An Archimedean solid is a convex polyhedron made from regular polygons in which every vertex has the same arrangement of faces. In more formal terms these shapes are part of geometry and are convex (see convex bodies) polyhedral figures built from regular polygons. They are a special class of polyhedron called uniform solids: faces are regular but may be of more than one type, while all vertices are equivalent under the symmetry group of the solid.

Defining properties

  • Each face is a regular polygon (triangles, squares, pentagons, etc.).
  • The figure is vertex-transitive: every corner has the same arrangement of faces around it.
  • The solid is convex and is neither one of the five Platonic solids, nor a simple prism or antiprism.
  • At least two different types of regular polygon appear among the faces (otherwise it would be Platonic).

Because of the vertex-transitive requirement, the local pattern of faces that meets at any vertex—called the vertex configuration—completely characterizes how the solid is built. Typical operations that produce Archimedean solids include truncation (cutting off vertices), rectification and alternation (creating snub forms), applied to Platonic or other regular polyhedra.

Common examples

The classical list contains thirteen distinct Archimedean solids if enantiomorphic (mirror-image) pairs are counted as one; counting both left- and right-handed forms raises the total to fifteen. Familiar examples include:

  • Cuboctahedron — alternating triangles and squares around each vertex.
  • Truncated tetrahedron — four triangular faces become hexagons and triangles remain.
  • Truncated cube and truncated octahedron — familiar truncations of Platonic solids.
  • Truncated dodecahedron and truncated icosahedron — the latter appears as the classical black-and-white soccer ball and as the C60 molecule in chemistry.
  • Rhombicuboctahedron and rhombicosidodecahedron — examples with three different face types at each vertex.
  • Truncated cuboctahedron and truncated icosidodecahedron — larger truncations with many faces.
  • Icosidodecahedron, snub cube and snub dodecahedron — the snub forms are chiral (left- and right-handed).

Each of these solids can be described by a vertex configuration such as (3,4,3,4) or (3,3,3,3,5) that lists the face sizes encountered around a vertex. The truncated icosahedron, for example, has five hexagons and twelve pentagons arranged so that every vertex joins two hexagons and one pentagon.

History and classification

The name honors the ancient mathematician Ancient Greek scholar mathematician Archimedes, who is credited with their discovery in antiquity. Archimedes' original treatise describing these solids has been lost; knowledge of his work survives through later summaries. In particular, the 4th-century scholar Pappus recorded descriptions that preserve the essential list. These forms were rediscovered and celebrated during the Renaissance by artists and mathematicians interested in symmetry and ideal form. The enumeration and study of the Archimedean solids were further developed by later figures such as Kepler and by modern geometers who placed them within the broader framework of uniform polyhedra.

Each Archimedean solid has a dual polyhedron called a Catalan solid; duals are face-transitive rather than vertex-transitive. Archimedean solids appear in practical and aesthetic contexts: architecture and sculpture exploit their symmetry, truncated icosahedral geometry models molecular cages like buckminsterfullerene (C60), and educational models help illustrate symmetry and group actions. Their construction by truncation and alternation also links them to operations used in polyhedral design and computer graphics.

Though the classical list is finite and well understood, Archimedean solids serve as an entry point to the larger family of uniform polyhedra (which include nonconvex and star-shaped forms) and to modern topics in geometric symmetry, tiling, and three-dimensional design.