Quasiregular polyhedron

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.

These forms representing a pair of a regular figure and its dual can be given a vertical Schläfli symbol

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 {p,q} can be considered quasiregular, with vertex configuration (p.p)q/2, if q is even.

Examples: The regular octahedron, with Schläfli symbol {3,4} 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.

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: There are two uniform convex quasiregular polyhedra: In addition, the octahedron, which is also regular,

, vertex configuration (3.3)2, can be considered quasiregular if alternate faces are given different colors.

The remaining convex 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.

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.

Two are based on dual pairs of regular Kepler–Poinsot solids, in the same way as for the convex examples: the great icosidodecahedron

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.

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.

Its vertex figure is the quasiregular tetratetrahedron (an octahedron with tetrahedral symmetry), .

[3] A related paracompact alternated order-6 cubic honeycomb, h{4,3,6} has alternating tetrahedral and hexagonal tiling cells with vertex figure is a quasiregular trihexagonal tiling, .