writing on Mathematics 2010|editor-first=Mircea|editor-last=Pitici|publisher=Princeton University Press|year=2011|pages=18–31}}.
* {{citation
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* {{citation
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* {{citation
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{{refend}}
=== Further reading ===
* {{citation
| last = Viana | first = Vera
| year = 2024
| doi = 10.1007/s00407-024-00331-7
| title = Archimedean solids in the fifteenth and sixteenth centuries
| journal = Archive for History of Exact Sciences
| doi-access = free
}}.
* {{citation
| last1 = Williams | first1 = Kim
| last2 = Monteleone | first2 = Cosimo
| year = 2021
| title = Daniele Barbaro’s Perspective of 1568
| url = https://books.google.com/books?id=w5RBEAAAQBAJ&pg=PA19
| page = 19&ndash;20
| doi = 10.1007/978-3-030-76687-0
| isbn = 978-3-030-76687-0
}}.
==External links==
* {{mathworld | urlname = ArchimedeanSolid | title = Archimedean solid }}
* [http://demonstrations.wolfram.com/ArchimedeanSolids/ Archimedean Solids] by [[Eric W. Weisstein]], [[Wolfram Demonstrations Project]].
* [http://www.software3d.com/Archimedean.php Paper models of Archimedean Solids and Catalan Solids]
* [http://www.korthalsaltes.com/cuadros.php?type=a Free paper models(nets) of Archimedean solids]
* [http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra] by Dr. R. Mäder
* [http://dmccooey.com/polyhedra/Archimedean.html Archimedean Solids] at Visual Polyhedra by David I. McCooey
* [http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra], ''The Encyclopedia of Polyhedra'' by George W. Hart
* [http://www.cs.utk.edu/~plank/plank/origami/penultimate/intro.html Penultimate Modular Origami] by James S. Plank
* [https://web.archive.org/web/20050403235101/http://ibiblio.org/e-notes/3Dapp/Convex.htm Interactive 3D polyhedra] in Java
* [https://archive.today/20130411004747/http://kovacsv.github.com/JSModeler/documentation/examples/solids.html Solid Body Viewer] is an interactive 3D polyhedron viewer which allows you to save the model in svg, stl or obj format.
* [http://www.software3d.com/Stella.php Stella: Polyhedron Navigator]: Software used to create many of the images on this page.
* [http://www.polyedergarten.de/ Paper Models of Archimedean (and other) Polyhedra]
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