The Archimedean solids are known to have been discussed by Archimedes, although the complete record is lost. During the Renaissance, artists and mathematicians valued pure forms and rediscovered all of these forms. This search culminated in the work of Johannes Kepler circa 1619, who defined prisms, antiprisms, and the non-convex solids known as Kepler solids.
All edges of an Archimedean solid have the same length, since the faces are regular polygons, and the edges of a regular polygon have the same length. The neighbours of a polygon must have the same edge length, therefore also the neighbours of the neighbours, and so on.
There are 13 Archimedean solids (15 if the mirror images of two enantiomorphs, see below, are counted separately).
|Name and picture||Faces||Edges||Vertices||Faces meeting at each vertex||Symmetry group|
|cuboctahedron||14 (8 triangles, 6 squares)||24||12||triangle-square-triangle-square||Oh|
|icosidodecahedron||32 (20 triangles, 12 pentagons)||60||30||triangle-pentagon-triangle-pentagon||Ih|
|truncated tetrahedron||8 (4 triangles, 4 hexagons)||18||12||triangle-hexagon-hexagon||Td|
|truncated cube or truncated hexahedron||14 (8 triangles, 6 octagons)||36||24||triangle-octagon-octagon||Oh|
|truncated octahedron||14 (8 squares, 6 hexagons)||36||24||square-hexagon-hexagon||Oh|
|truncated dodecahedron||32 (20 triangles, 12 decagons)||90||60||triangle-decagon-decagon||Ih|
|truncated icosahedron||32 (12 pentagons, 20 hexagons)||90||60||pentagon-hexagon-hexagon||Ih|
|rhombicuboctahedron or small rhombicuboctahedron||26 (8 triangles, 18 squares)||48||24||triangle-square-square-square||Oh|
|truncated cuboctahedron or great rhombicuboctahedron||26 (12 squares, 8 hexagons, 6 octagons)||72||48||square-hexagon-octagon||Oh|
|rhombicosidodecahedron or small rhombicosidodecahedron||62 (20 triangles, 30 squares, 12 pentagons)||120||60||triangle-square-pentagon-square||Ih|
|truncated icosidodecahedron or great rhombicosidodecahedron||62 (30 squares, 20 hexagons, 12 decagons)||180||120||square-hexagon-decagon||Ih|
|snub cube or snub cuboctahedron (2 chiral forms)||38 (32 triangles, 6 squares)||60||24||triangle-triangle-triangle-triangle-square||O|
|snub dodecahedron or snub icosidodecahedron||92 (80 triangles, 12 pentagons)||150||60||triangle-triangle-triangle-triangle-pentagon||I|
The first two solids (cuboctahedron and icosidodecahedron) are edge-uniform and are called quasi-regular.
The last two (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 chemical compounds).