Quasiregular polyhedron

Quasiregular figures
(3.3)2 (3.4)2 (3.5)2 (3.6)2 (3.7)2 (3.8)2 (3.∞)2
$\begin{Bmatrix} 3 \\ 3 \end{Bmatrix}$ $\begin{Bmatrix} 3 \\ 4 \end{Bmatrix}$ $\begin{Bmatrix} 3 \\ 5 \end{Bmatrix}$ $\begin{Bmatrix} 3 \\ 6 \end{Bmatrix}$ $\begin{Bmatrix} 3 \\ 7 \end{Bmatrix}$ $\begin{Bmatrix} 3 \\ 8 \end{Bmatrix}$ $\begin{Bmatrix} 3 \\ \infin \end{Bmatrix}$
r{3,3} r{3,4} r{3,5} r{3,6} r{3,7} r{3,8} r{3,∞}
A quasiregular polyhedron or tiling has exactly two kinds of regular face, which alternate around each vertex. Their vertex figures are rectangles.

In geometry, a quasiregular polyhedron is a semiregular polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are edge-transitive and hence a step closer to regularity than the semiregular which are merely vertex-transitive.

There are only two convex quasiregular polyhedra, the cuboctahedron and the icosidodecahedron. Their names, given by Kepler, come from recognizing their faces contain all the faces of the dual-pair cube and octahedron, in the first, and 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 $\begin{Bmatrix} p \\ q \end{Bmatrix}$ or r{p,q} to represent their containing the faces of both the regular {p,q} and dual regular {q,p}. A quasiregular polyhedron with this symbol will have a vertex configuration p.q.p.q (or (p.q)2).

More generally, a quasiregular figure can have a vertex configuration (p.q)r, representing r (2 or more) instances of the faces around the vertex.

Tilings of the plane can also be quasiregular, specifically the trihexagonal tiling, with vertex configuration (3.6)2. Other quasiregular tilings exist on the hyperbolic plane, like the triheptagonal tiling, (3.7)2. Or more generally, (p.q)2, with 1/p+1/q<1/2.

Regular and quasiregular figures
Right triangles (p p 2)
{3,4}
r{3,3}
{4,4}
r{4,4}
{5,4}
r{5,5}
{6,4}
r{6,6}
{7,4}
r{7,7}
{8,4}
r{8,8}
{∞,4}
r{∞,∞}
$\begin{Bmatrix} 3 \\ 3 \end{Bmatrix}$ $\begin{Bmatrix} 4 \\ 4 \end{Bmatrix}$ $\begin{Bmatrix} 5 \\ 5 \end{Bmatrix}$ $\begin{Bmatrix} 6 \\ 6 \end{Bmatrix}$ $\begin{Bmatrix} 7 \\ 7 \end{Bmatrix}$ $\begin{Bmatrix} 8 \\ 8 \end{Bmatrix}$ $\begin{Bmatrix} \infin \\ \infin \end{Bmatrix}$
(3.3)2 (4.4)2 (5.5)2 (6.6)2 (7.7)2 (8.8)2 (∞.∞)2

square tiling

order-4 pentagonal tiling

order-4 hexagonal tiling

order-4 heptagonal tiling

order-4 octagonal tiling

Order-4 apeirogonal tiling
General triangles (p p 3)
{3,6} {4,6} {5,6} {6,6} {7,6} {8,6} {∞,6}
(3.3)3 (4.4)3 (5.5)3 (6.6)3 (7.7)3 (8.8)3 (∞.∞)3
General triangles (p p 4)
{3,8} {4,8} {5,8} {6,8} {7,8} {8,8} {∞,8}
(3.3)4 (4.4)4 (5.5)4 (6.6)4 (7.7)4 (8.8)4 (∞.∞)4
A regular polyhedron or tiling can be considered quasiregular if it has an even number of faces around each vertex (and thus can have alternately colored faces).

Some regular polyhedra and tilings (those with an even number of faces at each vertex) 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 {p,q} can be quasiregular, with vertex configuration (p.p)q/2, if q is even.

The octahedron can be considered quasiregular as a tetratetrahedron (2 sets of 4 triangles of the tetrahedron), (3a.3b)2, alternating two colors of triangular faces. Similarly the square tiling (4a.4b)2 can be considered quasiregular, colored as a checkerboard. Also the triangular tiling can have alternately colored triangle faces, (3a.3b)3.

Wythoff construction

 Regular (p | 2 q) and quasiregular polyhedra (2 | p q) are created from a Wythoff construction with the generator point at one of 3 corners of the fundamental domain. This defines a single edge within the fundamental domain.
Quasiregular polyhedra are generated from all 3 corners of the fundamental domain for Schwarz triangles that have no right angles:
q | 2 p, p | 2 q, 2 | p q

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:

Schläfli symbol Coxeter diagram Wythoff symbol
$\begin{Bmatrix} p , q \end{Bmatrix}$ {p,q} q | 2 p
$\begin{Bmatrix} q , p \end{Bmatrix}$ {q,p} p | 2 q
$\begin{Bmatrix} p \\ q \end{Bmatrix}$ r{p,q} 2 | p q

The convex quasiregular polyhedra

There are two convex quasiregular polyhedra:

1. The cuboctahedron $\begin{Bmatrix} 3 \\ 4 \end{Bmatrix}$, vertex configuration (3.4)2, Coxeter-Dynkin diagram
2. The icosidodecahedron $\begin{Bmatrix} 3 \\ 5 \end{Bmatrix}$, vertex configuration (3.5)2, Coxeter-Dynkin diagram

In addition, the octahedron, which is also regular, $\begin{Bmatrix} 3 \\ 3 \end{Bmatrix}$, vertex configuration (3.3)2, 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.

Quasiregular tilings

This sequence continues as the trihexagonal tiling, vertex figure 3.6.3.6 - a quasiregular tiling based on the triangular tiling and hexagonal tiling.

Regular Dual regular Quasiregular Vertex figure

Hexagonal tiling
{6,3}

6 | 2 3

Triangular tiling
{3,6}

3 | 2 6

Trihexagonal tiling

2 | 3 6

3.6.3.6

The checkerboard pattern is a quasiregular coloring of the square tiling, vertex figure 4.4.4.4:

Regular Dual regular Quasiregular Vertex figure

{4,4}

4 | 2 4

{4,4}

4 | 2 4

2 | 4 4

4.4.4.4

The triangular tiling can also be considered quasiregular, with three sets of alternating triangles at each vertex, (3.3)3:

 3 | 3 3

In the hyperbolic plane, this sequence continues further, for example the triheptagonal tiling, vertex figure 3.7.3.7 - a quasiregular tiling based on the order-7 triangular tiling and heptagonal tiling.

Regular Dual regular Quasiregular Vertex figure

Heptagonal tiling
{7,3}

7 | 2 3

Triangular tiling
{3,7}

3 | 2 7

Triheptagonal tiling

2 | 3 7

3.7.3.7

Nonconvex examples

Coxeter, H.S.M. et al. (1954) 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 $\begin{Bmatrix} 3 \\ 5/2 \end{Bmatrix}$ and the dodecadodecahedron $\begin{Bmatrix} 5 \\ 5/2 \end{Bmatrix}$:

Regular Dual regular Quasiregular Vertex figure

great stellated dodecahedron
{5/2,3}

3 | 2 5/2

great icosahedron
{3,5/2}

5/2 | 2 3

Great icosidodecahedron

2 | 3 5/2

3.5/2.3.5/2

Small stellated dodecahedron
{5/2,5}

5 | 2 5/2

Great dodecahedron
{5,5/2}

5/2 | 2 5

2 | 5 5/2

5.5/2.5.5/2

Lastly there are three ditrigonal forms, whose vertex figures contain three alternations of the two face types:

Image Polyhedron name
Wythoff symbol
Coxeter diagram
Vertex figure
3 | 5/3 5

(5.5/3)3
Small ditrigonal icosidodecahedron
3 | 5/2 3

(3.5/2)3
Great ditrigonal icosidodecahedron
3/2 | 3 5

((3.5)3)/2

Quasiregular duals

Some authorities argue that, since the duals of the quasiregular solids share the same symmetries, these duals must be quasiregular too. But not everybody believes this to be true. 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:

1. The rhombic dodecahedron, with two types of alternating vertices, 8 with three rhombic faces, and 6 with four rhombic faces.
2. The rhombic triacontahedron, with two types of alternating vertices, 20 with three rhombic faces, and 12 with five rhombic faces.

In addition, by duality with the octahedron, the cube, which is usually regular, can be made quasiregular if alternate vertices are given different colors.

Their face configuration are of the form V3.n.3.n, and Coxeter-Dynkin diagram

 Cube V(3.3)2 Rhombic dodecahedron V(3.4)2 Rhombic triacontahedron V(3.5)2 Rhombille tiling V(3.6)2 V(3.7)2 V(3.8)2

These three quasiregular duals are also characterised by having rhombic faces.

This rhombic-faced pattern continues as V(3.6)2, the rhombille tiling.

Quasiregular polytopes and honeycombs

In Euclidean 4-space, the regular 16-cell can also be seen as quasiregular as an alternated tesseract, h{4,3,3}, Coxeter diagrams: = , composed of alternating tetrahedron and tetrahedron cells. Its vertex figure is the quasiregular tetratetrahedron (an octahedron with tetrahedral symmetry), .

The only quasiregular honeycomb in Euclidean 3-space is the alternated cubic honeycomb, h{4,3,4}, Coxeter diagrams: = , composed of alternating tetrahedral and octahedral cells. Its vertex figure is the quasiregular cuboctahedron, .[2]

In hyperbolic 3-space, one quasiregular honeycomb is the alternated order-5 cubic honeycomb, h{4,3,5}, 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{4,3,6} has alternating tetrahedral and hexagonal tiling cells with vertex figure is a quasiregular trihexagonal tiling, .

Quasiregular polychora and honeycombs: h{4,3,p}
Space Euclidean 4-space Euclidean 3-space Hyperbolic 3-space
Name h{4,3,3} = $\left\{3,{3\atop3}\right\}$ h{4,3,4} = $\left\{3,{3\atop4}\right\}$ h{4,3,5} = $\left\{3,{3\atop5}\right\}$ h{4,3,6} = $\left\{3,{3\atop6}\right\}$
Coxeter
diagram
= = = =
Image
Vertex
figure

r{p,3}

Regular polychora honeycombs of the form {p,3,4} or can have their symmetry cut in half as quasiregular form , creating alternately colored {p,3} cells. These cases include the Euclidean cubic honeycomb {4,3,4} with cubic cells, and compact hyperbolic {5,3,4} with dodecahedral cells, and paracompact {6,3,4} with infinite hexagonal tiling cells. They have four cells around each edge, alternating in 2 colors. Their vertex figures are quasiregular tetratetrahedra, .

Regular and Quasiregular honeycombs: {p,3,4} and {p,31,1}
Space Euclidean 4-space Euclidean 3-space Hyperbolic 3-space
Name {3,3,4}
{3,31,1} = $\left\{3,{3\atop3}\right\}$
{4,3,4}
{4,31,1} = $\left\{4,{3\atop3}\right\}$
{5,3,4}
{5,31,1} = $\left\{5,{3\atop3}\right\}$
{6,3,4}
{6,31,1} = $\left\{6,{3\atop3}\right\}$
Coxeter
diagram
and and and and
Image
Cells
{p,3}