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Archimedean solid

An Archimedean solid is a semi-regular (ie vertex-uniform, but not face-uniform) convex polyhedron with regular polygons for faces. Compare to Platonic solids, which are face-uniform, and Johnson solids, which need not be vertex-uniform. The prisms and antiprisms, though they meet the above criteria, are typically excluded from the Archimedean solids because they do not have a higher polyhedral symmetry.

These solids were known to be 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.

There are thirteen Archimedean solids, (ignoring that two are enantiomorphs).

  1. cuboctahedron (with a triangle, a square, a triangle, and a square in sequence at each vertex)
  2. icosidodecahedron (with a triangle, a pentagon, a triangle, and a pentagon in sequence at each vertex)
  3. truncated tetrahedron[?] (with a triangle, a hexagon, and a hexagon in sequence at each vertex)
  4. truncated octahedron[?] (with a square, a hexagon, and a hexagon in sequence at each vertex)
  5. truncated cube[?] (with a triangle, an octagon, and an octagon in sequence at each vertex)
  6. truncated icosahedron (with a pentagon, a hexagon,and a hexagon in sequence at each vertex)
  7. truncated dodecahedron[?] (with a triangle, a degagon, and a degagon in sequence at each vertex)
  8. rhombicuboctahedron, also named the small rhombicuboctahedron (with a triangle, a square, a square, and a square in sequence at each vertex)
  9. truncated cuboctahedron[?], also named the great rhombicuboctahedron (with a square, a hexagon, and an octagon in sequence at each vertex)
  10. rhombicosidodecahedron[?], also named the small rhombicosidodecahedron (with a triangle, a square, a pentagon, and a square in sequence at each vertex)
  11. truncated icosidodecahedron[?], also named the great rhombicosidodecahedron (with a square, a hexagon, and a degagon in sequence at each vertex)
  12. snub cube, also named the snub cuboctahedron (with a triangle, a triangle, a triangle, a triangle, and a square in sequence at each vertex)
  13. snub dodecahedron[?], also named the snub icosidodecahedron (with a triangle, a triangle, a triangle, a triangle, and a pentagon in sequence at each vertex)

As a reminder, a triangle has three sides, a square has four sides, a pentagon has five sides, a hexagon has six sides, an octagon has eight sides, and a decagon has ten sides. 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 based on rotation, these forms may be called enantiomorphs. (This nomeclature is also used for the forms of chemical compounds[?]). These forms are similar to reflections in a mirror, but are actual three-dimensional shapes.

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.