Math 2.0

Geometry Polygons, polyhedra and solid geometry Object

Archimedean solid

Builds on Regular polygonSymmetry group

Description

An Archimedean solid is a convex polyhedron with two properties. Its faces are regular polygons of at least two different kinds. Its symmetry group 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 44 triangles and 44 hexagons as faces, 1212 vertices and 1818 edges. Euler's polyhedron formula checks out: 12−18+8=212 - 18 + 8 = 2. The truncated icosahedron, the pattern of a classic football, has 1212 pentagons, 2020 hexagons, 6060 vertices and 9090 edges, and 60−90+32=260 - 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 are the vertex-transitive convex polyhedra with a single kind of regular face.

Reading path

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Regular polygon, concept, no evidence levelRegular polygonconceptSymmetry group, concept, no evidence levelSymmetry groupconceptArchimedean solid, object, no evidence levelArchimedean solidobject
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  1. Regular polygon concept
  2. Symmetry group concept
  3. Archimedean solid object

Further reading

  1. Web https://en.wikipedia.org/wiki/Archimedean_solid
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Kindobject
WritingW0 stub
MSC51M20, 52B10, 52B15
Instance ofPolyhedron
UsesRegular polygon, Symmetry group
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