# 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[edit | edit source]

The following is a list of quasiregular polytopes.

## See also[edit | edit source]

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