- Quasiregular polyhedron
A

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

quasiregular polyhedron can have faces of only two kinds and these must alternate around each vertex.They are given a vertical

Schläfli symbol $egin\{Bmatrix\}\; p\; \backslash \; q\; end\{Bmatrix\}$ to represent this combined form which contains the combined faces of the regular**{p,q}**and dual**{q,p}**. A quasiregular polyhedron with this symbol will have avertex configuration **p.q.p.q**.The

Coxeter-Dynkin diagram is another symbolic representation that shows the quasiregular relation between the two dual-regular forms:

* {p,q} :

* {q,p} :

***p.q.p.q**: .**The convex quasiregular polyhedra**There are two

convex quasiregular polyhedra:

#Thecuboctahedron $egin\{Bmatrix\}\; 3\; \backslash \; 4\; end\{Bmatrix\}$, vertex configuration**3.4.3.4**,Coxeter-Dynkin diagram

#Theicosidodecahedron $egin\{Bmatrix\}\; 3\; \backslash \; 5\; end\{Bmatrix\}$, vertex configuration**3.5.3.5**, "Coxeter-Dynkin diagram"In addition, the

octahedron , which is also regular, $egin\{Bmatrix\}\; 3\; \backslash \; 3\; end\{Bmatrix\}$, 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 pair of regular polyhedra. The names of two of these give clues to the associated dual pair, respectively the

cube +octahedron and theicosahedron +dodecahedron . Theoctahedron is the core of a dual pair of tetrahedra (an arrangement known as thestella octangula ), and when derived in this way is sometimes called the "tetratetrahedron".These three quasiregular duals are also characterised by having rhombic faces.

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

quasiregular rhombic tiling .**See also***

Rectification (geometry)

*Trihexagonal tiling - A quasiregular tiling based on thetriangular tiling andhexagonal tiling **References***Coxeter, H.S.M., 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.

*Cromwell, P. "Polyhedra", Cambridge University Press (1977).**External links***

* George Hart, [*http://www.georgehart.com/virtual-polyhedra/quasi-regular-info.html Quasiregular polyhedra*]

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