In [[geometry]] an '''Archimedean solid''' is a highly symmetric, semi-regular [[convex set|convex]] [[polyhedron]] composed of two or more types of [[regular polygon]]s meeting in identical [[wikt:vertex|vertices]]. They are distinct from the [[Platonic solid]]s, which are composed of only one type of polygon meeting in identical vertices, and from the [[Johnson solid]]s, whose regular polygonal faces do not meet in identical vertices. The symmetry of the Archimedean solids excludes the members of the [[dihedral group]], the [[prism (geometry)|prisms]] and [[antiprism]]s.
The '''Archimedean solids''' can all be made via [[Wythoff construction]]s from the [[Platonic solids]] with [[Tetrahedral symmetry|tetrahedral]], [[Octahedral symmetry|octahedral]] and [[Icosahedral symmetry|icosahedral]] symmetry. See [[Uniform polyhedron#Convex forms and fundamental vertex arrangements|Convex uniform polyhedron]].
==Origin of name==
The Archimedean solids take their name from [[Archimedes]], who discussed them in a now-lost work. During the [[Renaissance]], [[artist]]s and [[mathematician]]s valued ''pure forms'' and rediscovered all of these forms. This search was completed around [[1619]] by [[Johannes Kepler]], who defined prisms, antiprisms, and the non-convex solids known as the [[Kepler-Poinsot polyhedra]].
==Classification==
There are 13 Archimedean solids (15 if the [[mirror image]]s of two [[chirality (mathematics)|enantiomorphs]], see below, 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 of (4,6,8) means that a square, hexagon, and octagon meet at a vertex (with the order taken to be clockwise around the vertex).
The number of vertices is 720° divided by the vertex [[Defect (geometry)|angle defect]].
{| class="wikitable sortable" style="text-align:center"
|-
! Name<BR>([[Vertex configuration]])
! Transparent
! Solid
! [[Net (polyhedron)|Net]]
! Faces
! Faces<BR>(By type)
! Edges
! Vertices
! [[List of spherical symmetry groups|Symmetry group]]
|-
| [[truncated tetrahedron]]<BR>(3.6.6)
| [[image:truncatedtetrahedron.jpg|60px|Truncated tetrahedron]]<br /><small>([[:image:truncatedtetrahedron.gif|Animation]])</small>
| [[image:truncated_tetrahedron.png|80px]]
| [[image:truncated_tetrahedron_flat.svg|80px]]
| 8
| 4 triangles<br>4 [[hexagon]]s
| 18
| 12
| T<sub>d</sub>
|-
| [[cuboctahedron]]<BR>(3.4.3.4)
| [[image:cuboctahedron.svg|60px|Cuboctahedron]]<br /><small>([[:image:cuboctahedron.gif|Animation]])</small>
| [[image:cuboctahedron.png|80px]]
| [[image:cuboctahedron_flat.svg|80px]]
| 14
| 8 [[triangle (geometry)|triangle]]s<br>6 [[square (geometry)|squares]]
| 24
| 12
| O<sub>h</sub>
|-
| [[truncated cube]]<br />or truncated hexahedron<BR>(3.8.8)
| [[image:truncatedhexahedron.jpg|60px|Truncated hexahedron]]<br /><small>([[:image:truncatedhexahedron.gif|Animation]])</small>
| [[image:truncated_hexahedron.png|80px]]
| [[image:truncated_hexahedron_flat.svg|80px]]
| 14
| 8 triangles<br>6 [[octagon]]s
| 36
| 24
| O<sub>h</sub>
|-
| [[truncated octahedron]]<BR>(4.6.6)
| [[image:truncatedoctahedron.jpg|60px|Truncated octahedron]]<br /><small>
([[:image:truncatedoctahedron.gif|Animation]])</small>
| [[image:truncated_octahedron.png|80px]]
| [[image:truncated_octahedron_flat.png|80px]]
| 14
| 6 squares<br>8 hexagons
| 36
| 24
| O<sub>h</sub>
|-
| [[rhombicuboctahedron]]<br />or small rhombicuboctahedron<BR>(3.4.4.4 )
| [[image:rhombicuboctahedron.jpg|60px|Rhombicuboctahedron]]<br /><small>([[:image:rhombicuboctahedron.gif|Animation]])</small>
| [[image:small_rhombicuboctahedron.png|80px]]
| [[image:Rhombicuboctahedron_flat.png|80px]]
| 26
|8 triangles<br>18 squares
| 48
| 24
| O<sub>h</sub>
|-
| [[truncated cuboctahedron]]<br />or great rhombicuboctahedron<BR>(4.6.8)
| [[image:truncatedcuboctahedron.jpg|60px|Truncated cuboctahedron]]<br /><small>([[:image:truncatedcuboctahedron.gif|Animation]])</small>
| [[image:Great rhombicuboctahedron.png|80px]]
| [[image:Truncated_cuboctahedron_flat.svg|80px]]
| 26
| 12 squares<br>8 hexagons<br>6 octagons
| 72
| 48
| O<sub>h</sub>
|-
| [[snub cube]]<br />or snub hexahedron<br> or snub cuboctahedron<br>(2 [[chirality (mathematics)|chiral]] forms)<BR>(3.3.3.3.4)
| [[image:snubhexahedronccw.jpg|60px|Snub hexahedron (Ccw)]]<br /><small>([[:image:snubhexahedronccw.gif|Animation]])</small><br />[[image:snubhexahedroncw.jpg|60px|Snub hexahedron (Cw)]]<br /><small>([[:image:snubhexahedroncw.gif|Animation]])</small>
| [[image:snub_hexahedron.png|80px]]
| [[image:snub_cube_flat.png|80px]]
| 38
|32 triangles<br>6 squares
| 60
| 24
| O
|-
| [[icosidodecahedron]]<BR>(3.5.3.5)
| [[image:icosidodecahedron.jpg|60px|Icosidodecahedron]]<br /><small>([[:image:icosidodecahedron.gif|Animation]])</small>
| [[image:icosidodecahedron.png|80px]]
| [[image:icosidodecahedron_flat.png|80px]]
| 32
| 20 triangles<br>12 [[pentagon]]s
| 60
| 30
| I<sub>h</sub>
|-
| [[truncated dodecahedron]]<BR>(3.10.10)
| [[image:truncateddodecahedron.jpg|60px|Truncated dodecahedron]]<br /><small>([[:image:truncateddodecahedron.gif|Animation]])</small>
| [[image:truncated_dodecahedron.png|80px]]
| [[image:truncated_dodecahedron_flat.png|80px]]
| 32
|20 triangles<br>12 [[decagon]]s
| 90
| 60
| I<sub>h</sub>
|-
| [[truncated icosahedron]]<br />or [[Fullerene|buckyball]]<BR>or [[Football (ball)|football]]/soccer ball<BR>(5.6.6 )
| [[image:truncatedicosahedron.jpg|60px|Truncated icosahedron]]<br /><small>([[:image:truncatedicosahedron.gif|Animation]])</small>
| [[image:truncated_icosahedron.png|90px]]
| [[image:truncated_icosahedron_flat.png|90px]]
| 32
| 12 pentagons<br>20 hexagons
| 90
| 60
| I<sub>h</sub>
|-
| [[rhombicosidodecahedron]]<br />or small rhombicosidodecahedron<BR>(3.4.5.4)
| [[image:rhombicosidodecahedron.jpg|60px|Rhombicosidodecahedron]]<br /><small>([[:image:rhombicosidodecahedron.gif|Animation]])</small>
| [[image:small rhombicosidodecahedron.png|80px]]
| [[image:Rhombicosidodecahedron_flat.png|80px]]
| 62
| 20 triangles<br>30 squares<br>12 pentagons
| 120
| 60
| I<sub>h</sub>
|-
| [[truncated icosidodecahedron]]<br />or great rhombicosidodecahedron<BR>(4.6.10)
| [[image:truncatedicosidodecahedron.jpg|60px|Truncated icosidodecahedron]]<br /><small>([[:image:truncatedicosidodecahedron.gif|Animation]])</small>
| [[image:Great rhombicosidodecahedron.png|80px]]
| [[image:Truncated_icosidodecahedron_flat.png|80px]]
| 62
|30 squares<br>20 hexagons<br>12 decagons
| 180
| 120
| I<sub>h</sub>
|-
| [[snub dodecahedron]]<br />or snub icosidodecahedron<br>(2 [[chirality (mathematics)|chiral]] forms)<BR>(3.3.3.3.5)
| [[image:snubdodecahedronccw.jpg|60px|Snub dodecahedron (Ccw)]]<br /><small>([[:image:snubdodecahedronccw.gif|Animation]])</small><br />[[image:snubdodecahedroncw.jpg|60px|Snub dodecahedron (Cw)]]<br><small>([[:image:snubdodecahedroncw.gif|Animation]])</small>
| [[image:snub_dodecahedron_ccw.png|80px]]
| [[image:snub_dodecahedron_flat.svg|80px]]
| 92
| 80 triangles<br>12 pentagons
| 150
| 60
| I
|}
The cuboctahedron and icosidodecahedron are edge-uniform and are called [[quasiregular polyhedron|quasi-regular]].
The snub cube and snub dodecahedron are known as ''chiral'', as they come in a left-handed (Latin: levomorph or laevomorph) form and right-handed (Latin: dextromorph) form. When something comes in multiple forms which are each other's three-dimensional [[mirror image]], these forms may be called enantiomorphs. (This nomenclature is also used for the forms of certain [[chemical compound]]s).
The [[dual polyhedron|duals]] of the Archimedean solids are called the [[Catalan solid]]s. Together with the [[bipyramid]]s and [[trapezohedron|trapezohedra]], these are the face-uniform solids with regular vertices.
== See also ==
* [[semiregular polyhedron]]
* [[uniform polyhedron]]
* [[List of uniform polyhedra]]
== References ==
* {{cite book | first=Robert | last=Williams | authorlink=Robert Williams | title=The Geometrical Foundation of Natural Structure: A Source Book of Design | publisher=Dover Publications, Inc | year=1979 | id=ISBN 0-486-23729-X }} (Section 3-9)
==External links==
* {{mathworld | urlname = ArchimedeanSolid | title = Archimedean solid }}
*[http://www.software3d.com/Archimedean.html Paper models of Archimedean solids]
*[http://www.korthalsaltes.com/archimedean_solids_pictures.html Free paper models(nets) of Archimedean solids]
*[http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
*[http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] The Encyclopedia of Polyhedra
*[http://www.cs.utk.edu/~plank/plank/origami/penultimate/intro.html Penultimate Modular Origami]
*[http://ibiblio.org/e-notes/3Dapp/Convex.htm Interactive 3D polyhedra] in Java
*[http://video.google.com/videoplay?docid=7084140981126344386&q=tom+barber&hl=en Contemporary Archimedean Solid Surfaces] Designed by [[Tom Barber]]
[[Category:Archimedean solids]]
[[Category:Polyhedra]]
[[bg:Архимедово тяло]]
[[de:Archimedischer Körper]]
[[eo:Arĥimeda solido]]
[[es:Sólidos arquimedianos]]
[[fr:Solide d'Archimède]]
[[he:פאון ארכימדי]]
[[it:Solido archimedeo]]
[[mk:Архимедови тела]]
[[pt:Sólidos de Arquimedes]]
[[fi:Arkhimedeen kappaleet]]
[[zh:半正多面體]]