# Quasiregular Polyhedron - The Convex Quasiregular Polyhedra

The Convex Quasiregular Polyhedra

There are two convex quasiregular polyhedra:

1. The cuboctahedron, vertex configuration 3.4.3.4, Coxeter-Dynkin diagram
2. The icosidodecahedron, vertex configuration 3.5.3.5, Coxeter-Dynkin diagram

In addition, the octahedron, which is also regular, vertex configuration 3.3.3.3, can be considered quasiregular if alternate faces are given different colors. In this form it is sometimes known as the tetratetrahedron. 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. The names of two of these give clues to the associated dual pair, respectively the cube + octahedron and the icosahedron + dodecahedron. The octahedron is the core of a dual pair of tetrahedra (an arrangement known as the stella octangula), and when derived in this way is sometimes called the tetratetrahedron.

Regular Dual regular Quasiregular Vertex figure

Tetrahedron
{3,3}

3 | 2 3

Tetrahedron
{3,3}

3 | 2 3

Tetratetrahedron
(Octahedron)

2 | 3 3

3.3.3.3

Cube
{4,3}

3 | 2 4

Octahedron
{3,4}

4 | 2 3

Cuboctahedron

2 | 3 4

3.4.3.4

Dodecahedron
{5,3}

3 | 2 5

Icosahedron
{3,5}

5 | 2 3

Icosidodecahedron

2 | 3 5

3.5.3.5

Each of these quasiregular polyhedra can be constructed by a rectification operation on either regular parent, truncating the edges fully, until the original edges are reduced to a point.