Editing Archimedean solid (section)

== 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 <math> 3 \cdot 5 \cdot 3 \cdot 5 </math> 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 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)|chirality]], 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}}

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.<ref>{{multiref
|{{harvp|Chai|Yuan|Zamfirescu|2018}}
|{{harvp|Hoffmann|2019}}
|{{harvp|Lavau|2019}}
}}</ref>

The [[dual polyhedron]] of an Archimedean solid is a [[Catalan solid]].{{sfnp|Diudea|2018|p=[https://books.google.com/books?id=p_06DwAAQBAJ&pg=PA39 39]}}