# Quasiregular polytope (Coxeter's definition)

As originally defined by Coxeter, a quasiregular polyhedron is a polyhedron that has only regular faces, and its vertex figures are isogonal but not regular. It follows that quasiregular polyhedra are isogonal, isotoxal, two-orbit, and have two types of faces distinguishable by the polytope's symmetry group, with each edge joining the two kinds of faces. The quasiregular polyhedra are precisely the non-regular isotoxal uniform polyhedra.

In Regular Polytopes, Coxeter also defined a quasiregular honeycomb as a tiling of 3D space that is isogonal, has only regular cells, and its vertex figure is a quasiregular polyhedron.

Coxeter did not offer a definition of the "quasiregular" for general polytopes, so there is no official source that extends his definition to polytopes of arbitrary rank. More confusingly, the term "quasiregular" also branched off into a subtly different definition based on Wythoffian construction, namely a Coxeter-Dynkin diagram with a single ringed node. Due to these differences, it's wise to precisely define "quasiregular" in any context where it is used.

## List of quasiregular polyhedra

The following is a list of quasiregular polytopes.

Quasiregular polyhedra
Name Image Facets Vertex figure Rectification of
Tetrahemihexahedron 4 triangles, 3 squares Bowtie Petrial tetrahedron
Cuboctahedron 8 triangles, 6 squares Rectangle
Cubohemioctahedron 6 squares, 4 hexagons Bowtie Petrial octahedron
Octahemioctahedron 8 triangles, 4 hexagons Bowtie Petrial cube
Icosidodecahedron 20 triangles, 12 pentagons Rectangle
Small dodecahemidodecahedron 12 pentagons, 6 decagons Bowtie Petrial icosahedron
Small icosihemidodecahedron 20 triangles, 6 decagons Bowtie Petrial dodecahedron
Dodecadodecahedron 12 pentagons, 12 pentagrams Rectangle
Great dodecahemicosahedron 12 pentagons, 10 hexagons Bowtie Petrial small stellated dodecahedron
Small dodecahemicosahedron 12 pentagrams, 10 hexagons Bowtie Petrial great dodecahedron
Great icosidodecahedron 20 triangles, 12 pentagrams Rectangle
Great dodecahemidodecahedron 12 pentagrams, 6 decagrams Bowtie Petrial great icosahedron
Great icosihemidodecahedron 20 triangles, 6 decagrams Bowtie Petrial great stellated dodecahedron
Small ditrigonary icosidodecahedron 20 triangles, 12 pentagrams Ditrigon None
Ditrigonary dodecadodecahedron 12 pentagons, 12 pentagrams Propeller tripod None
Great ditrigonary icosidodecahedron 20 triangles, 12 pentagons Tripod None