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Their dual figures are face-transitive and edge-transitive; 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 dual-pair cube and octahedron, in the first case, and of the dual-pair icosahedron and dodecahedron, in the second case.
\begin{Bmatrix}p\ q\end{Bmatrix}
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 can be considered quasiregular, with vertex configuration (p.p)q/2, if q is even.
Examples:
The regular octahedron, with Schläfli symbol 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 = (3a.3b)2, alternating two colors of triangular faces.
The square tiling, with vertex configuration 44 and 4 being even, can be considered quasiregular, with vertex configuration (4.4)4/2 = (4a.4b)2, colored as a checkerboard.
The triangular tiling, with vertex configuration 36 and 6 being even, can be considered quasiregular, with vertex configuration (3.3)6/2 = (3a.3b)3, alternating two colors of triangular faces.
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 Coxeter-Dynkin diagram is another symbolic representation that shows the quasiregular relation between the two dual-regular forms:
| Schläfli symbol | Coxeter diagram | Wythoff symbol | ||
|---|---|---|---|---|
\begin{Bmatrix}p,q\end{Bmatrix} | q | 2 p | |||
\begin{Bmatrix}q,p\end{Bmatrix} | p | 2 q | |||
\begin{Bmatrix}p\ q\end{Bmatrix} | r | or | 2 | p q | |
There are two uniform convex quasiregular polyhedra:
\begin{Bmatrix}3\ 5\end{Bmatrix}
In addition, the octahedron, which is also regular,
\begin{Bmatrix}3\ 3\end{Bmatrix}
\cap
\cap
\cap
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.
The checkerboard pattern is a quasiregular coloring of the square tiling, vertex figure (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 order-7 triangular tiling and heptagonal tiling.
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:
\begin{Bmatrix}3\ 5/2\end{Bmatrix}
\begin{Bmatrix}5\ 5/2\end{Bmatrix}
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:
Lastly there are three ditrigonal forms, all facetings of the regular dodecahedron, whose vertex figures contain three alternations of the two face types:
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 edge-transitive 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.
These three quasiregular duals are also characterised by having rhombic faces.
This rhombic-faced 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.
In Euclidean 4-space, the regular 16-cell can also be seen as quasiregular as an alternated tesseract, h, 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 3-space is the alternated cubic honeycomb, h, Coxeter diagrams: =, composed of alternating tetrahedral and octahedral cells. Its vertex figure is the quasiregular cuboctahedron, .[2]
In hyperbolic 3-space, one quasiregular honeycomb is the alternated order-5 cubic honeycomb, h, Coxeter diagrams: =, composed of alternating tetrahedral and icosahedral cells. Its vertex figure is the quasiregular icosidodecahedron, . A related paracompact alternated order-6 cubic honeycomb, h has alternating tetrahedral and hexagonal tiling cells with vertex figure is a quasiregular trihexagonal tiling, .
Regular polychora or honeycombs of the form or can have their symmetry cut in half as into quasiregular form, creating alternately colored cells. These cases include the Euclidean cubic honeycomb with cubic cells, and compact hyperbolic with dodecahedral cells, and paracompact with infinite hexagonal tiling cells. They have four cells around each edge, alternating in 2 colors. Their vertex figures are quasiregular tetratetrahedra, = .
Similarly regular hyperbolic honeycombs of the form or can have their symmetry cut in half as into quasiregular form, creating alternately colored cells. They have six cells around each edge, alternating in 2 colors. Their vertex figures are quasiregular triangular tilings, .