Semiregular Polyhedron
   HOME

TheInfoList



OR:

In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, the term semiregular polyhedron (or semiregular polytope) is used variously by different authors.


Definitions

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: represents the icosidodecahedron, which alternates two
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colli ...
s and two
pentagon In geometry, a pentagon (from the Greek language, Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is ...
s around each vertex. 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 Emanuel Lodewijk Elte (16 March 1881 in Amsterdam – 9 April 1943 in Sobibór) Emanuël Lodewijk 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
dipyramid A (symmetric) -gonal bipyramid or dipyramid is a polyhedron formed by joining an -gonal pyramid and its mirror image base-to-base. An -gonal bipyramid has triangle faces, edges, and vertices. The "-gonal" in the name of a bipyramid does not ...
s, and the convex 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 The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instrum ...
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 most common failing. Coxeter, Cromwell, and Cundy & Rollett are all guilty of such slips.


General remarks

Johannes Kepler coined the category semiregular in his book 1619, Harmonices Mundi, includes the 13 Archimedean solids as well as the rhombic dodecahedron and rhombic triacontahedron (two Catalan solids). He also considered a
rhombus In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. Th ...
as semiregular, being equilateral and alternating two angles. Concave star polygons are also included, now called isotoxal figures which he used in planar tilings. Kepler didn't mention specifically, but the trigonal trapezohedron, a topological
cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the on ...
with congruent rhombic faces, would also qualify as semiregular. 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 Robert, Rob, Robbie, Bob or Bobby Williams may refer to: Entertainment Film * Robert Williams (actor, born 1894) (1894–1931), American stage and film actor * Robert B. Williams (actor) (1904–1978), American film actor * R. J. Williams (born ...
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). (Chapter 3: Polyhedra) Peter Cromwell (1997) writes in a footnote to Page 149 that, "in current terminology, 'semiregular polyhedra' refers to the Archimedean and
Catalan Catalan may refer to: Catalonia From, or related to Catalonia: * Catalan language, a Romance language * Catalans, an ethnic group formed by the people from, or with origins in, Northern or southern Catalonia Places * 13178 Catalan, asteroid ...
(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


External links

* {{MathWorld , urlname=SemiregularPolyhedron , title=Semiregular polyhedron
George Hart: Archimedean Semi-regular Polyhedra





Encyclopaedia of Mathematics: Semi-regular polyhedra, uniform polyhedra, Archimedean solids
Polyhedra