---
tag: X7K75G78C8
kind: object
name: Archimedean solid
msc: [51M20, 52B10, 52B15]
instance_of: [0325SGSJAR]
uses: [01KJJ9A400, 0ZG3Z5JZ85]
sources:
  - url: https://en.wikipedia.org/wiki/Archimedean_solid
same_as:
  wikidata: Q213486
  wikipedia: Archimedean solid
---

## Description

An *Archimedean solid* is a convex [[polyhedron|0325SGSJAR]] with two properties.
Its faces are [[regular polygons|01KJJ9A400]] of at least two different kinds.
Its [[symmetry group|0ZG3Z5JZ85]] acts transitively on its vertices.
The prisms and antiprisms also have these properties and are excluded.
Up to similarity there are thirteen Archimedean solids.
Seven have the symmetry of the tetrahedron or the cube: the truncated tetrahedron, cuboctahedron, truncated cube, truncated octahedron, rhombicuboctahedron, truncated cuboctahedron and snub cube.
Six have the symmetry of the icosahedron: the icosidodecahedron, truncated dodecahedron, truncated icosahedron, rhombicosidodecahedron, truncated icosidodecahedron and snub dodecahedron.

## Remarks

The truncated tetrahedron has $4$ triangles and $4$ hexagons as faces, $12$ vertices and $18$ edges.
[[Euler's polyhedron formula|09HT3P35YP]] checks out: $12 - 18 + 8 = 2$.
The truncated icosahedron, the pattern of a classic football, has $12$ pentagons, $20$ hexagons, $60$ vertices and $90$ edges, and $60 - 90 + 32 = 2$.
The snub cube and the snub dodecahedron come in two mirror image forms, and some authors count these separately.
Asking only that all vertices look alike locally is weaker: the elongated square gyrobicupola satisfies that condition but is not vertex-transitive.
The [[Platonic solids|017E5PRS1Q]] are the vertex-transitive convex polyhedra with a single kind of regular face.

## Explanations

### L1 human

The Archimedean solids are the next most regular solids after the five Platonic solids.
Their faces are regular polygons of more than one shape, such as triangles and squares.
Every corner looks exactly like every other corner, and a rotation or reflection of the solid can move any corner to any other.