View source for Archimedean solid

{{short description|Polyhedra in which all vertices are the same}}
{{multiple image
| align             = right
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| image1            = Polyhedron truncated 4a max.png
| width1            = 3927
| height1           = 3873
| image2            = Polyhedron 6-8 max.png
| width2            = 3850
| height2           = 3680
| image3            = Polyhedron great rhombi 12-20 max.png
| width3            = 3943
| height3           = 3977
| footer            = [[Truncated tetrahedron]], [[Cuboctahedron|cuboctahedron]] and [[Truncated icosidodecahedron|truncated icosidodecahedron]]. The first can be described as the smallest Archimedean solid, the last as the largest.
}}
{{multiple image
| align             = right
| total_width       = 400
| image1            = Polyhedron small rhombi 6-8, davinci.png
| image2            = Elongated square gyrobicupola, davinci.png
| footer            = [[Rhombicuboctahedron]] and [[Elongated square gyrobicupola|pseudo-rhombicuboctahedron]]
}}

In [[Geometry|geometry]], an '''Archimedean solid''' is one of 13 convex polyhedra whose faces are [[Regular polygon|regular polygon]]s and whose vertices are all symmetric to each other. They were first enumerated by [[Biography:Archimedes|Archimedes]]. They belong to the class of convex [[Uniform polyhedron|uniform polyhedra]], the convex polyhedra with regular faces and symmetric vertices, which is divided into the Archimedean solids, the five [[Platonic solid]]s (each with only one type of polygon face), and the two infinite families of [[Prism (geometry)|prisms]] and [[Antiprism|antiprism]]s. The pseudorhombicuboctahedron 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.<ref>{{cite web |last1=Steckles |first1=Katie |title=The Unwanted Shape |url=https://www.youtube.com/watch?v=wN9f-k9oRts |website=[[Company:YouTube|YouTube]] |access-date=20 January 2022}}</ref> An even larger class than the convex uniform polyhedra is the [[Johnson solid]]s, whose regular polygonal faces do not need to meet in identical vertices.

In these polyhedra, the vertices are identical, in the sense that a global [[Isometry|isometry]] of the entire solid takes any one vertex to any other. {{harvs|first=Branko|last=Grünbaum|year=2009|txt}} observed that a 14th polyhedron, the [[Elongated square gyrobicupola|elongated square gyrobicupola]] (or pseudo-rhombicuboctahedron), 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 14th polyhedron. If only 13 polyhedra are to be listed, the definition must use global symmetries of the polyhedron rather than local neighborhoods.

[[Prism (geometry)|Prisms]] and [[Antiprism|antiprism]]s, whose [[Symmetry group|symmetry group]]s are the [[Dihedral group|dihedral group]]s, are generally not considered to be Archimedean solids, even though their faces are regular polygons and their symmetry groups act transitively on their vertices. Excluding these two infinite families, there are 13 Archimedean solids. All the Archimedean solids (but not the elongated square gyrobicupola) can be made via [[Wythoff construction]]s from the Platonic solids with [[Tetrahedral symmetry|tetrahedral]], [[Octahedral symmetry|octahedral]] and [[Icosahedral symmetry|icosahedral symmetry]].

==Origin of name==
The Archimedean solids take their name from [[Biography:Archimedes|Archimedes]], who discussed them in a now-lost work. [[Biography:Pappus of Alexandria|Pappus]] refers to it, stating that Archimedes listed 13 polyhedra.<ref name="g09">{{harvtxt|Grünbaum|2009}}.</ref> During the [[Philosophy:Renaissance|Renaissance]], artists and [[Biography:Mathematician|mathematician]]s valued ''pure forms'' with high symmetry, and by around 1620 [[Biography:Johannes Kepler|Johannes Kepler]] had completed the rediscovery of the 13 polyhedra,<ref>Field J., Rediscovering the Archimedean Polyhedra: Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Dürer, Daniele Barbaro, and Johannes Kepler, ''Archive for History of Exact Sciences'', '''50''', 1997, 227</ref> as well as defining the [[Prism (geometry)|prisms]], antiprisms, and the non-convex solids known as Kepler-Poinsot polyhedra. (See {{harvnb|Schreiber|Fischer|Sternath|2008}} for more information about the rediscovery of the Archimedean solids during the renaissance.)

Kepler may have also found the [[Elongated square gyrobicupola|elongated square gyrobicupola]] (pseudorhombicuboctahedron): at least, he once stated that there were 14 Archimedean solids. However, his published enumeration only includes the 13 uniform polyhedra, and the first clear statement of the pseudorhombicuboctahedron's existence was made in 1905, by [[Biography:Duncan Sommerville|Duncan Sommerville]].<ref name="g09"/>

==Classification==
There are 13 Archimedean solids (not counting the [[Elongated square gyrobicupola|elongated square gyrobicupola]]; 15 if the [[Mirror image|mirror image]]s of two [[Chirality (mathematics)|enantiomorphs]], the snub cube and snub dodecahedron, are counted separately).

Here the ''vertex configuration'' refers to the type of regular polygons that meet at any given vertex. For example, a [[Vertex configuration|vertex configuration]] of 4.6.8 means that a [[Square|square]], [[Hexagon|hexagon]], and [[Octagon|octagon]] meet at a vertex (with the order taken to be clockwise around the vertex).

{| class="wikitable sortable" style="text-align:center"
|-
! Name/{{Break}}(alternative name)
![[Schläfli symbol|Schläfli]]{{Break}}Coxeter
!class="unsortable"| Transparent
!class="unsortable"| Solid
!class="unsortable"| [[Net (polyhedron)|Net]]
! Vertex<br>[[Vertex configuration|conf.]]/[[Vertex figure|fig.]]
! colspan="2" | Faces
! Edges
! Vert.
! Volume<BR>(unit edges)
! [[List of spherical symmetry groups#Polyhedral symmetry|Point<BR>group]]
! [[Sphericity]]
|-
| [[Truncated tetrahedron]]||t{3,3}<BR>{{CDD|node_1|3|node_1|3|node}}
| [[Image:truncatedtetrahedron.jpg|70px|Truncated tetrahedron]] &nbsp; [[File:Cog-scripted-svg-blue.svg|12px|link=File:truncatedtetrahedron.gif]]
| [[Image:Polyhedron truncated 4a max.png|70px]]
| [[Image:Polyhedron truncated 4a net.svg|60px]]
| 3.6.6<BR>[[Image:Polyhedron truncated 4a vertfig.png|50px]]
| 8
| 4 triangles<br>4 [[Hexagon|hexagon]]s
| 18
| 12
| {{val|2.710576}}
| T<sub>d</sub>
| {{val|0.7754132}}
|-
| [[Cuboctahedron]]<BR>(rhombitetratetrahedron, triangular gyrobicupola)||r{4,3} or rr{3,3}<BR>{{CDD|node|4|node_1|3|node}} or {{CDD|node_1|3|node|3|node_1}}
| [[Image:cuboctahedron.svg|70px|Cuboctahedron]] &nbsp; [[File:Cog-scripted-svg-blue.svg|12px|link=File:cuboctahedron.gif]]
| [[Image:Polyhedron 6-8 max.png|70px]]
| [[Image:Polyhedron 6-8 net.svg|60px]]
| 3.4.3.4<BR>[[Image:Polyhedron 6-8 vertfig.png|50px]]
| 14
| 8 [[Triangle|triangle]]s<br>6 [[Square|square]]s
| 24
| 12
| {{val|2.357023}}
| O<sub>h</sub>
| {{val|0.9049972}}
|-
| [[Truncated cube]]||t{4,3}<BR>{{CDD|node_1|4|node_1|3|node}}
| [[Image:truncatedhexahedron.svg|70px|Truncated hexahedron]] &nbsp; [[File:Cog-scripted-svg-blue.svg|12px|link=File:truncatedhexahedron.gif]]
| [[Image:Polyhedron truncated 6 max.png|70px]]
| [[Image:Polyhedron truncated 6 net.svg|60px]]
| 3.8.8<BR>[[Image:Polyhedron truncated 6 vertfig.png|50px]]
| 14
| 8 triangles<br>6 [[Octagon|octagon]]s
| 36
| 24
| {{val|13.599663}}
| O<sub>h</sub>
| {{val|0.8494937}}
|-
| [[Truncated octahedron]]<BR>(truncated tetratetrahedron)||t{3,4} or tr{3,3}<BR>{{CDD|node_1|3|node_1|4|node}} or {{CDD|node_1|3|node_1|3|node_1}}
| [[Image:truncatedoctahedron.jpg|70px|Truncated octahedron]] &nbsp; [[File:Cog-scripted-svg-blue.svg|12px|link=File:truncatedoctahedron.gif]]
| [[Image:Polyhedron truncated 8 max.png|70px]]
| [[Image:Polyhedron truncated 8 net.svg|60px]]
| 4.6.6<BR>[[Image:Polyhedron truncated 8 vertfig.png|50px]]
| 14
| 6 squares<br>8 hexagons
| 36
| 24
| {{val|11.313709}}
| O<sub>h</sub>
| {{val|0.9099178}}
|-
| [[Rhombicuboctahedron]]<BR>(small rhombicuboctahedron, elongated square orthobicupola)||rr{4,3}<BR>{{CDD|node_1|4|node|3|node_1}}
| [[Image:rhombicuboctahedron.jpg|70px|Rhombicuboctahedron]] &nbsp; [[File:Cog-scripted-svg-blue.svg|12px|link=File:rhombicuboctahedron.gif]]
| [[Image:Polyhedron small rhombi 6-8 max.png|70px]]
| [[Image:Polyhedron small rhombi 6-8 net.svg|60px]]
| 3.4.4.4<BR>[[Image:Polyhedron small rhombi 6-8 vertfig.png|50px]]
| 26
|8 triangles<br>18 squares
| 48
| 24
| {{val|8.714045}}
| O<sub>h</sub>
| {{val|0.9540796}}
|-
| [[Truncated cuboctahedron]]<BR>(great rhombicuboctahedron)||tr{4,3}<BR>{{CDD|node_1|4|node_1|3|node_1}}
| [[Image:truncatedcuboctahedron.jpg|70px|Truncated cuboctahedron]] &nbsp; [[File:Cog-scripted-svg-blue.svg|12px|link=File:truncatedcuboctahedron.gif]]
| [[Image:Polyhedron great rhombi 6-8 max.png|70px]]
| [[Image:Polyhedron great rhombi 6-8 net.svg|60px]]
| 4.6.8<BR>[[Image:Polyhedron great rhombi 6-8 vertfig light.png|50px]]
| 26
| 12 squares<br>8 hexagons<br>6 octagons
| 72
| 48
| {{val|41.798990}}
| O<sub>h</sub>
| {{val|0.9431657}}
|-
| [[Snub cube]]<BR>(snub cuboctahedron)||sr{4,3}<BR>{{CDD|node_h|4|node_h|3|node_h}}
| [[Image:snubhexahedronccw.jpg|70px|Snub hexahedron (Ccw)]] &nbsp; [[File:Cog-scripted-svg-blue.svg|12px|link=File:snubhexahedronccw.gif]]
| [[Image:Polyhedron snub 6-8 left max.png|70px]]
| [[Image:Polyhedron snub 6-8 left net.svg|60px]]
| 3.3.3.3.4<BR>[[Image:Polyhedron snub 6-8 left vertfig.png|50px]]
| 38
|32 triangles<br>6 squares
| 60
| 24
| {{val|7.889295}}
| O
| {{val|0.9651814}}
|-
| [[Icosidodecahedron]]<BR>(pentagonal gyrobirotunda)||r{5,3}<BR>{{CDD|node|5|node_1|3|node}}
| [[Image:icosidodecahedron.svg|70px|Icosidodecahedron]] &nbsp; [[File:Cog-scripted-svg-blue.svg|12px|link=File:icosidodecahedron.gif]]
| [[Image:Polyhedron 12-20 max.png|70px]]
| [[Image:Polyhedron 12-20 net.svg|60px]]
| 3.5.3.5<BR>[[Image:Polyhedron 12-20 vertfig.png|50px]]
| 32
| 20 triangles<br>12 [[Pentagon|pentagon]]s
| 60
| 30
| {{val|13.835526}}
| I<sub>h</sub>
| {{val|0.9510243}}
|-
| [[Truncated dodecahedron]]||t{5,3}<BR>{{CDD|node_1|5|node_1|3|node}}
| [[Image:truncateddodecahedron.jpg|70px|Truncated dodecahedron]] &nbsp; [[File:Cog-scripted-svg-blue.svg|12px|link=File:truncateddodecahedron.gif]]
| [[Image:Polyhedron truncated 12 max.png|70px]]
| [[Image:Polyhedron truncated 12 net.svg|60px]]
| 3.10.10<BR>[[Image:Polyhedron truncated 12 vertfig.png|50px]]
| 32
|20 triangles<br>12 [[Decagon|decagon]]s
| 90
| 60
| {{val|85.039665}}
| I<sub>h</sub>
| {{val|0.9260125}}
|-
| [[Truncated icosahedron]]||t{3,5}<BR>{{CDD|node_1|3|node_1|5|node}}
| [[Image:truncatedicosahedron.jpg|70px|Truncated icosahedron]] &nbsp; [[File:Cog-scripted-svg-blue.svg|12px|link=File:truncatedicosahedron.gif]]
| [[Image:Polyhedron truncated 20 max.png|70px]]
| [[Image:Polyhedron truncated 20 net compact.svg|60px]]
| 5.6.6<BR>[[Image:Polyhedron truncated 20 vertfig.png|50px]]
| 32
| 12 pentagons<br>20 hexagons
| 90
| 60
| {{val|55.287731}}
| I<sub>h</sub>
| {{val|0.9666219}}
|-
| [[Rhombicosidodecahedron]]<BR>(small rhombicosidodecahedron)||rr{5,3}<BR>{{CDD|node_1|5|node|3|node_1}}
| [[Image:rhombicosidodecahedron.jpg|70px|Rhombicosidodecahedron]] &nbsp; [[File:Cog-scripted-svg-blue.svg|12px|link=File:rhombicosidodecahedron.gif]]
| [[Image:Polyhedron small rhombi 12-20 max.png|70px]]
| [[Image:Polyhedron small rhombi 12-20 net.svg|60px]]
| 3.4.5.4<BR>[[Image:Polyhedron small rhombi 12-20 vertfig.png|50px]]
| 62
| 20 triangles<br>30 squares<br>12 pentagons
| 120
| 60
| {{val|41.615324}}
| I<sub>h</sub>
| {{val|0.9792370}}
|-
| [[Truncated icosidodecahedron]]<BR>(great rhombicosidodecahedron)||tr{5,3}<BR>{{CDD|node_1|5|node_1|3|node_1}}
| [[Image:truncatedicosidodecahedron.jpg|70px|Truncated icosidodecahedron]] &nbsp; [[File:Cog-scripted-svg-blue.svg|12px|link=File:truncatedicosidodecahedron.gif]]
| [[Image:Polyhedron great rhombi 12-20 max.png|70px]]
| [[Image:Polyhedron great rhombi 12-20 net.svg|60px]]
| 4.6.10<BR>[[Image:Polyhedron great rhombi 12-20 vertfig light.png|50px]]
| 62
|30 squares<br>20 hexagons<br>12 decagons
| 180
| 120
| {{val|206.803399}}
| I<sub>h</sub>
| {{val|0.9703127}}
|-
| [[Snub dodecahedron]]<BR>(snub icosidodecahedron)||sr{5,3}<BR>{{CDD|node_h|5|node_h|3|node_h}}
| [[Image:snubdodecahedroncw.jpg|70px|Snub dodecahedron (Cw)]] &nbsp; [[File:Cog-scripted-svg-blue.svg|12px|link=File:snubdodecahedroncw.gif]]
| [[Image:Polyhedron snub 12-20 left max.png|70px]]
| [[Image:Polyhedron snub 12-20 left net.svg|60px]]
| 3.3.3.3.5<BR>[[Image:Polyhedron snub 12-20 left vertfig.png|50px]]
| 92
| 80 triangles<br>12 pentagons
| 150
| 60
| {{val|37.616650}}
| I
| {{val|0.9820114}}
|}

Some definitions of [[Semiregular polyhedron|semiregular polyhedron]] include one more figure, the [[Elongated square gyrobicupola|elongated square gyrobicupola]] or "pseudo-rhombicuboctahedron".<ref>{{harvtxt|Malkevitch|1988}}, p.&nbsp;85</ref>

== Properties ==

The number of vertices is 720° divided by the vertex angle defect.

The cuboctahedron and icosidodecahedron are edge-uniform and are called [[Quasiregular polyhedron|quasi-regular]].

The [[Dual polyhedron|duals]] of the Archimedean solids are called the [[Catalan solid]]s. Together with the [[Bipyramid|bipyramid]]s and [[Trapezohedron|trapezohedra]], these are the face-uniform solids with regular vertices.

=== Chirality ===

The snub cube and snub dodecahedron are known as ''[[Chirality (mathematics)|chiral]]'', as they come in a left-handed form (Latin: levomorph or laevomorph) and right-handed form (Latin: dextromorph). When something comes in multiple forms which are each other's three-dimensional [[Mirror image|mirror image]], these forms may be called enantiomorphs.  (This nomenclature is also used for the forms of certain [[Chemistry:Chemical compound|chemical compound]]s.)

== Construction of Archimedean solids ==
{{Further|Uniform polyhedron|Conway polyhedron notation}}
[[File:Polyhedron truncation example3.png|thumb|The Archimedean solids can be constructed as generator positions in a kaleidoscope.]]

The different Archimedean and Platonic solids can be related to each other using a handful of general constructions.  Starting with a Platonic solid, [[Truncation (geometry)|truncation]] involves cutting away of 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. Depending on how much is truncated (see table below), different Platonic and Archimedean (and other) solids can be created. 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. An [[Expansion (geometry)|expansion]], or cantellation, involves moving each face away from the center (by the same distance so as to preserve the symmetry of the Platonic solid) and taking the convex hull. Expansion with twisting also involves rotating the faces, thus splitting each rectangle corresponding to an edge into two triangles by one of the diagonals of the rectangle. The last construction we use here is truncation of both corners and edges.  Ignoring scaling, expansion can also be viewed as the rectification of the rectification. Likewise, the cantitruncation can be viewed as the truncation of the rectification.
{{Clear|left}}
{| class="wikitable"
|+ Construction of Archimedean Solids
!colspan=2|Symmetry
![[Tetrahedral symmetry|Tetrahedral]]<br>[[File:Disdyakis 6 spherical.png|120px]]
!colspan=2|[[Octahedral symmetry|Octahedral]]<br>[[File:Disdyakis 12 spherical.png|120px]]
!colspan=2|[[Icosahedral symmetry|Icosahedral]]<br>[[File:Disdyakis 30 spherical.png|120px]]
|-
! Starting solid<br>Operation||Symbol<br>{p,q}<br>{{CDD|node_1|p|node|q|node}}
! [[Tetrahedron]]<br>{3,3}<br>[[File:Uniform polyhedron-33-t0.png|50px]] || [[Cube|Cube]]<br>{4,3}<br>50px || [[Octahedron|Octahedron]]<br>{3,4}<br>50px || [[Regular dodecahedron|Dodecahedron]]<br>{5,3}<br>50px || [[Regular icosahedron|Icosahedron]]<br>{3,5}<br>[[File:Uniform polyhedron-53-t2.svg|50px]]
|- align=center
! [[Truncation (geometry)|Truncation]] (t)||t{p,q}<br>{{CDD|node_1|p|node_1|q|node}}
| [[Truncated tetrahedron|truncated tetrahedron]]<br>[[File:Uniform polyhedron-33-t01.png|50px]] || [[Truncated cube|truncated cube]]<br>50px || [[Truncated octahedron|truncated octahedron]]<br>50px || [[Truncated dodecahedron|truncated dodecahedron]]<br>50px || [[Truncated icosahedron|truncated icosahedron]]<br>[[File:Uniform polyhedron-53-t12.svg|50px]]
|- align=center
! [[Rectification (geometry)|Rectification]] (r)<br>Ambo (a)||r{p,q}<br>{{CDD|node|p|node_1|q|node}}
| tetratetrahedron<br>(octahedron)<br>[[File:Uniform polyhedron-33-t1.png|50px]] ||colspan=2| [[Cuboctahedron|cuboctahedron]]<br>50px ||colspan=2| [[Icosidodecahedron|icosidodecahedron]]<br>[[File:Uniform polyhedron-53-t1.svg|50px]]
|- align=center
! [[Bitruncation]] (2t)<br>Dual kis (dk)||2t{p,q}<br>{{CDD|node|p|node_1|q|node_1}}
| truncated tetrahedron<br>[[File:Uniform polyhedron-33-t12.png|50px]] || truncated octahedron<br>50px || truncated cube<br>50px || truncated icosahedron<br>50px || truncated dodecahedron<br>[[File:Uniform polyhedron-53-t01.svg|50px]]
|- align=center
! Birectification (2r)<br>[[Dual polyhedron|Dual]] (d)||2r{p,q}<br>{{CDD|node|p|node|q|node_1}}
| tetrahedron<br>[[File:Uniform polyhedron-33-t2.png|50px]]|| octahedron<br>50px || cube<br>50px || icosahedron<br>50px || dodecahedron<br>[[File:Uniform polyhedron-53-t0.svg|50px]]
|- align=center
! Cantellation (rr)<br>[[Expansion (geometry)|Expansion]] (e)||rr{p,q}<br>{{CDD|node_1|p|node|q|node_1}}
| rhombitetratetrahedron<br><small>(cuboctahedron)</small><br>[[File:Uniform polyhedron-33-t02.png|50px]] ||colspan=2| [[Rhombicuboctahedron|rhombicuboctahedron]]<br>50px || colspan=2|[[Rhombicosidodecahedron|rhombicosidodecahedron]] <br>[[File:Uniform polyhedron-53-t02.png|50px]]
|- align=center
! Snub rectified (sr)<br>[[Snub (geometry)|Snub]] (s)||sr{p,q}<br>{{CDD|node_h|p|node_h|q|node_h}}
| snub tetratetrahedron<br><small>(icosahedron)</small><br>[[File:Uniform polyhedron-33-s012.svg|50px]] ||colspan=2| snub cuboctahedron<br>50px ||colspan=2| snub icosidodecahedron<br>[[File:Uniform polyhedron-53-s012.png|50px]]
|- align=center
! Cantitruncation (tr)<br>Bevel (b)||tr{p,q}<br>{{CDD|node_1|p|node_1|q|node_1}}
| truncated tetratetrahedron<br><small>(truncated octahedron)</small><br>[[File:Uniform polyhedron-33-t012.png|50px]] ||colspan=2| [[Truncated cuboctahedron|truncated cuboctahedron]]<br>50px ||colspan=2| [[Truncated icosidodecahedron|truncated icosidodecahedron]]<br>[[File:Uniform polyhedron-53-t012.png|50px]]
|}

Note the duality between the cube and the octahedron, and between the dodecahedron and the icosahedron. Also, partially because the tetrahedron is self-dual, only one Archimedean solid that has at most tetrahedral symmetry. (All Platonic solids have at least tetrahedral symmetry, as tetrahedral symmetry is a symmetry operation of (i.e. is included in) octahedral and isohedral symmetries, which is demonstrated by the fact that an octahedron can be viewed as a rectified tetrahedron, and an icosahedron can be used as a snub tetrahedron.)

==Stereographic projection==
{|class=wikitable
|- align=center valign=top
!colspan=2|[[Truncated tetrahedron|truncated tetrahedron]]||colspan=2|[[Truncated cube|truncated cube]]||colspan=2|[[Truncated octahedron|truncated octahedron]]||colspan=2|[[Truncated dodecahedron|truncated dodecahedron]]||colspan=2|[[Truncated icosahedron|truncated icosahedron]]
|-
|[[Image:truncated tetrahedron stereographic projection triangle.png|160px]]<br>[[Triangle|triangle]]-centered
|[[Image:truncated tetrahedron stereographic projection hexagon.png|160px]]<br>[[Hexagon|hexagon]]-centered
|[[Image:truncated cube stereographic projection octagon.png|160px]]<br>[[Octagon|octagon]]-centered
|[[Image:truncated cube stereographic projection triangle.png|160px]]<br>[[Triangle|triangle]]-centered
|[[Image:truncated octahedron stereographic projection square.png|160px]]<br/>[[Square|square]]-centered
|[[Image:truncated octahedron stereographic projection hexagon.png|160px]]<br/>[[Hexagon|hexagon]]-centered
|[[Image:truncated dodecahedron stereographic projection decagon.png|160px]]<br>[[Decagon]]-centered
|[[Image:truncated dodecahedron stereographic projection triangle.png|160px]]<br>[[Triangle]]-centered
|[[File:Truncated icosahedron stereographic projection pentagon.png|160px]]<br>[[Pentagon|pentagon]]-centered
|[[File:Truncated icosahedron stereographic projection hexagon.png|160px]]<br>[[Hexagon|hexagon]]-centered

|-
|}

{|class=wikitable
|-
!colspan=3|[[Cuboctahedron|cuboctahedron]]||colspan=2|[[Icosidodecahedron|icosidodecahedron]]||colspan=3|[[Rhombicuboctahedron|rhombicuboctahedron]]||colspan=3|[[Rhombicosidodecahedron|rhombicosidodecahedron]]||colspan=3|
|- align=center valign=top
|[[File:cuboctahedron stereographic projection square.png|155px]]<br>[[Square|square]]-centered
|[[File:cuboctahedron stereographic projection triangle.png|160px]]<br>[[Triangle|triangle]]-centered
|[[File:Cuboctahedron_stereographic_projection_vertex.png|160px]]<br>[[Vertex (geometry)|vertex]]-centered
|[[Image:icosidodecahedron stereographic projection pentagon.png|160px]]<br>[[Pentagon|pentagon]]-centered
|[[Image:icosidodecahedron stereographic projection triangle.png|160px]]<br>[[Triangle|triangle]]-centered
|[[File:rhombicuboctahedron stereographic projection square.png|160px]]<br>[[Square|square]]-centered
|[[File:Rhombicuboctahedron stereographic projection square2.png|160px]]<br>[[Square|square]]-centered
|[[File:rhombicuboctahedron stereographic projection triangle.png|160px]]<br> [[Triangle|triangle]]-centered
|[[Image:rhombicosidodecahedron stereographic projection pentagon'.png|160px]]<br>[[Pentagon]]-centered
|[[Image:rhombicosidodecahedron stereographic projection triangle.png|160px]]<br>[[Triangle]]-centered
|[[Image:rhombicosidodecahedron stereographic projection square.png|160px]]<br>[[Square]]-centered

|}

{|class=wikitable
|- align=center valign=top
!colspan=3|[[Truncated cuboctahedron|truncated cuboctahedron]]||colspan=3|[[Truncated icosidodecahedron|truncated icosidodecahedron]]||[[Snub cube|snub cube]]
|-
|[[Image:truncated cuboctahedron stereographic projection square.png|165px]]<br>[[Square|square]]-centered
|[[Image:truncated cuboctahedron stereographic projection hexagon.png|152px]]<br>[[Hexagon|hexagon]]-centered
|[[Image:truncated cuboctahedron stereographic projection octagon.png|160px]]<br>[[Octagon|octagon]]-centered
|[[Image:Truncated icosidodecahedron stereographic projection decagon.png|160px]]<br>[[Decagon|decagon]]-centered
|[[Image:Truncated icosidodecahedron stereographic projection hexagon.png|168px]]<br>[[Hexagon|hexagon]]-centered
|[[Image:Truncated icosidodecahedron stereographic projection square.png|172px]]<br>[[Square|square]]-centered
|[[File:Snub_cube_stereographic_projection.png|160px]]<br>[[Square|square]]-centered
|-
|}

== See also ==
* [[Aperiodic tiling]]
* [[Archimedean graph]]
* [[Physics:Icosahedral twins|Icosahedral twins]]
* [[List of uniform polyhedra]]
* Prince Rupert's cube#Generalizations
* [[Physics:Quasicrystal|Quasicrystal]]
* [[Regular polyhedron]]
* [[Semiregular polyhedron]]
* Toroidal polyhedron
* [[Uniform polyhedron]]

== Citations ==
{{reflist}}

=== Works cited ===
*{{citation
 | last = Grünbaum | first = Branko
 | doi = 10.4171/EM/120
 | issue = 3
 | journal = Elemente der Mathematik
 | mr = 2520469
 | pages = 89–101
 | title = An enduring error
 | 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
 | last = Malkevitch | first = Joseph
 | contribution = Milestones in the history of polyhedra
 | pages = 80–92
 | title = Shaping Space: A Polyhedral Approach
 | year = 1988
 | editor2-last = Fleck | editor2-first = G.
 | publisher = Birkhäuser
 | location = Boston
}}.

== General references ==
*{{cite journal
 | last = Jayatilake | first = Udaya
 | title = Calculations on face and vertex regular polyhedra
 | journal = Mathematical Gazette
 | date = March 2005
 | volume = 89
 | issue = 514
 | pages = 76–81
| doi = 10.1017/S0025557200176818
 | s2cid = 125675814
 }}.
* {{cite book
 | last = Pugh | first = Anthony
 | year= 1976
 | title= Polyhedra: A visual approach
 | publisher= University of California Press Berkeley
 | location= California
 | isbn= 0-520-03056-7
}} Chapter 2
* {{The Geometrical Foundation of Natural Structure (book)}} (Section 3–9)
*{{cite journal
 | last1 = Schreiber | first1 = Peter
 | last2 = Fischer | first2 = Gisela
 | last3 = Sternath | first3 = Maria Luise
 | title = New light on the rediscovery of the Archimedean solids during the renaissance
 | journal = Archive for History of Exact Sciences
 | issn = 0003-9519
 | volume = 62
 | issue = 4
 | pages = 457–467
 | year = 2008
 | doi = 10.1007/s00407-008-0024-z
| bibcode = 2008AHES...62..457S
 | s2cid = 122216140
 }}.

==External links==
* {{mathworld | urlname = ArchimedeanSolid | title = Archimedean solid }}
* [http://demonstrations.wolfram.com/ArchimedeanSolids/ Archimedean Solids] by [[Biography:Eric W. Weisstein|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]

{{Polyhedron navigator}}

{{DEFAULTSORT:Archimedean Solid}}
[[Category:Archimedean solids| ]]

{{Sourceattribution|Archimedean solid}}