Quasiregular polyhedron

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1 Definition
2 The convex quasiregular polyhedra
3 Nonconvex examples
4 Quasiregular duals
5 See also
- 5.1 References
- 5.2 External links

[edit] Definition

A polyhedron which has regular faces and is transitive on its edges 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 $\begin{Bmatrix} p \\ q \end{Bmatrix}$ 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.

[edit] The convex quasiregular polyhedra

There are three convex quasiregular polyhedra:

The octahedron, which is also regular, $\begin{Bmatrix} 3 \\ 3 \end{Bmatrix}$ , vertex configuration 3.3.3.3.
The cuboctahedron $\begin{Bmatrix} 3 \\ 4 \end{Bmatrix}$ , vertex configuration 3.4.3.4.
The icosidodecahedron $\begin{Bmatrix} 3 \\ 5 \end{Bmatrix}$ , vertex configuration 3.5.3.5.

Each of these forms the common core of a dual pair of regular polyhedra. The names of the last two listed 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.

Regular	Dual regular	Quasiregular
Tetrahedron {3,3}	Tetrahedron {3,3}	Tetratetrahedron 3.3.3.3
Cube {4,3}	Octahedron {3,4}	Cuboctahedron 3.4.3.4
Dodecahedron {5,3}	Icosahedron {3,5}	Icosidodecahedron 3.5.3.5

Each of these quasiregular polyhedra can be constructed by a rectification operation on either regular parent, truncating the edges fully, until the original edges are reduced to a point.

[edit] Nonconvex examples

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

Two are based on the regular Kepler-Poinsot solids, in the same way as for the convex examples:
1. Great icosidodecahedron $\begin{Bmatrix} 3 \\ 5/2 \end{Bmatrix}$ - Combining the great icosahedron and great stellated dodecahedron.
2. Dodecadodecahedron $\begin{Bmatrix} 5 \\ 5/2 \end{Bmatrix}$ - Combining the great dodecahedron and small stellated dodecahedron.

Three ditrigonal forms, whose vertex figures contain three alternatations of the two face types:

[edit] 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. The convex ones are, in corresponding order as above:

These three quasiregular duals are also characterised by having rhombic faces.

[edit] See also

Trihexagonal tiling - A quasiregular tiling based on the triangular tiling and hexagonal tiling

[edit] 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).