Quasiregular polyhedron


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.

They are given a vertical Schläfli symbol egin{Bmatrix} p \ 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 a vertex 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:
#The cuboctahedron egin{Bmatrix} 3 \ 4 end{Bmatrix}, vertex configuration 3.4.3.4, Coxeter-Dynkin diagram
#The icosidodecahedron egin{Bmatrix} 3 \ 5 end{Bmatrix}, vertex configuration 3.5.3.5, "Coxeter-Dynkin diagram"

In addition, the octahedron, which is also regular, egin{Bmatrix} 3 \ 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 the icosahedron + dodecahedron. The octahedron is the core of a dual pair of tetrahedra (an arrangement known as the stella 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 the triangular tiling and hexagonal 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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