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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-Poinsot 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 groupThe symmetry group of a geometric figure is the group of congruencies under which it is invariant, with composition as the operation. The article on group theory also contains an explanation of the concept. In Euclidean geometry, discrete symmetry groups |
|---|---|---|---|---|---|
| cuboctahedronA cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it i () | 14 (8 triangles, 6 squareIn plane geometry, a square is a polygon with four equal sides and equal angles. Those angles are then necessarily right angles. Squares are regular quadrilaterals, rectangles, rhombi, kites, parallelograms, and isosceles trapezoids/isosceles trapezia.s) | 24 | 12 | triangle-square-triangle-square | Oh |
| icosidodecahedronAn icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a () | 32 (20 triangles, 12 pentagonThis is an article about the geometrical shape. See The Pentagon for an article about the building near Washington, DC. See also: Pentagon (disambiguation). In geometry, a pentagon is any five-sided polygon. However, the term is commonly used to mean a res) | 60 | 30 | triangle-pentagon-triangle-pentagon | Ih |
| truncated tetrahedronThe truncated tetrahedron is an Archimedean solid. Canonical coordinates for the vertices of a truncated tetrahedron centered at the origin are (±3, ±1, ±1), (±1, ±3, ±1), (±1, ±1, ±3), where the ± has the same parity for each coordinate, that is, all coo () | 8 (4 triangles, 4 hexagonA regular hexagon A hexagon (also known as "sexagon") is a polygon with six edges and six vertices. Its Schlafli symbol is {6}. The internal angles of a regular hexagon (one where all sides and all angles are equal) are all 120 °. Like squares and equilats) | 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 (6 squares, 8 hexagons) | 36 | 24 | square-hexagon-hexagon | Oh |
| truncated dodecahedron () | 32 (20 triangles, 12 decagons) | 90 | 60 | triangle-decagon-decagon | Ih |
| truncated icosahedron or commonly soccer ball () | 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 (2 chiral forms) () () | 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).
The duals of the Archimedean solids are called the Catalan solids. Together with the bipyramids and trapezohedra, these are the face-uniform solids with regular vertices.