Quasiregular polyhedron

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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 $\begin{Bmatrix} p \\ q \end{Bmatrix}$ 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.

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: .

1 The convex quasiregular polyhedra
2 Nonconvex examples
3 Quasiregular duals
4 See also
5 References
6 External links

[edit] The convex quasiregular polyhedra

There are two convex quasiregular polyhedra:

The cuboctahedron $\begin{Bmatrix} 3 \\ 4 \end{Bmatrix}$ , vertex configuration 3.4.3.4, Coxeter-Dynkin diagram
The icosidodecahedron $\begin{Bmatrix} 3 \\ 5 \end{Bmatrix}$ , vertex configuration 3.5.3.5, Coxeter-Dynkin diagram

In addition, the octahedron, which is also regular, $\begin{Bmatrix} 3 \\ 3 \end{Bmatrix}$ , 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.

Regular	Dual regular	Quasiregular	Vertex figure
Tetrahedron {3,3}	Tetrahedron {3,3}	Tetratetrahedron (Octahedron)	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 dual pairs of regular Kepler-Poinsot solids, in the same way as for the convex examples.

The great icosidodecahedron $\begin{Bmatrix} 3 \\ 5/2 \end{Bmatrix}$ and the dodecadodecahedron $\begin{Bmatrix} 5 \\ 5/2 \end{Bmatrix}$ :

Regular	Dual regular	Quasiregular	Vertex figure
great stellated dodecahedron {⁵/₂,3}	great icosahedron {3,⁵/₂}	Great icosidodecahedron	3.⁵/₂.3.⁵/₂
Small stellated dodecahedron {⁵/₂,5}	Great dodecahedron {5,⁵/₂}	Dodecadodecahedron	5.⁵/₂.5.⁵/₂

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:

The cube, which is also regular, with 2 types of alternating vertices, each set with three squares.
The rhombic dodecahedron, with two types of alternating vertices, 8 with three rhombic faces, and 6 with four rhombic faces.
The rhombic triacontahedron, with two types of alternating vertices, 20 with three rhombic faces, and 12 with five rhombic faces.

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

[edit] See also

Rectification (geometry)
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).