Semiregular Polyhedron - General Remarks

General Remarks

In many works semiregular polyhedron is used as a synonym for Archimedean solid. For example Cundy & Rollett (1961).

We can distinguish between the facially-regular and vertex-transitive figures based on Gosset, and their vertically-regular (or versi-regular) and facially-transitive duals.

Coxeter et al. (1954) use the term semiregular polyhedra to classify uniform polyhedra with Wythoff symbol of the form p q | r, a definition encompassing only six of the Archimedean solids, as well as the regular prisms (but not the regular antiprisms) and numerous nonconvex solids. Later, Coxeter (1973) would quote Gosset's definition without comment, thus accepting it by implication.

Eric Weisstein, Robert Williams and others use the term to mean the convex uniform polyhedra excluding the five regular polyhedra – including the Archimedean solids, the uniform prisms, and the uniform antiprisms (overlapping with the cube as a prism and regular octahedron as an antiprism).

Peter Cromwell (1997) writes in a footnote to Page 149 that, "in current terminology, 'semiregular polyhedra' refers to the Archimedean and Catalan (Archimedean dual) solids". On Page 80 he describes the thirteen Archimedeans as semiregular, while on Pages 367 ff. he discusses the Catalans and their relationship to the 'semiregular' Archimedeans. By implication this treats the Catalans as not semiregular, thus effectively contradicting (or at least confusing) the definition he provided in the earlier footnote. He ignores nonconvex polyhedra.