Quasiregular polyhedron

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
${\displaystyle {\begin{Bmatrix}p,q\end{Bmatrix}}}$	{p,q}		q | 2 p
${\displaystyle {\begin{Bmatrix}q,p\end{Bmatrix}}}$	{q,p}		p | 2 q
${\displaystyle {\begin{Bmatrix}p\\q\end{Bmatrix}}}$	r{p,q}	or	2 | p q

There are two uniform convex quasiregular polyhedra:

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

In addition, the octahedron, which is also regular, ${\displaystyle {\begin{Bmatrix}3\\3\end{Bmatrix}}}$, 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 cube ${\displaystyle \cap }$ octahedron, and icosahedron ${\displaystyle \cap }$ dodecahedron. The octahedron is the common core of a dual pair of tetrahedra (a compound known as the stella octangula); when derived in this way, the octahedron is sometimes called the tetratetrahedron, as tetrahedron ${\displaystyle \cap }$ tetrahedron.

Regular	Dual regular	Quasiregular common core	Vertex figure
Tetrahedron {3,3} 3 | 2 3	Tetrahedron {3,3} 3 | 2 3	Tetratetrahedron r{3,3} 2 | 3 3	3.3.3.3
Cube {4,3} 3 | 2 4	Octahedron {3,4} 4 | 2 3	Cuboctahedron r{3,4} 2 | 3 4	3.4.3.4
Dodecahedron {5,3} 3 | 2 5	Icosahedron {3,5} 5 | 2 3	Icosidodecahedron r{3,5} 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 vertices fully, until each original edge is reduced to its midpoint.

Quasiregular tilings

This sequence continues as the trihexagonal tiling, vertex figure (3.6)² - a quasiregular tiling based on the triangular tiling and hexagonal tiling.

Regular	Dual regular	Quasiregular combination	Vertex figure
Hexagonal tiling {6,3} 6 | 2 3	Triangular tiling {3,6} 3 | 2 6	Trihexagonal tiling r{6,3} 2 | 3 6	(3.6)2

The checkerboard pattern is a quasiregular coloring of the square tiling, vertex figure (4.4)²:

Regular	Dual regular	Quasiregular combination	Vertex figure
{4,4} 4 | 2 4	{4,4} 4 | 2 4	r{4,4} 2 | 4 4	(4.4)2

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

h{6,3} 3 | 3 3 =

In the hyperbolic plane, this sequence continues further, for example the triheptagonal tiling, vertex figure (3.7)² - a quasiregular tiling based on the order-7 triangular tiling and heptagonal tiling.

Regular	Dual regular	Quasiregular combination	Vertex figure
Heptagonal tiling {7,3} 7 | 2 3	Triangular tiling {3,7} 3 | 2 7	Triheptagonal tiling r{3,7} 2 | 3 7	(3.7)2

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 ${\displaystyle {\begin{Bmatrix}3\\5/2\end{Bmatrix}}}$, and the dodecadodecahedron ${\displaystyle {\begin{Bmatrix}5\\5/2\end{Bmatrix}}}$:

Regular	Dual regular	Quasiregular common core	Vertex figure
Great stellated dodecahedron {5/2,3} 3 | 2 5/2	Great icosahedron {3,5/2} 5/2 | 2 3	Great icosidodecahedron r{3,5/2} 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	Dodecadodecahedron r{5,5/2} 2 | 5 5/2	5.5/2.5.5/2

Nine more are the hemipolyhedra, which are faceted forms of the aforementioned quasiregular polyhedra derived from rectification of regular polyhedra. These include equatorial faces passing through the centre of the polyhedra:

Quasiregular (rectified)	Tetratetrahedron	Cuboctahedron	Icosidodecahedron	Great icosidodecahedron	Dodecadodecahedron
Quasiregular (hemipolyhedra)	Tetrahemihexahedron 3/2 3 | 2	Octahemioctahedron 3/2 3 | 3	Small icosihemidodecahedron 3/2 3 | 5	Great icosihemidodecahedron 3/2 3 | 5/3	Small dodecahemicosahedron 5/3 5/2 | 3
Vertex figure	3.4.3/2.4	3.6.3/2.6	3.10.3/2.10	3.10/3.3/2.10/3	5/2.6.5/3.6
Quasiregular (hemipolyhedra)		Cubohemioctahedron 4/3 4 | 3	Small dodecahemidodecahedron 5/4 5 | 5	Great dodecahemidodecahedron 5/3 5/2 | 5/3	Great dodecahemicosahedron 5/4 5 | 3
Vertex figure		4.6.4/3.6	5.10.5/4.10	5/2.10/3.5/3.10/3	5.6.5/4.6

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

Image	Faceted form Wythoff symbol Coxeter diagram	Vertex figure
	Ditrigonal dodecadodecahedron 3 | 5/3 5 or	(5.5/3)3
	Small ditrigonal icosidodecahedron 3 | 5/2 3 or	(3.5/2)3
	Great ditrigonal icosidodecahedron 3/2 | 3 5 or	((3.5)3)/2

In the Euclidean plane, the sequence of hemipolyhedra continues with the following four star tilings, where apeirogons appear as the aforementioned equatorial polygons:

Original rectified tiling	Edge diagram	Solid	Vertex Config	Wythoff	Symmetry group
Square tiling			4.∞.4/3.∞ 4.∞.-4.∞	4/3 4 | ∞	p4m
Triangular tiling			(3.∞.3.∞.3.∞)/2	3/2 | 3 ∞	p6m
Trihexagonal tiling			6.∞.6/5.∞ 6.∞.-6.∞	6/5 6 | ∞
			∞.3.∞.3/2 ∞.3.∞.-3	3/2 3 | ∞

Some authorities argue that, since the duals of the quasiregular solids share the same symmetries, these duals should be called quasiregular too. But not everybody uses this terminology. 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 configurations 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.

In higher dimensions, Coxeter defined a quasiregular polytope or honeycomb to have regular facets and quasiregular vertex figures. It follows that all vertex figures are congruent and that there are two kinds of facets, which alternate.[2]

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,p,q}
Space	Finite	Affine	Compact	Paracompact
Schläfli symbol	h{4,3,3}	h{4,3,4}	h{4,3,5}	h{4,3,6}	h{4,4,3}	h{4,4,4}
	${\displaystyle \left\{3,{3 \atop 3}\right\}}$	${\displaystyle \left\{3,{4 \atop 3}\right\}}$	${\displaystyle \left\{3,{5 \atop 3}\right\}}$	${\displaystyle \left\{3,{6 \atop 3}\right\}}$	${\displaystyle \left\{4,{4 \atop 3}\right\}}$	${\displaystyle \left\{4,{4 \atop 4}\right\}}$
Coxeter diagram	↔	↔	↔	↔	↔	↔
	↔					↔
Image
Vertex figure r{p,3}

Regular polychora or honeycombs of the form {p,3,4} or can have their symmetry cut in half as into 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, = .

Common vertex figure is the quasiregular tetratetrahedron, , same as regular octahedron

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} = ${\displaystyle \left\{3,{3 \atop 3}\right\}}$	{4,3,4} {4,31,1} = ${\displaystyle \left\{4,{3 \atop 3}\right\}}$	{5,3,4} {5,31,1} = ${\displaystyle \left\{5,{3 \atop 3}\right\}}$	{6,3,4} {6,31,1} = ${\displaystyle \left\{6,{3 \atop 3}\right\}}$
Coxeter diagram	=	=	=	=
Image
Cells {p,3}

Similarly regular hyperbolic honeycombs of the form {p,3,6} or can have their symmetry cut in half as into quasiregular form , creating alternately colored {p,3} cells. They have six cells around each edge, alternating in 2 colors. Their vertex figures are quasiregular triangular tilings, .

The common vertex figure is a quasiregular triangular tiling, =

Hyperbolic uniform honeycombs: {p,3,6} and {p,3^[3]}

Form	Paracompact				Noncompact
Name	{3,3,6} {3,3[3]}	{4,3,6} {4,3[3]}	{5,3,6} {5,3[3]}	{6,3,6} {6,3[3]}	{7,3,6} {7,3[3]}	{8,3,6} {8,3[3]}	... {∞,3,6} {∞,3[3]}
Image
Cells	{3,3}	{4,3}	{5,3}	{6,3}	{7,3}	{8,3}	{∞,3}

• Chiral polytope
• Rectification (geometry)

• Cromwell, P. Polyhedra, Cambridge University Press (1977).
• Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, 2.3 Quasi-Regular Polyhedra. (p. 17), Quasi-regular honeycombs p.69

• Weisstein, Eric W. "Quasiregular polyhedron". MathWorld.
• Weisstein, Eric W. "Uniform polyhedron". MathWorld. Quasi-regular polyhedra: (p.q)^r
• George Hart, Quasiregular polyhedra