Semiregular Polyhedron

The term semiregular polyhedron (or semiregular polytope) is used variously by different authors.

In its original definition, it is a polyhedron with regular faces and a symmetry group which is transitive on its vertices, which is more commonly referred to today 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.
• 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. 3.3.3.5 in contrast 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 antidipyramids or trapezohedra, and their nonconvex analogues.

A further source of confusion lies in the way that the Archimedean solids are defined, again with different interpretations appearing.

Gosset's definition of semiregular includes figures of higher symmetry, the regular and quasiregular polyhedra. Some later authors prefer to say that these are not semiregular, because they are more regular than that - the uniform polyhedra are then said to include the regular, quasiregular and semiregular ones. This naming system works well, and reconciles many (but by no means all) of the confusions.

In practice even the most eminent authorities can get themselves confused, defining a given set of polyhedra as semiregular and/or Archimedean, and then assuming (or even stating) a different set in subsequent discussions. Assuming that one's stated definition applies only to convex polyhedra is probably the commonest failing. Coxeter, Cromwell and Cundy & Rollett are all guilty of such slips.