Copositive matrix

Quasiregular figures

(3.3)2	(3.4)2	(3.5)2	(3.6)2	(3.7)2	(3.8)2	(3.∞)2
$\left\{\begin{array}{c}3\\ 3\end{array}\right\}$	$\left\{\begin{array}{c}3\\ 4\end{array}\right\}$	$\left\{\begin{array}{c}3\\ 5\end{array}\right\}$	$\left\{\begin{array}{c}3\\ 6\end{array}\right\}$	$\left\{\begin{array}{c}3\\ 7\end{array}\right\}$	$\left\{\begin{array}{c}3\\ 8\end{array}\right\}$	$\left\{\begin{array}{c}3\\ \mathrm{\infty }\end{array}\right\}$
r{3,3}	r{3,4}	r{3,5}	r{3,6}	r{3,7}	r{3,8}	r{3,∞}
Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD
File:Uniform polyhedron-33-t1.png	File:Uniform polyhedron-43-t1.png	File:Uniform polyhedron-53-t1.png	File:Uniform polyhedron-63-t1.png	File:Uniform tiling 73-t1.png	File:Uniform tiling 83-t1.png	File:Uniform tiling infin32-t1.png
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 $\left\{\begin{array}{c}p\\ q\end{array}\right\}$ 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{∞,∞}
$\left\{\begin{array}{c}3\\ 3\end{array}\right\}$	$\left\{\begin{array}{c}4\\ 4\end{array}\right\}$	$\left\{\begin{array}{c}5\\ 5\end{array}\right\}$	$\left\{\begin{array}{c}6\\ 6\end{array}\right\}$	$\left\{\begin{array}{c}7\\ 7\end{array}\right\}$	$\left\{\begin{array}{c}8\\ 8\end{array}\right\}$	$\left\{\begin{array}{c}\mathrm{\infty }\\ \mathrm{\infty }\end{array}\right\}$
(3.3)2	(4.4)2	(5.5)2	(6.6)2	(7.7)2	(8.8)2	(∞.∞)2
Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD
File:Uniform polyhedron-33-t1.png	File:Uniform tiling 44-t1.png square tiling	File:H2 tiling 255-2.png order-4 pentagonal tiling	File:H2 tiling 266-2.png order-4 hexagonal tiling	File:H2 tiling 277-2.png order-4 heptagonal tiling	File:H2 tiling 288-2.png order-4 octagonal tiling	File:H2 tiling 2ii-2.png 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
Template:CDD	Template:CDD	Template:CDD				Template:CDD
File:Uniform tiling 333-t1.png	File:H2 tiling 344-2.png	File:H2 tiling 355-2.png	File:H2 tiling 366-2.png	File:H2 tiling 377-2.png	File:H2 tiling 388-2.png	File:H2 tiling 3ii-2.png
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
Template:CDD	Template:CDD	Template:CDD				Template:CDD
File:H2 tiling 334-4.png	File:H2 tiling 444-2.png	File:H2 tiling 455-2.png	File:H2 tiling 466-2.png	File:H2 tiling 477-2.png	File:H2 tiling 488-2.png	File:H2 tiling 4ii-2.png
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), (3_a.3_b)², alternating two colors of triangular faces. Similarly the square tiling (4_a.4_b)² can be considered quasiregular, colored as a checkerboard. Also the triangular tiling can have alternately colored triangle faces, (3_a.3_b)³.

Wythoff construction

File:Wythoffian construction diagram.png 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

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.

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

The convex quasiregular polyhedra

Template:See There are two 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 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
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.png Tetratetrahedron (Octahedron) Template:CDD 2 | 3 3	File:Tetratetrahedron vertfig.png 3.3.3.3
File:Uniform polyhedron-43-t0.png Cube {4,3} Template:CDD 3 | 2 4	File:Uniform polyhedron-43-t2.png Octahedron {3,4} Template:CDD 4 | 2 3	File:Uniform polyhedron-43-t1.png Cuboctahedron Template:CDD 2 | 3 4	File:Cuboctahedron vertfig.png 3.4.3.4
File:Uniform polyhedron-53-t0.png Dodecahedron {5,3} Template:CDD 3 | 2 5	File:Uniform polyhedron-53-t2.png Icosahedron {3,5} Template:CDD 5 | 2 3	File:Uniform polyhedron-53-t1.png Icosidodecahedron 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 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
File:Uniform tiling 63-t0.png Hexagonal tiling {6,3} Template:CDD 6 | 2 3	File:Uniform tiling 63-t2.png Triangular tiling {3,6} Template:CDD 3 | 2 6	File:Uniform tiling 63-t1.png Trihexagonal tiling Template:CDD 2 | 3 6	File:Trihexagonal tiling vertfig.png 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
File:Uniform tiling 44-t0.png {4,4} Template:CDD 4 | 2 4	File:Uniform tiling 44-t2.png {4,4} Template:CDD 4 | 2 4	File:Uniform tiling 44-t1.png Template:CDD 2 | 4 4	File:Square tiling vertfig.png 4.4.4.4

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.png 3 | 3 3 Template:CDD

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
File:Uniform tiling 73-t0.png Heptagonal tiling {7,3} Template:CDD 7 | 2 3	File:Uniform tiling 73-t2.png Triangular tiling {3,7} Template:CDD 3 | 2 7	File:Uniform tiling 73-t1.png Triheptagonal tiling Template:CDD 2 | 3 7	File:Triheptagonal tiling vertfig.png 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 $\left\{\begin{array}{c}3\\ 5/2\end{array}\right\}$ and the dodecadodecahedron $\left\{\begin{array}{c}5\\ 5/2\end{array}\right\}$:

Regular	Dual regular	Quasiregular	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   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   Template:CDD 2 | 5 5/2	File:Dodecadodecahedron vertfig.png 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
File:Ditrigonal dodecadodecahedron.png	Ditrigonal dodecadodecahedron 3 | 5/3 5 File:Ditrigonal dodecadodecahedron cd.png	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	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	File:Great ditrigonal icosidodecahedron vertfig.png ((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 Template:CDD

File:Hexahedron.svg	File:Rhombicdodecahedron.jpg	File:Rhombictriacontahedron.svg	File:Rhombic star tiling.png	File:Order73 qreg rhombic til.png	File:Uniform dual tiling 433-t01-yellow.png
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 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.

Quasiregular polychora and honeycombs: h{4,p,q}

Space	Euclidean 4-space	Euclidean 3-space	Hyperbolic 3-space
Name	h{4,3,3} = $\{3,\genfrac{}{}{0ex}{}{3}{3}\}$	h{4,3,4} = $\{3,\genfrac{}{}{0ex}{}{3}{4}\}$	h{4,3,5} = $\{3,\genfrac{}{}{0ex}{}{3}{5}\}$	h{4,3,6} = $\{3,\genfrac{}{}{0ex}{}{3}{6}\}$	h{4,4,3} = $\{4,\genfrac{}{}{0ex}{}{3}{4}\}$	h{4,4,4} = $\{4,\genfrac{}{}{0ex}{}{4}{4}\}$
Regular	{3,3,4}	-	-	-	-	{4,4,4}
Coxeter diagram	Template:CDD = Template:CDD	Template:CDD = Template:CDD	Template:CDD = Template:CDD	Template:CDD = Template:CDD	Template:CDD and Template:CDD	Template:CDD and Template:CDD
Image	File:16-cell nets.png	File:Tetrahedral-octahedral honeycomb.png	File:Alternated order 5 cubic honeycomb.png			File:H3 444 FC boundary.png
Vertex figure r{p,3}	File:Uniform polyhedron-33-t1.png Template:CDD	File:Uniform polyhedron-43-t1.png Template:CDD	File:Uniform polyhedron-53-t1.png Template:CDD	File:Uniform polyhedron-63-t1.png Template:CDD	File:Uniform polyhedron-43-t1.png Template:CDD	File:Square tiling uniform coloring 7.png Template:CDD

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

File:Uniform polyhedron-33-t1.png

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

Regular and Quasiregular honeycombs: {p,3,4} and {p,3^1,1}

Space	Euclidean 4-space	Euclidean 3-space	Hyperbolic 3-space
Name	{3,3,4} {3,31,1} = $\{3,\genfrac{}{}{0ex}{}{3}{3}\}$	{4,3,4} {4,31,1} = $\{4,\genfrac{}{}{0ex}{}{3}{3}\}$	{5,3,4} {5,31,1} = $\{5,\genfrac{}{}{0ex}{}{3}{3}\}$	{6,3,4} {6,31,1} = $\{6,\genfrac{}{}{0ex}{}{3}{3}\}$
Coxeter diagram	Template:CDD and Template:CDD	Template:CDD and Template:CDD	Template:CDD and Template:CDD	Template:CDD and Template:CDD
Image	File:16-cell nets.png	File:Bicolor cubic honeycomb.png	File:H3 534 CC center.png	File:H3 634 FC boundary.png
Cells {p,3}	File:Uniform polyhedron-33-t0.png Template:CDD	File:Uniform polyhedron-43-t0.png Template:CDD	File:Uniform polyhedron-53-t0.png Template:CDD	File:Uniform polyhedron-63-t0.png Template:CDD

Similarly regular hyperbolic honeycombs of the form {p,3,6} or Template:CDD can have their symmetry cut in half as 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.png

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

Regular and quasiregular hyperbolic honeycombs: {p,3,6} and {p,3^[3]}

Name	{3,3,6} and {3,3[3]}	{4,3,6} and {4,3[3]}	{5,3,6} and {5,3[3]}	{6,3,6} and {6,3[3]}
Coxeter diagram	Template:CDD and Template:CDD	Template:CDD and Template:CDD	Template:CDD and Template:CDD	Template:CDD and Template:CDD
Image	File:H3 336 CC center.png	File:H3 436 CC center.png	File:H3 536 CC center.png	File:H3 636 FC boundary.png
Cells {p,3}	File:Uniform polyhedron-33-t0.png Template:CDD	File:Uniform polyhedron-43-t0.png Template:CDD	File:Uniform polyhedron-53-t0.png Template:CDD	File:Uniform polyhedron-63-t0.png Template:CDD

See also

• Chiral polytope
• Rectification (geometry)

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

• 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)

External links

• 22 year-old Systems Analyst Rave from Merrickville-Wolford, has lots of hobbies and interests including quick cars, property developers in singapore and baking. Always loves visiting spots like Historic Monuments Zone of Querétaro.
  
  Here is my web site - cottagehillchurch.com
• 22 year-old Systems Analyst Rave from Merrickville-Wolford, has lots of hobbies and interests including quick cars, property developers in singapore and baking. Always loves visiting spots like Historic Monuments Zone of Querétaro.
  
  Here is my web site - cottagehillchurch.com Quasi-regular polyhedra: (p.q)^r
• George Hart, Quasiregular polyhedra