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Concept
Semiregular polyhedron
Summary
In geometry, the term semiregular polyhedron (or semiregular polytope) is used variously by different authors.
In its original definition, it is a polyhedron with regular polygonal faces, and a symmetry group which is transitive on its vertices; today, this is more commonly referred to as a uniform polyhedron (this follows from Thorold Gosset's 1900 definition of the more general semiregular polytope). These polyhedra include:
The thirteen Archimedean solids.
The elongated square gyrobicupola, also called a pseudo-rhombicuboctahedron, a Johnson solid, has identical vertex figures 3.4.4.4, but is not vertex-transitive including a twist has been argued for inclusion as a 14th Archimedean solid by Branko Grünbaum.
An infinite series of convex prisms.
An infinite series of convex antiprisms (their semiregular nature was first observed by Kepler).
These semiregular solids can be fully specified by a vertex configuration: a listing of the faces by number of sides, in order as they occur around a vertex. For example: 3.5.3.5 represents the icosidodecahedron, which alternates two triangles and two pentagons around each vertex. In contrast: 3.3.3.5 is a pentagonal antiprism. These polyhedra are sometimes described as vertex-transitive.
Since Gosset, other authors have used the term semiregular in different ways in relation to higher dimensional polytopes. E. L. Elte provided a definition which Coxeter found too artificial. Coxeter himself dubbed Gosset's figures uniform, with only a quite restricted subset classified as semiregular.
Yet others have taken the opposite path, categorising more polyhedra as semiregular. These include:
Three sets of star polyhedra which meet Gosset's definition, analogous to the three convex sets listed above.
The duals of the above semiregular solids, arguing that since the dual polyhedra share the same symmetries as the originals, they too should be regarded as semiregular. These duals include the Catalan solids, the convex dipyramids, and the convex antidipyramids or trapezohedra, and their nonconvex analogues.
Official source
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