Quasiregular polyhedron

Quasiregular figures Right triangle domains (p q 2), Template:CDD = r{p,q} r{4,3} r{5,3} r{6,3} r{7,3}... r{∞,3} Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD File:Uniform polyhedron-43-t1.svg (3.4)2 File:Uniform polyhedron-53-t1.svg (3.5)2 File:Uniform tiling 63-t1.svg (3.6)2 File:Triheptagonal tiling.svg (3.7)2 File:H2 tiling 23i-2.png (3.∞)2 Isosceles triangle domains (p p 3), Template:CDD = Template:CDD = h{6,p} h{6,4} h{6,5} h{6,6} h{6,7}... h{6,∞} Template:CDD = Template:CDD Template:CDD = Template:CDD Template:CDD = Template:CDD Template:CDD = Template:CDD Template:CDD = Template:CDD File:H2 tiling 344-4.png (4.3)4 File:H2 tiling 355-4.png (5.3)5 File:H2 tiling 366-4.png (6.3)6 File:H2 tiling 377-4.png (7.3)7 File:H2 tiling 3ii-4.png (∞.3)∞ Isosceles triangle domains (p p 4), Template:CDD = Template:CDD = h{8,p} h{8,3} h{8,5} h{8,6} h{8,7}... h{8,∞} Template:CDD =Template:CDD Template:CDD =Template:CDD Template:CDD =Template:CDD Template:CDD =Template:CDD Template:CDD =Template:CDD File:H2 tiling 334-1.png (4.3)3 File:H2 tiling 455-1.png (4.5)5 File:H2 tiling 466-1.png (4.6)6 File:H2 tiling 477-1.png (4.7)7 File:H2 tiling 4ii-1.png (4.∞)∞ Scalene triangle domain (5 4 3), Template:CDD Template:CDD Template:CDD Template:CDD File:H2 tiling 345-1.png (3.5)4 File:H2 tiling 345-2.png (4.5)3 File:H2 tiling 345-4.png (3.4)5 A quasiregular polyhedron or tiling has exactly two kinds of regular face, which alternate around each vertex. Their vertex figures are isogonal polygons.

Regular and quasiregular figures Right triangle domains (p p 2), Template:CDD = Template:CDD = r{p,p} = {p,4}1⁄2 {3,4}1⁄2 r{3,3} {4,4}1⁄2 r{4,4} {5,4}1⁄2 r{5,5} {6,4}1⁄2 r{6,6}... {∞,4}1⁄2 r{∞,∞} Template:CDD = Template:CDD Template:CDD = Template:CDD Template:CDD = Template:CDD Template:CDD = Template:CDD Template:CDD = Template:CDD File:Uniform polyhedron-33-t1.svg (3.3)2 File:Uniform tiling 44-t1.svg (4.4)2 File:H2 tiling 255-2.png (5.5)2 File:H2 tiling 266-2.png (6.6)2 File:H2 tiling 2ii-2.png (∞.∞)2 Isosceles triangle domains (p p 3), Template:CDD = Template:CDD = {p,6}1⁄2 {3,6}1⁄2 {4,6}1⁄2 {5,6}1⁄2 {6,6}1⁄2... {∞,6}1⁄2 Template:CDD = Template:CDD Template:CDD = Template:CDD Template:CDD = Template:CDD Template:CDD = Template:CDD Template:CDD = Template:CDD File:Uniform tiling 333-t1.svg (3.3)3 File:H2 tiling 344-2.png (4.4)3 File:H2 tiling 355-2.png (5.5)3 File:H2 tiling 366-2.png (6.6)3 File:H2 tiling 3ii-2.png (∞.∞)3 Isosceles triangle domains (p p 4), Template:CDD = Template:CDD = {p,8}1⁄2 {3,8}1⁄2 {4,8}1⁄2 {5,8}1⁄2 {6,8}1⁄2... {∞,8}1⁄2 Template:CDD =Template:CDD Template:CDD =Template:CDD Template:CDD =Template:CDD Template:CDD =Template:CDD Template:CDD =Template:CDD File:H2 tiling 334-4.png (3.3)4 File:H2 tiling 444-2.png (4.4)4 File:H2 tiling 455-2.png (5.5)4 File:H2 tiling 466-2.png (6.6)4 File:H2 tiling 4ii-2.png(∞.∞)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).

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

Their dual figures are face-transitive and edge-transitive; they have exactly two kinds of regular vertex figures, which alternate around each face. They are sometimes also considered quasiregular.

There are only two convex quasiregular polyhedra: the cuboctahedron and the icosidodecahedron. Their names, given by Kepler, come from recognizing that their faces are all the faces (turned differently) of the dual-pair cube and octahedron, in the first case, and of 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 $\left\{\begin{array}{c}p\\ q\end{array}\right\}$ or r{p,q}, to represent that their faces are all the faces (turned differently) of both the regular {p,q} and the 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) sequences 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 polyhedra and tilings with an even number of faces at each vertex can also be considered quasiregular by differentiating between faces of the same order, by representing them differently, like coloring them alternately (without defining any surface orientation). A regular figure with Schläfli symbol {p,q} can be considered quasiregular, with vertex configuration (p.p)^q/2, if q is even.

Examples:

The regular octahedron, with Schläfli symbol {3,4} and 4 being even, can be considered quasiregular as a tetratetrahedron (2 sets of 4 triangles of the tetrahedron), with vertex configuration (3.3)^4/2 = (3_a.3_b)², alternating two colors of triangular faces.

The square tiling, with vertex configuration 4⁴ and 4 being even, can be considered quasiregular, with vertex configuration (4.4)^4/2 = (4_a.4_b)², colored as a checkerboard.

The triangular tiling, with vertex configuration 3⁶ and 6 being even, can be considered quasiregular, with vertex configuration (3.3)^6/2 = (3_a.3_b)³, alternating two colors of triangular faces.

Wythoff construction

File:Wythoffian construction diagram.svg 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.

File:Wythoff construction-pqr.png

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.

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
$\left\{\begin{array}{c}p,q\end{array}\right\}$	{p,q}	Template:CDD	2 p
$\left\{\begin{array}{c}q,p\end{array}\right\}$	{q,p}	Template:CDD	2 q
$\left\{\begin{array}{c}p\\ q\end{array}\right\}$	r{p,q}	Template:CDD or Template:CDD	p q

The convex quasiregular polyhedra

There are two uniform convex quasiregular polyhedra:

1. The cuboctahedron $\left\{\begin{array}{c}3\\ 4\end{array}\right\}$, vertex configuration (3.4)², Coxeter-Dynkin diagram Template:CDD
2. The icosidodecahedron $\left\{\begin{array}{c}3\\ 5\end{array}\right\}$, vertex configuration (3.5)², Coxeter-Dynkin diagram Template:CDD

In addition, the octahedron, which is also regular, $\left\{\begin{array}{c}3\\ 3\end{array}\right\}$, 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 Template:CDD

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 $\cap$ octahedron, and icosahedron $\cap$ 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 $\cap$ tetrahedron.

Regular	Dual regular	Quasiregular common core	Vertex figure
File:Uniform polyhedron-33-t0.png Tetrahedron {3,3} Template:CDD 3 | 2 3	File:Uniform polyhedron-33-t2.png Tetrahedron {3,3} Template:CDD 3 | 2 3	File:Uniform polyhedron-33-t1.svg Tetratetrahedron r{3,3} Template:CDD 2 | 3 3	File:Tetratetrahedron vertfig.png 3.3.3.3
File:Uniform polyhedron-43-t0.svg Cube {4,3} Template:CDD 3 | 2 4	File:Uniform polyhedron-43-t2.svg Octahedron {3,4} Template:CDD 4 | 2 3	File:Uniform polyhedron-43-t1.svg Cuboctahedron r{3,4} Template:CDD 2 | 3 4	File:Cuboctahedron vertfig.png 3.4.3.4
File:Uniform polyhedron-53-t0.svg Dodecahedron {5,3} Template:CDD 3 | 2 5	File:Uniform polyhedron-53-t2.svg Icosahedron {3,5} Template:CDD 5 | 2 3	File:Uniform polyhedron-53-t1.svg Icosidodecahedron r{3,5} Template:CDD 2 | 3 5	File:Icosidodecahedron vertfig.png 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
File:Uniform tiling 63-t0.svg Hexagonal tiling {6,3} Template:CDD 6 | 2 3	File:Uniform tiling 63-t2.svg Triangular tiling {3,6} Template:CDD 3 | 2 6	File:Uniform tiling 63-t1.svg Trihexagonal tiling r{6,3} Template:CDD 2 | 3 6	File:Trihexagonal tiling vertfig.svg (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
File:Uniform tiling 44-t0.svg {4,4} Template:CDD 4 | 2 4	File:Uniform tiling 44-t2.svg {4,4} Template:CDD 4 | 2 4	File:Uniform tiling 44-t1.svg r{4,4} Template:CDD 2 | 4 4	File:Square tiling vertfig.svg (4.4)2

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

File:Uniform tiling 333-t1.svg h{6,3} 3 | 3 3 Template:CDD = Template:CDD

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
File:Heptagonal tiling.svg Heptagonal tiling {7,3} Template:CDD 7 | 2 3	File:Order-7 triangular tiling.svg Triangular tiling {3,7} Template:CDD 3 | 2 7	File:Triheptagonal tiling.svg Triheptagonal tiling r{3,7} Template:CDD 2 | 3 7	File:Triheptagonal tiling vertfig.png (3.7)2

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 $\left\{\begin{array}{c}3\\ 5/2\end{array}\right\}$, and the dodecadodecahedron $\left\{\begin{array}{c}5\\ 5/2\end{array}\right\}$:

Regular	Dual regular	Quasiregular common core	Vertex figure
File:Great stellated dodecahedron.png Great stellated dodecahedron {5/2,3} Template:CDD 3 | 2 5/2	File:Great icosahedron.png Great icosahedron {3,5/2} Template:CDD 5/2 | 2 3	File:Great icosidodecahedron.png Great icosidodecahedron r{3,5/2} Template:CDD 2 | 3 5/2	File:Great icosidodecahedron vertfig.png 3.5/2.3.5/2
File:Small stellated dodecahedron.png Small stellated dodecahedron {5/2,5} Template:CDD 5 | 2 5/2	File:Great dodecahedron.png Great dodecahedron {5,5/2} Template:CDD 5/2 | 2 5	File:Dodecadodecahedron.png Dodecadodecahedron r{5,5/2} Template:CDD 2 | 5 5/2	File:Dodecadodecahedron vertfig.png 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)	File:Rectified tetrahedron.png Tetratetrahedron	File:Cuboctahedron.png Cuboctahedron	File:Icosidodecahedron.png Icosidodecahedron	File:Great icosidodecahedron.png Great icosidodecahedron	File:Dodecadodecahedron.png Dodecadodecahedron
Quasiregular (hemipolyhedra)	File:Tetrahemihexahedron.png Tetrahemihexahedron 3/2 3 | 2	File:Octahemioctahedron.png Octahemioctahedron 3/2 3 | 3	File:Small icosihemidodecahedron.png Small icosihemidodecahedron 3/2 3 | 5	File:Great icosihemidodecahedron.png Great icosihemidodecahedron 3/2 3 | 5/3	File:Small dodecahemicosahedron.png Small dodecahemicosahedron 5/3 5/2 | 3
Vertex figure	File:Tetrahemihexahedron vertfig.png 3.4.3/2.4	File:Octahemioctahedron vertfig.png 3.6.3/2.6	File:Small icosihemidodecahedron vertfig.png 3.10.3/2.10	File:Great icosihemidodecahedron vertfig.png 3.10/3.3/2.10/3	File:Small dodecahemicosahedron vertfig.png 5/2.6.5/3.6
Quasiregular (hemipolyhedra)		File:Cubohemioctahedron.png Cubohemioctahedron 4/3 4 | 3	File:Small dodecahemidodecahedron.png Small dodecahemidodecahedron 5/4 5 | 5	File:Great dodecahemidodecahedron.png Great dodecahemidodecahedron 5/3 5/2 | 5/3	File:Great dodecahemicosahedron.png Great dodecahemicosahedron 5/4 5 | 3
Vertex figure		File:Cubohemioctahedron vertfig.png 4.6.4/3.6	File:Small dodecahemidodecahedron vertfig.png 5.10.5/4.10	File:Great dodecahemidodecahedron vertfig.png 5/2.10/3.5/3.10/3	File:Great dodecahemicosahedron vertfig.png 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
File:Ditrigonal dodecadodecahedron.png	Ditrigonal dodecadodecahedron 3 | 5/3 5 File:Ditrigonal dodecadodecahedron cd.png or Template:CDD	File:Ditrigonal dodecadodecahedron vertfig.png (5.5/3)3
File:Small ditrigonal icosidodecahedron.png	Small ditrigonal icosidodecahedron 3 | 5/2 3 File:Small ditrigonal icosidodecahedron cd.png or Template:CDD	File:Small ditrigonal icosidodecahedron vertfig.png (3.5/2)3
File:Great ditrigonal icosidodecahedron.png	Great ditrigonal icosidodecahedron 3/2 | 3 5 File:Great ditrigonal icosidodecahedron cd.png or Template:CDD	File:Great ditrigonal icosidodecahedron vertfig.png ((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
File:Uniform tiling 44-t1.svg Square tiling	File:4.oo.4-3.oo tiling frame.png	File:Star tiling sha.gif	4.∞.4/3.∞ 4.∞.-4.∞	∞	p4m
File:Uniform tiling 333-t1.svg Triangular tiling	File:3.oo.3.oo.3oo tiling-frame.png	File:Star tiling ditatha.gif	(3.∞.3.∞.3.∞)/2	3 ∞	p6m
File:Uniform tiling 63-t1.svg Trihexagonal tiling	File:6.oo.6-5.oo tiling-frame.png	File:Star tiling hoha.gif	6.∞.6/5.∞ 6.∞.-6.∞	∞
		File:Star tiling tha.gif	∞.3.∞.3/2 ∞.3.∞.-3	∞

Quasiregular duals

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 Template:CDD

File:Hexahedron.svg	File:Rhombicdodecahedron.jpg	File:Rhombictriacontahedron.svg	File:Rhombic star tiling.svg	File:7-3 rhombille tiling.svg	File:H2-8-3-rhombic.svg
Cube V(3.3)2 Template:CDD	Rhombic dodecahedron V(3.4)2 Template:CDD	Rhombic triacontahedron V(3.5)2 Template:CDD	Rhombille tiling V(3.6)2 Template:CDD	V(3.7)2 Template:CDD	V(3.8)2 Template:CDD

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 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.

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

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

In hyperbolic 3-space, one quasiregular honeycomb is the alternated order-5 cubic honeycomb, h{4,3,5}, Coxeter diagrams: Template:CDD = Template:CDD, composed of alternating tetrahedral and icosahedral cells. Its vertex figure is the quasiregular icosidodecahedron, Template:CDD. 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, Template:CDD.

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

File:Uniform polyhedron-33-t1.svg

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

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

File:Uniform tiling 333-t1.svg

The common vertex figure is a quasiregular triangular tiling, Template:CDD = Template:CDD

See also

• Chiral polytope
• Rectification (geometry)

References

• Cromwell, P. Polyhedra, Cambridge University Press (1977).
• Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, Template:Isbn, 2.3 Quasi-Regular Polyhedra. (p. 17), Quasi-regular honeycombs p.69

External links

• Quasi-regular polyhedra: (p.q)^r
• George Hart, Quasiregular polyhedra