Editing Semiregular polyhedron (section)

==General remarks==
{| class="wikitable"  align=right width=240
|+ Rhombic semiregular polyhedra (Kepler)
|- align=center
|[[File:TrigonalTrapezohedron.svg|35px]]<BR>[[Trigonal trapezohedron]]<BR>(V(3.3)<sup>2</sup>)
|[[File:Rhombicdodecahedron.jpg|95px]]<BR>[[Rhombic dodecahedron]]<BR>[[Face_configuration|V(3.4)<sup>2</sup>]]
|[[File:Rhombictriacontahedron.svg|80px]]<BR>[[Rhombic triacontahedron]]<BR>V(3.5)<sup>2</sup>
|}

[[Johannes Kepler]] coined the category semiregular in his book ''[[Harmonice Mundi]]'' (1619), including the 13 [[Archimedean solid]]s, two infinite families ([[Prism (geometry)|prisms]] and [[antiprism]]s on regular bases), and two [[edge-transitive]] [[Catalan solid]]s, the [[rhombic dodecahedron]] and [[rhombic triacontahedron]]. He also considered a [[rhombus]] as a semiregular polygon (being equilateral and alternating two angles) as well as [[Star_polygon#Simple_isotoxal_star_polygons|star polygon]]s, now called [[isotoxal figure]]s which he used in [[Star_polygon#Examples_in_tilings|planar tilings]]. The [[trigonal trapezohedron]], a topological [[cube]] with congruent rhombic faces, would also qualify as semiregular, though Kepler did not mention it specifically.

In many works ''semiregular polyhedron'' is used as a synonym for [[Archimedean solid]].<ref>"Archimedes". (2006). In ''Encyclopædia Britannica''.  Retrieved [[19 Dec]] 2006, from [http://www.search.eb.com/eb/article-21480 Encyclopædia Britannica Online] (subscription required).</ref> 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 construction|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 (geometer)|Robert Williams]] and others use the term to mean the [[Convex polytope|convex]] [[Uniform polyhedron|uniform polyhedra]] excluding the five [[regular polyhedron|regular polyhedra]] – including the Archimedean solids, the uniform [[prism (geometry)|prisms]], and the uniform [[antiprism]]s (overlapping with the cube as a prism and regular octahedron as an antiprism).<ref>{{MathWorld | urlname=SemiregularPolyhedron | title=Semiregular polyhedron}} The definition here does not exclude the case of all faces being congruent, but the [[Platonic solid]]s are not included in the article's enumeration.</ref><ref>{{The Geometrical Foundation of Natural Structure (book)}} (Chapter 3: Polyhedra)</ref>

Peter Cromwell (1997) writes in a footnote to Page 149 that, "in current terminology, 'semiregular polyhedra' refers to the Archimedean and [[Catalan solid|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.