 

In geometry, a quasiregular polyhedron is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are vertextransitive and edgetransitive, hence a step closer to regular polyhedra than the semiregular, which are merely vertextransitive.
Their dual figures are facetransitive and edgetransitive; they have exactly two kinds of regular vertex figures, which alternate around each face. They are sometimes also considered quasiregular.
There are only two convex quasiregular polyhedra: the cuboctahedron and the icosidodecahedron. Their names, given by Kepler, come from recognizing that their faces are all the faces (turned differently) of the dualpair cube and octahedron, in the first case, and of the dualpair icosahedron and dodecahedron, in the second case.
These forms representing a pair of a regular figure and its dual can be given a vertical Schläfli symbol or r{p,q}, to represent that their faces are all the faces (turned differently) of both the regular {p,q} and the dual regular {q,p}. A quasiregular polyhedron with this symbol will have a vertex configuration p.q.p.q (or (p.q)^{2}).
More generally, a quasiregular figure can have a vertex configuration (p.q)^{r}, representing r (2 or more) sequences of the faces around the vertex.
Tilings of the plane can also be quasiregular, specifically the trihexagonal tiling, with vertex configuration (3.6)^{2}. Other quasiregular tilings exist on the hyperbolic plane, like the triheptagonal tiling, (3.7)^{2}. Or more generally: (p.q)^{2}, with 1/p + 1/q < 1/2.
Regular polyhedra and tilings with an even number of faces at each vertex can also be considered quasiregular by differentiating between faces of the same order, by representing them differently, like coloring them alternately (without defining any surface orientation). A regular figure with Schläfli symbol {p,q} can be considered quasiregular, with vertex configuration (p.p)^{q/2}, if q is even.
Examples:
The regular octahedron, with Schläfli symbol {3,4} and 4 being even, can be considered quasiregular as a tetratetrahedron (2 sets of 4 triangles of the tetrahedron), with vertex configuration (3.3)^{4/2} = (3_{a}.3_{b})^{2}, alternating two colors of triangular faces.
The square tiling, with vertex configuration 4^{4} and 4 being even, can be considered quasiregular, with vertex configuration (4.4)^{4/2} = (4_{a}.4_{b})^{2}, colored as a checkerboard.
The triangular tiling, with vertex configuration 3^{6} and 6 being even, can be considered quasiregular, with vertex configuration (3.3)^{6/2} = (3_{a}.3_{b})^{3}, alternating two colors of triangular faces.
Regular (p  2 q) and quasiregular polyhedra (2  p q) are created from a Wythoff construction with the generator point at one of 3 corners of the fundamental domain. This defines a single edge within the fundamental domain. 
Coxeter defines a quasiregular polyhedron as one having a Wythoff symbol in the form p  q r, and it is regular if q=2 or q=r.^{[1]}
The CoxeterDynkin diagram is another symbolic representation that shows the quasiregular relation between the two dualregular forms:
Schläfli symbol  Coxeter diagram  Wythoff symbol  

{p,q}  q  2 p  
{q,p}  p  2 q  
r{p,q}  or  2  p q 
There are two uniform convex quasiregular polyhedra:
In addition, the octahedron, which is also regular, , vertex configuration (3.3)^{2}, can be considered quasiregular if alternate faces are given different colors. In this form it is sometimes known as the tetratetrahedron. The remaining convex regular polyhedra have an odd number of faces at each vertex so cannot be colored in a way that preserves edge transitivity. It has CoxeterDynkin diagram
Each of these forms the common core of a dual pair of regular polyhedra. The names of two of these give clues to the associated dual pair: respectively cube octahedron, and icosahedron dodecahedron. The octahedron is the common core of a dual pair of tetrahedra (a compound known as the stella octangula); when derived in this way, the octahedron is sometimes called the tetratetrahedron, as tetrahedron tetrahedron.
Regular  Dual regular  Quasiregular common core  Vertex figure 

Tetrahedron {3,3} 3  2 3 
Tetrahedron {3,3} 3  2 3 
Tetratetrahedron r{3,3} 2  3 3 
3.3.3.3 
Cube {4,3} 3  2 4 
Octahedron {3,4} 4  2 3 
Cuboctahedron r{3,4} 2  3 4 
3.4.3.4 
Dodecahedron {5,3} 3  2 5 
Icosahedron {3,5} 5  2 3 
Icosidodecahedron r{3,5} 2  3 5 
3.5.3.5 
Each of these quasiregular polyhedra can be constructed by a rectification operation on either regular parent, truncating the vertices fully, until each original edge is reduced to its midpoint.
This sequence continues as the trihexagonal tiling, vertex figure (3.6)^{2}  a quasiregular tiling based on the triangular tiling and hexagonal tiling.
Regular  Dual regular  Quasiregular combination  Vertex figure 

Hexagonal tiling {6,3} 6  2 3 
Triangular tiling {3,6} 3  2 6 
Trihexagonal tiling r{6,3} 2  3 6 
(3.6)^{2} 
The checkerboard pattern is a quasiregular coloring of the square tiling, vertex figure (4.4)^{2}:
Regular  Dual regular  Quasiregular combination  Vertex figure 

{4,4} 4  2 4 
{4,4} 4  2 4 
r{4,4} 2  4 4 
(4.4)^{2} 
The triangular tiling can also be considered quasiregular, with three sets of alternating triangles at each vertex, (3.3)^{3}:
In the hyperbolic plane, this sequence continues further, for example the triheptagonal tiling, vertex figure (3.7)^{2}  a quasiregular tiling based on the order7 triangular tiling and heptagonal tiling.
Regular  Dual regular  Quasiregular combination  Vertex figure 

Heptagonal tiling {7,3} 7  2 3 
Triangular tiling {3,7} 3  2 7 
Triheptagonal tiling r{3,7} 2  3 7 
(3.7)^{2} 
Coxeter, H.S.M. et al. (1954) also classify certain star polyhedra, having the same characteristics, as being quasiregular.
Two are based on dual pairs of regular KeplerPoinsot solids, in the same way as for the convex examples:
the great icosidodecahedron , and the dodecadodecahedron :
Regular  Dual regular  Quasiregular common core  Vertex figure 

Great stellated dodecahedron {^{5}/_{2},3} 3  2 5/2 
Great icosahedron {3,^{5}/_{2}} 5/2  2 3 
Great icosidodecahedron r{3,^{5}/_{2}} 2  3 5/2 
3.^{5}/_{2}.3.^{5}/_{2} 
Small stellated dodecahedron {^{5}/_{2},5} 5  2 5/2 
Great dodecahedron {5,^{5}/_{2}} 5/2  2 5 
Dodecadodecahedron r{5,^{5}/_{2}} 2  5 5/2 
5.^{5}/_{2}.5.^{5}/_{2} 
Nine more are the hemipolyhedra, which are faceted forms of the aforementioned quasiregular polyhedra derived from rectification of regular polyhedra. These include equatorial faces passing through the centre of the polyhedra:
Quasiregular (rectified)  Tetratetrahedron 
Cuboctahedron 
Icosidodecahedron 
Great icosidodecahedron 
Dodecadodecahedron 

Quasiregular (hemipolyhedra)  Tetrahemihexahedron ^{3}/_{2} 3  2 
Octahemioctahedron ^{3}/_{2} 3  3 
Small icosihemidodecahedron ^{3}/_{2} 3  5 
Great icosihemidodecahedron ^{3}/_{2} 3  ^{5}/_{3} 
Small dodecahemicosahedron ^{5}/_{3}^{5}/_{2}  3 
Vertex figure  3.4.^{3}/_{2}.4 
3.6.^{3}/_{2}.6 
3.10.^{3}/_{2}.10 
3.^{10}/_{3}.^{3}/_{2}.^{10}/_{3} 
^{5}/_{2}.6.^{5}/_{3}.6 
Quasiregular (hemipolyhedra)  Cubohemioctahedron ^{4}/_{3} 4  3 
Small dodecahemidodecahedron ^{5}/_{4} 5  5 
Great dodecahemidodecahedron ^{5}/_{3}^{5}/_{2}  ^{5}/_{3} 
Great dodecahemicosahedron ^{5}/_{4} 5  3  
Vertex figure  4.6.^{4}/_{3}.6 
5.10.^{5}/_{4}.10 
^{5}/_{2}.^{10}/_{3}.^{5}/_{3}.^{10}/_{3} 
5.6.^{5}/_{4}.6 
Lastly there are three ditrigonal forms, all facetings of the regular dodecahedron, whose vertex figures contain three alternations of the two face types:
Image  Faceted form Wythoff symbol Coxeter diagram 
Vertex figure 

Ditrigonal dodecadodecahedron 3  5/3 5 or 
(5.5/3)^{3}  
Small ditrigonal icosidodecahedron 3  5/2 3 or 
(3.5/2)^{3}  
Great ditrigonal icosidodecahedron 3/2  3 5 or 
((3.5)^{3})/2 
In the Euclidean plane, the sequence of hemipolyhedra continues with the following four star tilings, where apeirogons appear as the aforementioned equatorial polygons:
Some authorities argue that, since the duals of the quasiregular solids share the same symmetries, these duals should be called quasiregular too. But not everybody uses this terminology. These duals are transitive on their edges and faces (but not on their vertices); they are the edgetransitive Catalan solids. The convex ones are, in corresponding order as above:
In addition, by duality with the octahedron, the cube, which is usually regular, can be made quasiregular if alternate vertices are given different colors.
Their face configuration are of the form V3.n.3.n, and CoxeterDynkin diagram
Cube V(3.3)^{2} 
Rhombic dodecahedron V(3.4)^{2} 
Rhombic triacontahedron V(3.5)^{2} 
Rhombille tiling V(3.6)^{2} 
V(3.7)^{2} 
V(3.8)^{2} 
These three quasiregular duals are also characterised by having rhombic faces.
This rhombicfaced pattern continues as V(3.6)^{2}, the rhombille tiling.
In higher dimensions, Coxeter defined a quasiregular polytope or honeycomb to have regular facets and quasiregular vertex figures. It follows that all vertex figures are congruent and that there are two kinds of facets, which alternate.^{[2]}
In Euclidean 4space, the regular 16cell can also be seen as quasiregular as an alternated tesseract, h{4,3,3}, Coxeter diagrams: = , composed of alternating tetrahedron and tetrahedron cells. Its vertex figure is the quasiregular tetratetrahedron (an octahedron with tetrahedral symmetry), .
The only quasiregular honeycomb in Euclidean 3space is the alternated cubic honeycomb, h{4,3,4}, Coxeter diagrams: = , composed of alternating tetrahedral and octahedral cells. Its vertex figure is the quasiregular cuboctahedron, .^{[2]}
In hyperbolic 3space, one quasiregular honeycomb is the alternated order5 cubic honeycomb, h{4,3,5}, Coxeter diagrams: = , composed of alternating tetrahedral and icosahedral cells. Its vertex figure is the quasiregular icosidodecahedron, . A related paracompact alternated order6 cubic honeycomb, h{4,3,6} has alternating tetrahedral and hexagonal tiling cells with vertex figure is a quasiregular trihexagonal tiling, .
Quasiregular polychora and honeycombs: h{4,p,q}  

Space  Finite  Affine  Compact  Paracompact  
Schläfli symbol 
h{4,3,3}  h{4,3,4}  h{4,3,5}  h{4,3,6}  h{4,4,3}  h{4,4,4}  
Coxeter diagram 

Image  
Vertex figure r{p,3} 
Regular polychora or honeycombs of the form {p,3,4} or can have their symmetry cut in half as into quasiregular form , creating alternately colored {p,3} cells. These cases include the Euclidean cubic honeycomb {4,3,4} with cubic cells, and compact hyperbolic {5,3,4} with dodecahedral cells, and paracompact {6,3,4} with infinite hexagonal tiling cells. They have four cells around each edge, alternating in 2 colors. Their vertex figures are quasiregular tetratetrahedra, = .
Regular and Quasiregular honeycombs: {p,3,4} and {p,3^{1,1}}  

Space  Euclidean 4space  Euclidean 3space  Hyperbolic 3space  
Name  {3,3,4} {3,3^{1,1}} = 
{4,3,4} {4,3^{1,1}} = 
{5,3,4} {5,3^{1,1}} = 
{6,3,4} {6,3^{1,1}} =  
Coxeter diagram 
=  =  =  =  
Image  
Cells {p,3} 
Similarly regular hyperbolic honeycombs of the form {p,3,6} or can have their symmetry cut in half as into quasiregular form , creating alternately colored {p,3} cells. They have six cells around each edge, alternating in 2 colors. Their vertex figures are quasiregular triangular tilings, .
Form  Paracompact  Noncompact  

Name  {3,3,6} {3,3^{[3]}} 
{4,3,6} {4,3^{[3]}} 
{5,3,6} {5,3^{[3]}} 
{6,3,6} {6,3^{[3]}} 
{7,3,6} {7,3^{[3]}} 
{8,3,6} {8,3^{[3]}} 
... {∞,3,6} {∞,3^{[3]}} 
Image  
Cells  {3,3} 
{4,3} 
{5,3} 
{6,3} 
{7,3} 
{8,3} 
{∞,3} 