Quasiregular polyhedron
In geometry, a quasiregular polyhedron is a uniform polyhedron which has exactly two kinds of regular faces, which alternate around each vertex. They are edge-transitive, and hence a step closer to regular polyhedra than the semiregular, which are merely vertex-transitive. Their dual figures are also sometimes considered quasiregular, except that they are edge-transitive, are face-transitive, and alternate between two regular vertex figures.
There are only two convex quasiregular polyhedra: the cuboctahedron and the icosidodecahedron. Their names, given by Kepler, come from recognizing that their faces correspond to all the faces of the dual-pair cube and octahedron, in the first case, and of the dual-pair icosahedron and dodecahedron, in the second case.
These forms representing a pair of a regular figure and its dual can be given a vertical Schläfli symbol or r to represent their corresponding to the faces of both the regular ' and dual regular '. A quasiregular polyhedron with this symbol will have a vertex configuration p.q.p.q.
More generally, a quasiregular figure can have a vertex configuration r, representing r sequences of the faces around the vertex.
Tilings of the plane can also be quasiregular, specifically the trihexagonal tiling, with vertex configuration 2. Other quasiregular tilings exist on the hyperbolic plane, like the triheptagonal tiling, 2. Or more generally, 2, with 1/p+1/q<1/2.
Some regular polyhedra and tilings can also be considered quasiregular by differentiating between faces of the same number of sides, but representing them differently, like having different colors, but no surface features defining their orientation. A regular figure with Schläfli symbol can be quasiregular, with vertex configuration q/2, if q is even.
The octahedron can be considered quasiregular as a tetratetrahedron, 2, alternating two colors of triangular faces. Similarly the square tiling 2 can be considered quasiregular, colored as a checkerboard. Also the triangular tiling can have alternately colored triangle faces, 3.
Wythoff construction
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.The Coxeter-Dynkin diagram is another symbolic representation that shows the quasiregular relation between the two dual-regular forms:
The convex quasiregular polyhedra
There are two uniform convex quasiregular polyhedra:- The cuboctahedron, vertex configuration 2, Coxeter-Dynkin diagram
- The icosidodecahedron, vertex configuration 2, 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 cube ^ octahedron, and icosahedron ^ dodecahedron. The octahedron is the common core of a dual pair of tetrahedra. When derived in this way, the octahedron is sometimes called the tetratetrahedron, as tetrahedron ^ tetrahedron.
| Regular | Dual regular | Quasiregular common core | Vertex figure |
| 2 3 | 2 3 | 3 3 | 3.3.3.3 |
| 2 4 | 2 3 | 3 4 | 3.4.3.4 |
| 2 5 | 2 3 | 3 5 | 3.5.3.5 |
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.
Quasiregular tilings
This sequence continues as the trihexagonal tiling, vertex figure 2 - a quasiregular tiling based on the triangular tiling and hexagonal tiling.| Regular | Dual regular | Quasiregular combination | Vertex figure |
| 2 3 | 2 6 | 3 6 | 2 |
The checkerboard pattern is a quasiregular coloring of the square tiling, vertex figure 2:
| Regular | Dual regular | Quasiregular combination | Vertex figure |
| 2 4 | 2 4 | 4 4 | 2 |
The triangular tiling can also be considered quasiregular, with three sets of alternating triangles at each vertex, 3:
h 3 | 3 3 = |
In the hyperbolic plane, this sequence continues further, for example the triheptagonal tiling, vertex figure 2 - a quasiregular tiling based on the order-7 triangular tiling and heptagonal tiling.
| Regular | Dual regular | Quasiregular combination | Vertex figure |
| 2 3 | 2 7 | 3 7 | 2 |
Nonconvex examples
et al. 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, and the dodecadodecahedron :
| Regular | Dual regular | Quasiregular common core | Vertex figure |
| 2 5/2 | 2 3 | 3 5/2 | 3.5/2.3.5/2 |
| 2 5/2 | 2 5 | 5 5/2 | 5.5/2.5.5/2 |
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:
| Quasiregular | Tetratetrahedron | Cuboctahedron | Icosidodecahedron | Great icosidodecahedron | Dodecadodecahedron |
| Quasiregular | Tetrahemihexahedron 3/2 3 2 | Octahemioctahedron 3/2 3 3 | Small icosihemidodecahedron 3/2 3 5 | Great icosihemidodecahedron 3/2 3 5/3 | Small dodecahemicosahedron 5/3 5/2 3 |
| Vertex figure | 3.4.3/2.4 | 3.6.3/2.6 | 3.10.3/2.10 | 3.10/3.3/2.10/3 | 5/2.6.5/3.6 |
| Quasiregular | Cubohemioctahedron 4/3 4 3 | Small dodecahemidodecahedron 5/4 5 5 | Great dodecahemidodecahedron 5/3 5/2 5/3 | Great dodecahemicosahedron 5/4 5 3 | |
| Vertex figure | 4.6.4/3.6 | 5.10.5/4.10 | 5/2.10/3.5/3.10/3 | 5.6.5/4.6 |
Lastly there are three ditrigonal forms, all facetings of the regular dodecahedron, whose vertex figures contain three alternations of the two face types:
| Image | Faceted form Wythoff symbol Coxeter diagram | Vertex figure |
| Ditrigonal dodecadodecahedron 3 | 5/3 5 or | 3 | |
| Small ditrigonal icosidodecahedron 3 | 5/2 3 or | 3 | |
| Great ditrigonal icosidodecahedron 3/2 | 3 5 or | /2 |
In the Euclidean plane, the sequence of hemipolyhedra continues with the following four star tilings, where apeirogons appear as the aforementioned equatorial polygons:
| Original rectified tiling | Edge diagram | Solid | Vertex Config | Wythoff | Symmetry group |
Square tiling | 4.∞.4/3.∞ 4.∞.-4.∞ | 4/3 4 | ∞ | p4m | ||
Triangular tiling | /2 | 3/2 | 3 ∞ | p6m | ||
Trihexagonal tiling | 6.∞.6/5.∞ 6.∞.-6.∞ | 6/5 6 | ∞ | p6m | ||
Trihexagonal tiling | ∞.3.∞.3/2 ∞.3.∞.-3 | 3/2 3 | ∞ | p6m |
Quasiregular duals
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 ; they are the edge-transitive Catalan solids. The convex ones are, in corresponding order as above:- 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.
Their face configuration are of the form V3.n.3.n, and Coxeter-Dynkin diagram
| Cube V2 | Rhombic dodecahedron V2 | Rhombic triacontahedron V2 | Rhombille tiling V2 | V2 | V2 |
These three quasiregular duals are also characterised by having rhombic faces.
This rhombic-faced pattern continues as V2, the rhombille tiling.
Quasiregular polytopes and honeycombs
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,.
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,.
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,.
is a quasiregular triangular tiling, =