Quasiregular polyhedron

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Quasiregular polyhedron

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....
which has regular faces and is transitive on its edges but not transitive on its faces is said to be quasiregular.

A 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....
can have faces of only two kinds and these must alternate around each vertex.

They are given a vertical Schläfli symbol

Schläfli symbol

In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....
to represent this combined form which contains the combined faces of the regular and dual . A quasiregular polyhedron with this symbol will have 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....
p.q.p.q.

The Coxeter-Dynkin diagram

Coxeter-Dynkin diagram

In geometry, a Coxeter-Dynkin diagram is a Graph with labelled edges. It represents the spatial relations between a collection of mirrors , and describes a Kaleidoscope construction....
is another symbolic representation that shows the quasiregular relation between the two dual-regular forms:

The cuboctahedron

Cuboctahedron

In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....
, vertex configuration 3.4.3.4, Coxeter-Dynkin diagram

Coxeter-Dynkin diagram

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....
, vertex configuration 3.5.3.5, Coxeter-Dynkin diagram In addition, the octahedron

Octahedron

An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....
, which is also regular

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....
, , vertex configuration 3.3.3.3, can be considered quasiregular if alternate faces are given different colors.

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Quasiregular polyhedron

Schläfli symbol

Vertex configuration

Coxeter-Dynkin diagram

: * : * p.q.p.q: .

The convex quasiregular polyhedra

There are two convex

Convex

The word convex means curving out or bulging outward.Convex or convexity may refer to:Mathematics:* Convex set, a set of points containing all line segments between each pair of its points...
quasiregular polyhedra:

The cuboctahedron
Cuboctahedron

In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....
, vertex configuration 3.4.3.4, Coxeter-Dynkin diagram
Coxeter-Dynkin diagram

In geometry, a Coxeter-Dynkin diagram is a Graph with labelled edges. It represents the spatial relations between a collection of mirrors , and describes a Kaleidoscope construction....
#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....
, vertex configuration 3.5.3.5, Coxeter-Dynkin diagram In addition, the octahedron
Octahedron

An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....
, which is also regular
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....
, , vertex configuration 3.3.3.3, can be considered quasiregular if alternate faces are given different colors. The remaining regular polyhedra have an odd number of faces at each vertex so cannot be colored in a way that preserves edge transitivity. It has Coxeter-Dynkin diagram Each of these forms the common core of a dual
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....
pair of 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....
. The names of two of these give clues to the associated dual pair, respectively the cube
Cube

A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....
+ octahedron
Octahedron

An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....
and the icosahedron
Icosahedron

In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....
+ dodecahedron
Dodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
. The octahedron
Octahedron

An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....
is the core of a dual pair of tetrahedra
Tetrahedron

A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....
(an arrangement known as the stella octangula
Stella octangula

The stella octangula, also known as the stellated octahedron, Star Tetrahedron, eight-pointed star, or 2D geometric model as the Star of David....
), and when derived in this way is sometimes called the tetratetrahedron.

Regular	Dual regular	Quasiregular	Vertex figure Vertex figure In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off....
Tetrahedron Tetrahedron A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....	Tetrahedron Tetrahedron A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....	Tetratetrahedron (Octahedron) Octahedron An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....	3.3.3.3
Cube Cube A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....	Octahedron Octahedron An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....	Cuboctahedron Cuboctahedron In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....	3.4.3.4
Dodecahedron Dodecahedron A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....	Icosahedron Icosahedron In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....	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....	3.5.3.5

Each of these quasiregular polyhedra can be constructed by a rectification

Rectification (geometry)

In Euclidean geometry, rectification is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points....
operation on either regular parent, truncating

Truncation (geometry)

In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new Facet in place of each vertex....
the edges fully, until the original edges are reduced to a point.

Nonconvex examples

Coxeter, H.S.M. et.al. (1954) also classify certain 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:...
having the same characteristics as being quasiregular:

Two are based on dual pairs of regular Kepler-Poinsot solid

Kepler-Poinsot solid

The Kepler-Poinsot polyhedra are the four Regular polyhedron Star polyhedron. They may be obtained by stellation the regular convex or Platonic solids, and differ from these in having regular star polygons for their faces or vertex figures....
s, in the same way as for the convex examples.

The great icosidodecahedron

Great icosidodecahedron

In geometry, the great icosidodecahedron is a nonconvex uniform polyhedron, indexed as U54.This polyhedron can be considered a Rectification great icosahedron....
and the dodecadodecahedron

Dodecadodecahedron

In geometry, the dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U36.This polyhedron can be considered a Rectification great dodecahedron....
:

Regular	Dual regular	Quasiregular	Vertex figure Vertex figure In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off....
great stellated dodecahedron Great stellated dodecahedron In geometry, the great stellated dodecahedron is a Kepler-Poinsot polyhedra. It is one of four nonconvex regular polyhedra.It is composed of 12 intersecting pentagrammic faces, with three pentagrams meeting at each vertex....	great icosahedron Great icosahedron In geometry, the great icosahedron is a Kepler-Poinsot polyhedra. It is one of four nonconvex regular polyhedra. It is composed of 20 intersecting triangular faces, with five triangles meeting at each vertex in a pentagrammic sequence....	Great icosidodecahedron Great icosidodecahedron In geometry, the great icosidodecahedron is a nonconvex uniform polyhedron, indexed as U54.This polyhedron can be considered a Rectification great icosahedron....	3.⁵/₂.3.⁵/₂
Small stellated dodecahedron Small stellated dodecahedron In geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedra. It is one of four nonconvex regular polyhedra. It is composed of 12 pentagrammic faces, with five pentagrams meeting at each vertex....	Great dodecahedron Great dodecahedron In geometry, the great dodecahedron is a Kepler-Poinsot polyhedra. It is one of four nonconvex regular polyhedra. It is composed of 12 pentagonal faces , with five pentagons meeting at each vertex, intersecting each other making a pentagrammic path....	Dodecadodecahedron Dodecadodecahedron In geometry, the dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U36.This polyhedron can be considered a Rectification great dodecahedron....	5.⁵/₂.5.⁵/₂

Three ditrigonal forms, whose vertex figures contain three alternatations of the two face types:
1. Ditrigonal dodecadodecahedron
  Ditrigonal dodecadodecahedron
  
  In geometry, the Ditrigonal dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U41.It shares its vertex arrangement with the regular dodecahedron....
2. Small ditrigonal icosidodecahedron
  Small ditrigonal icosidodecahedron
  
  In geometry, the small ditrigonal icosidodecahedron is a nonconvex uniform polyhedron, indexed as U30.It shares the vertex arrangement with the regular dodecahedron....
3. Great ditrigonal icosidodecahedron
  Great ditrigonal icosidodecahedron
  
  In geometry, the great ditrigonal icosidodecahedron is a nonconvex uniform polyhedron, indexed as U47.It shares the vertex arrangement with the regular dodecahedron, which is therefore its convex hull....

Quasiregular duals

Some authorities argue that, since the duals of the quasiregular solids share the same symmetries, these duals must be quasiregular too. But not everybody accepts this view. These duals have regular vertices and are transitive on their edges (but not on their vertices). The convex ones are, in corresponding order as above:

The rhombic dodecahedron
Rhombic dodecahedron

The rhombic dodecahedron is a convex set polyhedron with 12 rhombus faces. It is an Archimedean solid solid, or a Catalan solid. Its dual is the cuboctahedron....
, with two types of alternating vertices, 8 with three rhombic faces, and 6 with four rhombic faces.
The rhombic triacontahedron
Rhombic triacontahedron

In geometry, the rhombic triacontahedron is a convex set polyhedron with 30 rhombus faces. It is an Archimedean solid solid, or a Catalan solid....
, with two types of alternating vertices, 20 with three rhombic faces, and 12 with five rhombic faces.

In addition, by duality with the octahedron, the cube

Cube

A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....
, which is usually regular

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....
, can be made quasiregular if alternate vertices are given different colors.

Their face configuration

Face configuration

In geometry, a face configuration is notational description of a face-transitive polyhedron. It represents a sequential count of the number of faces that exist at each vertex around a face ....
are of the form V3.n.3.n:

Cube V3.3.3.3	rhombic dodecahedron V3.4.3.4	rhombic triacontahedron V3.5.3.5

These three quasiregular duals are also characterised by having rhombic

Rhombus

In geometry, a rhombus , or rhomb is an equilateral polygon parallelogram. In other words, it is a four-sided polygon in which every side has the same length....
faces.

This rhombic-faced pattern continues as V3.6.3.6, the quasiregular rhombic tiling

Quasiregular rhombic tiling

In geometry, the quasiregular rhombic tiling is a tiling of identical 60° rhombi polygons on the Euclidean plane. There are two types of vertices, one with three rhombi and one with six rhombi....
.

External links

George Hart,

Quasiregular polyhedron

The convex quasiregular polyhedra

Nonconvex examples

Quasiregular duals

See also

External links