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Semiregular polyhedron
 
 
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A semiregular polyhedron is a polyhedron
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
 with regular
Regular polygon

A regular polygon is a polygon which is Equiangular polygon and equilateral . Regular polygons may be convex or Star polygon....
 faces and a symmetry group which is transitive on its vertices. Or at least, that is what follows from Thorold Gosset
Thorold Gosset

Thorold Gosset was an England lawyer and an amateur mathematician. In mathematics, he is noted for discovering and classifying the semiregular polytopes in dimensions four and higher....
's 1900 definition of the more general semiregular polytope
Polytope

In geometry, polytope is a generic term that can refer to a two-dimensional polygon, a three-dimensional polyhedron, or any of the various generalizations thereof, including generalizations to higher dimensions and other abstractions ....
. These polyhedra include:

These semiregular solids can be fully specified by a vertex configuration
Vertex configuration

In polyhedral geometry a vertex configuration is a short-hand notation for representing a polyhedron vertex figure as the sequence of faces around a vertex....
, 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
Icosidodecahedron

An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon....
 which alternates two triangle
Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....
s and two pentagon
Pentagon

In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540?....
s around each vertex.






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A semiregular polyhedron is a polyhedron
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
 with regular
Regular polygon

A regular polygon is a polygon which is Equiangular polygon and equilateral . Regular polygons may be convex or Star polygon....
 faces and a symmetry group which is transitive on its vertices. Or at least, that is what follows from Thorold Gosset
Thorold Gosset

Thorold Gosset was an England lawyer and an amateur mathematician. In mathematics, he is noted for discovering and classifying the semiregular polytopes in dimensions four and higher....
's 1900 definition of the more general semiregular polytope
Polytope

In geometry, polytope is a generic term that can refer to a two-dimensional polygon, a three-dimensional polyhedron, or any of the various generalizations thereof, including generalizations to higher dimensions and other abstractions ....
. These polyhedra include:
  • The thirteen Archimedean solid
    Archimedean solid

    In geometry an Archimedean solid is a highly symmetric, semi-regular convex set polyhedron composed of two or more types of regular polygons meeting in identical vertex ....
    s    .
  • An infinite series of convex prism
    Prism (geometry)

    In geometry, an n-sided prism is a polyhedron made of an n-sided polygon base, a Translation copy, and n faces joining corresponding sides....
    s    .
  • An infinite series of convex antiprism
    Antiprism

    An n-sided antiprism is a polyhedron composed of 2 parallel copies of some particular n-sided polygon, connected by an alternating band of triangles....
    s     (their semiregular nature was first observed by Kepler).


These semiregular solids can be fully specified by a vertex configuration
Vertex configuration

In polyhedral geometry a vertex configuration is a short-hand notation for representing a polyhedron vertex figure as the sequence of faces around a vertex....
, 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
Icosidodecahedron

An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon....
 which alternates two triangle
Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....
s and two pentagon
Pentagon

In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540?....
s around each vertex. 3.3.3.5 in contrast is a pentagonal antiprism
Pentagonal antiprism

In geometry, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps....
. These polyhedra are sometimes described as vertex-transitive
Vertex-transitive

In geometry, a polytope is isogonal or vertex-transitive if all its vertex are the same. That is, each vertex is surrounded by the same kinds of face in the same order, and with the same angles between corresponding faces....
.

Since Gosset, other authors have used the term semiregular in different ways. E. L. Elte provided a definition which Coxeter found too artificial. Coxeter himself dubbed Gosset's figures uniform
Uniform polyhedron

A Uniform polytope polyhedron is a polyhedron which has regular polygons as Face and is transitive on its vertex . It follows that all vertices are Congruence , and the polyhedron has a high degree of reflectional and rotational symmetry....
, 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
    Star polyhedron

    In geometry, a star polyhedron is a polyhedron which has some repetitive quality of nonconvex polygon giving it a star-like visual quality.There are two general kinds of star polyhedron:...
     which meet Gosset's definition, analogous to the three convex sets listed above.
  • The duals
    Dual polyhedron

    In geometry, polyhedron are associated into pairs called duals, where the wikt:vertex of one correspond to the face s of the other. The dual of the dual is the original polyhedron....
     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
    Catalan solid

    In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. The Catalan solids are named for the Belgium mathematician, Eug?ne Catalan, who first described them in 1865....
    s    , the convex dipyramids and antidipyramids or trapezohedra
    Trapezohedron

    The n-gonal trapezohedron, antidipyramid or deltohedron is the dual polyhedron of an n-gonal antiprism. Its 2n faces are congruent kite ....
    , and their nonconvex analogues.


A further source of confusion lies in the way that the Archimedean solid
Archimedean solid

In geometry an Archimedean solid is a highly symmetric, semi-regular convex set polyhedron composed of two or more types of regular polygons meeting in identical vertex ....
s are defined, again with different interpretations appearing.

Gosset's definition of semiregular includes figures of higher symmetry, the regular
Platonic solid

In geometry, a Platonic solid is a convex set polyhedron that is regular polyhedron, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruence regular polygons, with the same number of faces meeting at each vertex....
 and quasiregular
Quasiregular polyhedron

A polyhedron which has regular faces and is transitive on its edges but not transitive on its faces is said to be quasiregular.A quasiregular polyhedron can have faces of only two kinds and these must alternate around each vertex....
 polyhedra. Some later authors prefer to say that these are not semiregular, because they are more regular than that - the uniform polyhedra
Uniform polyhedron

A Uniform polytope polyhedron is a polyhedron which has regular polygons as Face and is transitive on its vertex . It follows that all vertices are Congruence , and the polyhedron has a high degree of reflectional and rotational symmetry....
 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
Archimedean solid

In geometry an Archimedean solid is a highly symmetric, semi-regular convex set polyhedron composed of two or more types of regular polygons meeting in identical vertex ....
, 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.

General remarks


In many works semiregular polyhedron is used as a synonym for Archimedean solid
Archimedean solid

In geometry an Archimedean solid is a highly symmetric, semi-regular convex set polyhedron composed of two or more types of regular polygons meeting in identical vertex ....
. For example Cundy & Rollett (1961).

We can distinguish between the facially-regular and vertex-transitive
Vertex-transitive

In geometry, a polytope is isogonal or vertex-transitive if all its vertex are the same. That is, each vertex is surrounded by the same kinds of face in the same order, and with the same angles between corresponding faces....
 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
Wythoff construction

File:Wythoffian_construction_diagram.pngIn geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling....
 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
Convex set

In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object....
 uniform polyhedra
Uniform polyhedron

A Uniform polytope polyhedron is a polyhedron which has regular polygons as Face and is transitive on its vertex . It follows that all vertices are Congruence , and the polyhedron has a high degree of reflectional and rotational symmetry....
 excluding the five regular polyhedra
Regular polyhedron

A regular polyhedron is a polyhedron whose faces are Congruence regular polygons which are assembled in the same way around each vertex. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive - i.e....
-- including the Archimedean solids, the uniform prisms
Prism (geometry)

In geometry, an n-sided prism is a polyhedron made of an n-sided polygon base, a Translation copy, and n faces joining corresponding sides....
, and the uniform antiprism
Antiprism

An n-sided antiprism is a polyhedron composed of 2 parallel copies of some particular n-sided polygon, connected by an alternating band of triangles....
s (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
Catalan solid

In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. The Catalan solids are named for the Belgium mathematician, Eug?ne Catalan, who first described them in 1865....
 (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.

External references