Editing Semiregular polyhedron

{{Short description|Variously-defined concept in geometry}}
{| class="wikitable" width=320 align=right
|+ Semiregular polyhedra:<BR>[[Archimedean solid]]s, [[Prism (geometry)|prisms]], and [[antiprism]]s
|-
| [[Image:truncated tetrahedron.png|80px]]
| [[Image:cuboctahedron.png|80px]]
| [[Image:truncated hexahedron.png|80px]]
| [[Image:truncated octahedron.png|80px]]
|-
| [[Image:small rhombicuboctahedron.png|80px]]
| [[Image:Great rhombicuboctahedron.png|80px]]
| [[Image:snub hexahedron.png|80px]]
| [[Image:icosidodecahedron.png|80px]]
|-
| [[Image:truncated dodecahedron.png|80px]]
| [[Image:truncated icosahedron.png|90px]]
| [[Image:small rhombicosidodecahedron.png|80px]]
| [[Image:Great rhombicosidodecahedron.png|80px]]
|-
| [[Image:snub dodecahedron ccw.png|80px]]
| [[File:Triangular prism.png|80px]]
| [[File:Pentagonal prism.png|80px]]
| [[File:Hexagonal prism.png|80px]]
|-
| [[File:Prism 7.png|80px]]
| [[File:Square antiprism.png|80px]]
| [[File:Pentagonal antiprism.png|80px]]
| [[File:Hexagonal antiprism.png|80px]]
|}
In [[geometry]], the term '''semiregular polyhedron''' (or '''semiregular polytope''') is used variously by different authors.

== Definitions==
In its original definition, it is a [[polyhedron]] with [[regular polygon]]al faces, and a [[symmetry group]] which is [[Transitive action|transitive]] on its [[Vertex (geometry)|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]]).<ref>[[Thorold Gosset]] ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', [[Messenger of Mathematics]], Macmillan, 1900</ref><ref>[[Coxeter|Coxeter, H.S.M.]] ''Regular polytopes'', 3rd Edn, Dover (1973)</ref> These polyhedra include:
*The thirteen [[Archimedean solid]]s.
** The [[elongated square gyrobicupola]] (also called a pseudo-rhombicuboctahedron), a [[Johnson solid]], has identical [[vertex figure]]s (3.4.4.4) but because of a twist it is not [[vertex-transitive]].  [[Branko Grünbaum]] argued for including it as a 14th Archimedean solid.
*An infinite series of convex [[Prism (geometry)|prism]]s.
*An infinite series of convex [[antiprism]]s (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: {{math|3.5.3.5}} represents the [[icosidodecahedron]], which alternates two [[triangle]]s and two [[pentagon]]s around each vertex. In contrast: {{math|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]]<ref>{{citation | last = Elte | first = E. L. | title = The Semiregular Polytopes of the Hyperspaces | publisher = University of Groningen | location = Groningen | year = 1912}}</ref> provided a definition which [[Coxeter]] found too artificial. Coxeter himself dubbed Gosset's figures ''uniform'', with only a quite restricted subset classified as semiregular.<ref>[[Coxeter|Coxeter, H.S.M.]], [[Michael S. Longuet-Higgins|Longuet-Higgins, M.S.]] and Miller, J.C.P. Uniform Polyhedra, ''Philosophical Transactions of the Royal Society of London'' '''246 A''' (1954), pp. 401-450. ([https://www.jstor.org/stable/91532 JSTOR archive], subscription required).</ref>

Yet others have taken the opposite path, categorising more polyhedra as semiregular. These include:
*Three sets of [[star polyhedron|star polyhedra]] which meet Gosset's definition, analogous to the three convex sets listed above.
*The [[Dual polyhedron|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 solid]]s, the convex [[dipyramid]]s, and the convex antidipyramids or [[trapezohedron|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 [[Platonic solid|regular]] and [[Quasiregular polyhedron|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 most common failing. Coxeter, Cromwell,<ref>Cromwell, P. ''Polyhedra'', Cambridge University Press (1977)</ref> and Cundy & Rollett<ref>Cundy H.M and Rollett, A.P. ''[[Mathematical Models (Cundy and Rollett)|Mathematical models]]'', 2nd Edn. Oxford University Press (1961)</ref> are all guilty of such slips.

==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.

==See also==
* [[Semiregular polytope]]
* [[Regular polyhedron]]

==References==
<references />

== External links ==
* {{MathWorld | urlname=SemiregularPolyhedron | title=Semiregular polyhedron }}
* [http://www.georgehart.com/virtual-polyhedra/archimedean-info.html George Hart: Archimedean Semi-regular Polyhedra]
* [https://web.archive.org/web/20061209233653/http://daviddarling.info/encyclopedia/S/semi-regular_polyhedron.html David Darling: semi-regular polyhedron]
* [http://polyhedra.mathmos.net/entry/semiregularpolyhedron.html polyhedra.mathmos.net: Semi-Regular Polyhedron]
* [https://encyclopediaofmath.org/wiki/Semi-regular_polyhedra Encyclopaedia of Mathematics: Semi-regular polyhedra, uniform polyhedra, Archimedean solids]

[[Category:Polyhedra]]