Semiregular polyhedron

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.

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Uniform polyhedron

In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent. Uniform polyhedra may be regular (if also face- and edge-transitive), quasi-regular (if also edge-transitive but not face-transitive), or semi-regular (if neither edge- nor face-transitive). The faces and vertices need not be convex, so many of the uniform polyhedra are also star polyhedra.

Elongated square gyrobicupola

In geometry, the elongated square gyrobicupola or pseudo-rhombicuboctahedron is one of the Johnson solids (J_37). It is not usually considered to be an Archimedean solid, even though its faces consist of regular polygons that meet in the same pattern at each of its vertices, because unlike the 13 Archimedean solids, it lacks a set of global symmetries that map every vertex to every other vertex (though Grünbaum has suggested it should be added to the traditional list of Archimedean solids as a 14th example).

Johnson solid

In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johnson solid is the square-based pyramid with equilateral sides (J_1); it has 1 square face and 4 triangular faces. Some authors require that the solid not be uniform (i.e., not Platonic solid, Archimedean solid, uniform prism, or uniform antiprism) before they refer to it as a "Johnson solid".

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