View source for Quasiregular polyhedron

{{Short description|Polyhedron with two kinds of faces}}
{| align=right
|
{| class=wikitable width=425
|+ Quasiregular figures
|-
!colspan=6|Right triangle domains (p q 2), {{CDD|node|p|node_1|q|node}} = r{p,q}
|-
![[cuboctahedron|r{4,3}]]
![[icosidodecahedron|r{5,3}]]
![[trihexagonal tiling|r{6,3}]]
![[triheptagonal tiling|r{7,3}]]...
!r{&infin;,3}
|-
!{{CDD|node|4|node_1|3|node}}
!{{CDD|node|5|node_1|3|node}}
!{{CDD|node|6|node_1|3|node}}
!{{CDD|node|7|node_1|3|node}}
!{{CDD|node|infin|node_1|3|node}}
|- align=center
|[[File:Uniform polyhedron-43-t1.svg|60px]]<BR>(3.4)<sup>2</sup>
|[[File:Uniform polyhedron-53-t1.svg|60px]]<BR>(3.5)<sup>2</sup>
|[[File:Uniform_tiling_63-t1.svg|60px]]<BR>(3.6)<sup>2</sup>
|[[File:Triheptagonal tiling.svg|60px]]<BR>(3.7)<sup>2</sup>
|[[File:H2 tiling 23i-2.png|60px]]<BR>(3.&infin;)<sup>2</sup>
|-
!colspan=6| Isosceles triangle domains (p p 3), {{CDD|branch_10ru|split2-pp|node}} = {{CDD|node_h|6|node|p|node}} = h{6,p}
|-
![[Alternated order-4 hexagonal tiling|h{6,4}]]
!h{6,5}
!h{6,6}
!h{6,7}...
!h{6,&infin;}
|-
!{{CDD|node_h|6|node|4|node}} = {{CDD|branch_10ru|split2-44|node}}
!{{CDD|node_h|6|node|5|node}} = {{CDD|branch_10ru|split2-55|node}}
!{{CDD|node_h|6|node|6|node}} = {{CDD|branch_10ru|split2-66|node}}
!{{CDD|node_h|6|node|7|node}} = {{CDD|branch_10ru|split2-77|node}}
!{{CDD|node_h|6|node|infin|node}} = {{CDD|branch_10ru|split2-ii|node}}
|- align=center
|[[File:H2 tiling 344-4.png|60px]]<BR>(4.3)<sup>4</sup>
|[[File:H2 tiling 355-4.png|60px]]<BR>(5.3)<sup>5</sup>
|[[File:H2 tiling 366-4.png|60px]]<BR>(6.3)<sup>6</sup>
|[[File:H2 tiling 377-4.png|60px]]<BR>(7.3)<sup>7</sup>
|[[File:H2 tiling 3ii-4.png|60px]]<BR>(&infin;.3)<sup>&infin;</sup>

|-
!colspan=6| Isosceles triangle domains (p p 4), {{CDD|label4|branch_10ru|split2-pp|node}} = {{CDD|node_h|8|node|p|node}} = h{8,p}
|-
![[Alternated octagonal tiling|h{8,3}]]
!h{8,5}
!h{8,6}
!h{8,7}...
!h{8,&infin;}
|-
!{{CDD|node_h|8|node|3|node}} ={{CDD|label4|branch_10ru|split2|node}}
!{{CDD|node_h|8|node|5|node}} ={{CDD|label4|branch_10ru|split2-55|node}}
!{{CDD|node_h|8|node|6|node}} ={{CDD|label4|branch_10ru|split2-66|node}}
!{{CDD|node_h|8|node|7|node}} ={{CDD|label4|branch_10ru|split2-77|node}}
!{{CDD|node_h|8|node|infin|node}} ={{CDD|label4|branch_10ru|split2-ii|node}}
|- align=center
|[[File:H2 tiling 334-1.png|60px]]<BR>(4.3)<sup>3</sup>
|[[File:H2 tiling 455-1.png|60px]]<BR>(4.5)<sup>5</sup>
|[[File:H2 tiling 466-1.png|60px]]<BR>(4.6)<sup>6</sup>
|[[File:H2 tiling 477-1.png|60px]]<BR>(4.7)<sup>7</sup>
|[[File:H2 tiling 4ii-1.png|60px]]<BR>(4.&infin;)<sup>&infin;</sup>

|-
!colspan=6| Scalene triangle domain (5 4 3), {{CDD|branch|split2-45|node}}
|-
!{{CDD|branch_01rd|split2-45|node}}
!{{CDD|branch|split2-45|node_1}}
!{{CDD|branch_10ru|split2-45|node}}
!
!
|- align=center
|[[File:H2 tiling 345-1.png|60px]]<BR>(3.5)<sup>4</sup>
|[[File:H2 tiling 345-2.png|60px]]<BR>(4.5)<sup>3</sup>
|[[File:H2 tiling 345-4.png|60px]]<BR>(3.4)<sup>5</sup>
|
|
|-
|colspan=7|A '''quasiregular polyhedron''' or '''tiling''' has exactly two kinds of regular face, which alternate around each vertex. Their [[vertex figure]]s are [[Isogonal figure|isogonal polygons]].
|}
|-
|
{| class=wikitable width=425
|+ Regular and quasiregular figures
|-
!colspan=5|Right triangle domains (p p 2), {{CDD|node_1|split1-pp|nodes}} = {{CDD|node_1|p|node|4|node_h0}} = r{p,p} = {p,4}<sub>{{frac|1|2}}</sub>
|-
![[octahedron|{3,4}<sub>{{frac|1|2}}</sub>]]<BR>r{3,3}
![[square tiling|{4,4}<sub>{{frac|1|2}}</sub>]]<br>r{4,4}
![[order-4 pentagonal tiling|{5,4}<sub>{{frac|1|2}}</sub>]]<br>r{5,5}
![[order-4 hexagonal tiling|{6,4}<sub>{{frac|1|2}}</sub>]]<br>r{6,6}...
![[order-4 apeirogonal tiling|{&infin;,4}<sub>{{frac|1|2}}</sub>]]<br>r{&infin;,&infin;}
|-
!{{CDD|node_1|3|node|4|node_h0}} = {{CDD|node_1|split1|nodes}}
!{{CDD|node_1|4|node|4|node_h0}} = {{CDD|node_1|split1-44|nodes}}
!{{CDD|node_1|5|node|4|node_h0}} = {{CDD|node_1|split1-55|nodes}}
!{{CDD|node_1|6|node|4|node_h0}} = {{CDD|node_1|split1-66|nodes}}
!{{CDD|node_1|infin|node|4|node_h0}} = {{CDD|node_1|split1-ii|nodes}}
|- align=center
|[[File:Uniform polyhedron-33-t1.svg|60px]]<BR>(3.3)<sup>2</sup>
|[[File:Uniform tiling 44-t1.svg|60px]]<BR>(4.4)<sup>2</sup>
|[[File:H2 tiling 255-2.png|60px]]<BR>(5.5)<sup>2</sup>
|[[File:H2 tiling 266-2.png|60px]]<BR>(6.6)<sup>2</sup>
|[[File:H2 tiling 2ii-2.png|60px]]<BR>(&infin;.&infin;)<sup>2</sup>
|-
!colspan=5| Isosceles triangle domains (p p 3), {{CDD|node_1|split1-pp|branch}} = {{CDD|node_1|p|node|6|node_h0}} = {p,6}<sub>{{frac|1|2}}</sub>
|-
![[triangular tiling|{3,6}<sub>{{frac|1|2}}</sub>]]
![[order-6 square tiling|{4,6}<sub>{{frac|1|2}}</sub>]]
![[order-6 pentagonal tiling|{5,6}<sub>{{frac|1|2}}</sub>]]
![[order-6 hexagonal tiling|{6,6}<sub>{{frac|1|2}}</sub>]]...
![[order-6 apeirogonal tiling|{&infin;,6}<sub>{{frac|1|2}}</sub>]]
|-
!{{CDD|node_1|3|node|6|node_h0}} = {{CDD|node_1|split1|branch}}
!{{CDD|node_1|4|node|6|node_h0}} = {{CDD|node_1|split1-44|branch}}
!{{CDD|node_1|5|node|6|node_h0}} = {{CDD|node_1|split1-55|branch}}
!{{CDD|node_1|6|node|6|node_h0}} = {{CDD|node_1|split1-66|branch}}
!{{CDD|node_1|infin|node|6|node_h0}} = {{CDD|node_1|split1-ii|branch}}
|- align=center
|[[File:Uniform tiling 333-t1.svg|60px]]<BR>(3.3)<sup>3</sup>
|[[File:H2 tiling 344-2.png|60px]]<BR>(4.4)<sup>3</sup>
|[[File:H2 tiling 355-2.png|60px]]<BR>(5.5)<sup>3</sup>
|[[File:H2 tiling 366-2.png|60px]]<BR>(6.6)<sup>3</sup>
|[[File:H2 tiling 3ii-2.png|60px]]<BR>(&infin;.&infin;)<sup>3</sup>
|-
!colspan=6| Isosceles triangle domains (p p 4), {{CDD|node_1|split1-pp|branch|label4}} = {{CDD|node_1|p|node|8|node_h0}} = {p,8}<sub>{{frac|1|2}}</sub>
|-
![[order-8 triangular tiling|{3,8}<sub>{{frac|1|2}}</sub>]]
![[order-8 square tiling|{4,8}<sub>{{frac|1|2}}</sub>]]
![[order-8 pentagonal tiling|{5,8}<sub>{{frac|1|2}}</sub>]]
![[order-8 hexagonal tiling|{6,8}<sub>{{frac|1|2}}</sub>]]...
![[order-8 apeirogonal tiling|{&infin;,8}<sub>{{frac|1|2}}</sub>]]
|-
!{{CDD|node_1|3|node|8|node_h0}} ={{CDD|node_1|split1|branch|label4}}
!{{CDD|node_1|4|node|8|node_h0}} ={{CDD|node_1|split1-44|branch|label4}}
!{{CDD|node_1|5|node|8|node_h0}} ={{CDD|node_1|split1-55|branch|label4}}
!{{CDD|node_1|6|node|8|node_h0}} ={{CDD|node_1|split1-66|branch|label4}}
!{{CDD|node_1|infin|node|8|node_h0}} ={{CDD|node_1|split1-ii|branch|label4}}
|- align=center
|[[File:H2 tiling 334-4.png|60px]]<BR>(3.3)<sup>4</sup>
|[[File:H2 tiling 444-2.png|60px]]<BR>(4.4)<sup>4</sup>
|[[File:H2 tiling 455-2.png|60px]]<BR>(5.5)<sup>4</sup>
|[[File:H2 tiling 466-2.png|60px]]<BR>(6.6)<sup>4</sup>
|[[File:H2 tiling 4ii-2.png|60px]](&infin;.&infin;)<sup>4</sup>

|-
|colspan=6|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 polygon|regular faces]], which alternate around each [[vertex (geometry)|vertex]]. They are [[vertex-transitive]] and [[edge-transitive]], hence a step closer to [[regular polyhedra]] than the [[Semiregular polyhedron|semiregular]], which are merely vertex-transitive.

Their [[#Quasiregular_duals|dual figures]] are [[face-transitive]] and edge-transitive; they have exactly two kinds of regular [[vertex figure]]s, which alternate around each [[face (geometry)|face]]. They are sometimes also considered quasiregular.

There are only two [[Convex polytope|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 polyhedron|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]] <math>\begin{Bmatrix} p \\ q \end{Bmatrix}</math> 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)<sup>2</sup>''''').

More generally, a quasiregular figure can have a [[vertex configuration]] '''''(p.q)<sup>r</sup>''''', representing '''''r''''' (2 or more) sequences of the faces around the vertex.

[[Tessellation|Tilings]] of the plane can also be quasiregular, specifically the [[trihexagonal tiling]], with vertex configuration (3.6)<sup>2</sup>. [[Uniform tilings in hyperbolic plane|Other quasiregular tilings]] exist on the hyperbolic plane, like the [[triheptagonal tiling]], (3.7)<sup>2</sup>. Or more generally: '''''(p.q)<sup>2</sup>''''', 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)<sup>q/2</sup>''''', 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)<sup>4/2</sup> = (3<sub>a</sub>.3<sub>b</sub>)<sup>2</sup>, alternating two colors of triangular faces.

The [[square tiling]], with vertex configuration 4<sup>4</sup> and 4 being even, can be considered quasiregular, with vertex configuration (4.4)<sup>4/2</sup> = (4<sub>a</sub>.4<sub>b</sub>)<sup>2</sup>, colored as a ''checkerboard''.

The [[triangular tiling]], with vertex configuration 3<sup>6</sup> and 6 being even, can be considered quasiregular, with vertex configuration (3.3)<sup>6/2</sup> = (3<sub>a</sub>.3<sub>b</sub>)<sup>3</sup>, alternating two colors of triangular faces.

== Wythoff construction ==
{| class=wikitable width=320
|[[File:Wythoffian construction diagram.svg|320px]]<br>Regular (''p {{pipe}} 2 q'') and quasiregular polyhedra (''2 {{pipe}} 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|400px|thumb|Quasiregular polyhedra are generated from all 3 corners of the fundamental domain for [[Schwarz triangle]]s that have no right angles:<br>'''q {{pipe}} 2 p''', '''p {{pipe}} 2 q''', '''2 {{pipe}} 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.<ref>[[Coxeter|Coxeter, H.S.M.]], [[Michael S. Longuet-Higgins|Longuet-Higgins, M.S.]] and Miller, J.C.P. Uniform Polyhedra, ''Philosophical Transactions of the Royal Society of London'' '''246 A''' (1954), pp. 401–450. (Section 7, The regular and quasiregular polyhedra ''p | q r'')</ref>

The [[Coxeter-Dynkin diagram]] is another symbolic representation that shows the quasiregular relation between the two dual-regular forms: 
{| class=wikitable
!colspan=2|[[Schläfli symbol]]
![[Coxeter diagram]]
![[Wythoff symbol]]
|- align=center
|<math>\begin{Bmatrix} p , q \end{Bmatrix}</math>||{p,q}||{{CDD|node_1|p|node|q|node}}||''q {{pipe}} 2 p''
|- align=center
|<math>\begin{Bmatrix} q , p \end{Bmatrix}</math>||{q,p}|| {{CDD|node|p|node|q|node_1}}||''p {{pipe}} 2 q''
|- align=center
| <math>\begin{Bmatrix} p \\ q \end{Bmatrix}</math>||r{p,q}|| {{CDD|node|p|node_1|q|node}} or {{CDD|node_1|split1-pq|nodes}}||''2 {{pipe}} p q''
|}

==The convex quasiregular polyhedra==
{{Further|Rectification (geometry)}}
There are two uniform [[Convex polyhedron|convex]] quasiregular polyhedra:
#The [[cuboctahedron]] <math>\begin{Bmatrix} 3 \\ 4 \end{Bmatrix}</math>, vertex configuration '''(3.4)<sup>2</sup>''', [[Coxeter-Dynkin diagram]] {{CDD|node|4|node_1|3|node}}
#The [[icosidodecahedron]] <math>\begin{Bmatrix} 3 \\ 5 \end{Bmatrix}</math>, vertex configuration '''(3.5)<sup>2</sup>''', ''Coxeter-Dynkin diagram'' {{CDD|node|5|node_1|3|node}}

In addition, the [[octahedron]], which is also [[regular polyhedron|regular]], <math>\begin{Bmatrix} 3 \\ 3 \end{Bmatrix}</math>, vertex configuration '''(3.3)<sup>2</sup>''', 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'' {{CDD|node|3|node_1|3|node}}

Each of these forms the common core of a [[dual polyhedron|dual]] pair of [[regular polyhedron|regular polyhedra]]. The names of two of these give clues to the associated dual pair: respectively [[cube]] <math>\cap</math> [[octahedron]], and [[icosahedron]] <math>\cap</math> [[dodecahedron]]. The [[octahedron]] is the common core of a dual pair of [[tetrahedron|tetrahedra]] (a compound known as the [[stella octangula]]); when derived in this way, the [[octahedron]] is sometimes called the ''tetratetrahedron'', as [[tetrahedron]] <math>\cap</math> [[tetrahedron]].

{| class="wikitable"
!Regular
!Dual regular
!Quasiregular common core
![[Vertex figure]]
|- valign=top align=center
|[[File:Uniform polyhedron-33-t0.png|75px]]<br>[[Tetrahedron]]<br>{3,3}<br>{{CDD|node_1|3|node|3|node}}<br>3 | 2 3
|[[File:Uniform polyhedron-33-t2.png|75px]]<br>[[Tetrahedron]]<br>{3,3}<br>{{CDD|node|3|node|3|node_1}}<br>3 | 2 3
|[[File:Uniform polyhedron-33-t1.svg|75px]]<br>[[Octahedron|Tetratetrahedron]]<BR>r{3,3}<br>{{CDD|node|3|node_1|3|node}}<br>2 | 3 3
|[[File:Tetratetrahedron vertfig.png|75px]]<br>''3.3.3.3''
|- valign=top align=center
|[[File:Uniform polyhedron-43-t0.svg|75px]]<br>[[Cube]]<br>{4,3}<br>{{CDD|node_1|4|node|3|node}}<br>3 | 2 4
|[[File:Uniform polyhedron-43-t2.svg|75px]]<br>[[Octahedron]]<br>{3,4}<br>{{CDD|node|4|node|3|node_1}}<br>4 | 2 3
|[[File:Uniform polyhedron-43-t1.svg|75px]]<br>[[Cuboctahedron]]<br>r{3,4}<br>{{CDD|node|4|node_1|3|node}}<br>2 | 3 4
|[[File:Cuboctahedron vertfig.png|75px]]<br>''3.4.3.4''
|- valign=top align=center
|[[File:Uniform polyhedron-53-t0.svg|75px]]<br>[[Dodecahedron]]<br>{5,3}<br>{{CDD|node_1|5|node|3|node}}<br>3 | 2 5
|[[File:Uniform polyhedron-53-t2.svg|75px]]<br>[[Icosahedron]]<br>{3,5}<br>{{CDD|node|5|node|3|node_1}}<br>5 | 2 3
|[[File:Uniform polyhedron-53-t1.svg|75px]]<br>[[Icosidodecahedron]]<br>r{3,5}<br>{{CDD|node|5|node_1|3|node}}<br>2 | 3 5
|[[File:Icosidodecahedron vertfig.png|75px]]<br>''3.5.3.5''
|}

Each of these quasiregular polyhedra can be constructed by a [[Rectification (geometry)|rectification]] operation on either regular parent, [[Truncation (geometry)|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)<sup>2</sup>'' - a '''quasiregular tiling''' based on the [[triangular tiling]] and [[hexagonal tiling]].

{| class="wikitable"
!Regular
!Dual regular
!Quasiregular combination
![[Vertex figure]]
|- valign=top align=center
|[[File:Uniform tiling 63-t0.svg|75px]]<br>[[Hexagonal tiling]]<br>{6,3}<br>{{CDD|node|6|node|3|node_1}}<br>6 | 2 3
|[[File:Uniform tiling 63-t2.svg|75px]]<br>[[Triangular tiling]]<br>{3,6}<br>{{CDD|node_1|6|node|3|node}}<br>3 | 2 6
|[[File:Uniform tiling 63-t1.svg|75px]]<br>[[Trihexagonal tiling]]<br>r{6,3}<BR>{{CDD|node|6|node_1|3|node}}<br>2 | 3 6
|[[File:Trihexagonal tiling vertfig.svg|75px]]<br>''(3.6)<sup>2</sup>''
|}

The [[checkerboard]] pattern is a quasiregular coloring of the [[square tiling]], [[vertex figure]] ''(4.4)<sup>2</sup>'':
{| class="wikitable"
!Regular
!Dual regular
!Quasiregular combination
![[Vertex figure]]
|- valign=top align=center
|[[File:Uniform tiling 44-t0.svg|75px]]<br>{4,4}<br>{{CDD|node|4|node|4|node_1}}<br>4 | 2 4
|[[File:Uniform tiling 44-t2.svg|75px]]<br>{4,4}<br>{{CDD|node_1|4|node|4|node}}<br>4 | 2 4
|[[File:Uniform tiling 44-t1.svg|75px]]<br>r{4,4}<br>{{CDD|node|4|node_1|4|node}}<br>2 | 4 4
|[[File:square tiling vertfig.svg|75px]]<br>''(4.4)<sup>2</sup>''
|}

The [[triangular tiling]] can also be considered quasiregular, with three sets of alternating triangles at each vertex, (3.3)<sup>3</sup>:
{| class="wikitable"
|- align=center
|[[File:Uniform tiling 333-t1.svg|60px]]<BR>h{6,3}<br>3 {{pipe}} 3 3<br>{{CDD|branch_10ru|split2|node}} = {{CDD|node_h|6|node|3|node}}
|}

In the hyperbolic plane, this sequence continues further, for example the [[triheptagonal tiling]], [[vertex figure]] ''(3.7)<sup>2</sup>'' - a '''quasiregular tiling''' based on the ''order-7 triangular tiling'' and ''heptagonal tiling''.
{| class="wikitable"
!Regular
!Dual regular
!Quasiregular combination
![[Vertex figure]]
|- valign=top align=center
|[[File:Heptagonal tiling.svg|75px]]<br>Heptagonal tiling<br>{7,3}<br>{{CDD|node|7|node|3|node_1}}<br>7 | 2 3
|[[File:Order-7 triangular tiling.svg|75px]]<br>Triangular tiling<br>{3,7}<br>{{CDD|node_1|7|node|3|node}}<br>3 | 2 7
|[[File:Triheptagonal tiling.svg|75px]]<br>[[Triheptagonal tiling]]<br>r{3,7}<br>{{CDD|node|7|node_1|3|node}}<br>2 | 3 7
|[[File:Triheptagonal tiling vertfig.png|75px]]<br>''(3.7)<sup>2</sup>''
|}

==Nonconvex examples==

[[Coxeter|Coxeter, H.S.M.]] et al. (1954) also classify certain [[star polyhedron|star polyhedra]], having the same characteristics, as being quasiregular.

Two are based on dual pairs of regular [[Kepler–Poinsot solid]]s, in the same way as for the convex examples:

the [[great icosidodecahedron]] <math>\begin{Bmatrix} 3 \\ 5/2 \end{Bmatrix}</math>, and the  [[dodecadodecahedron]] <math>\begin{Bmatrix} 5 \\ 5/2 \end{Bmatrix}</math>:

{| class="wikitable"
!Regular
!Dual regular
!Quasiregular common core
![[Vertex figure]]
|- valign=top align=center
|[[File:Great stellated dodecahedron.png|75px]]<br>[[Great stellated dodecahedron]]<br>{<sup>5</sup>/<sub>2</sub>,3}<br>{{CDD|node_1|5|rat|d2|node|3|node}}<br>3 | 2 5/2
|[[File:Great icosahedron.png|75px]]<br>[[Great icosahedron]]<br>{3,<sup>5</sup>/<sub>2</sub>}<br>{{CDD|node|5|rat|d2|node|3|node_1}}<br>5/2 | 2 3
|[[File:Great icosidodecahedron.png|75px]]<br>[[Great icosidodecahedron]]<br>r{3,<sup>5</sup>/<sub>2</sub>}<br>{{CDD|node|5|rat|d2|node_1|3|node}}<br>2 | 3 5/2
|[[File:Great icosidodecahedron vertfig.png|75px]]<br>''3.<sup>5</sup>/<sub>2</sub>.3.<sup>5</sup>/<sub>2</sub>''
|- valign=top align=center
|[[File:Small stellated dodecahedron.png|75px]]<br>[[Small stellated dodecahedron]]<br>{<sup>5</sup>/<sub>2</sub>,5}<br>{{CDD|node_1|5|rat|d2|node|5|node}}<br>5 | 2 5/2
|[[File:Great dodecahedron.png|75px]]<br>[[Great dodecahedron]]<br>{5,<sup>5</sup>/<sub>2</sub>}<br>{{CDD|node|5|rat|d2|node|5|node_1}}<br>5/2 | 2 5
|[[File:Dodecadodecahedron.png|75px]]<br>[[Dodecadodecahedron]]<br>r{5,<sup>5</sup>/<sub>2</sub>}<br>{{CDD|node|5|rat|d2|node_1|5|node}}<br>2 | 5 5/2
|[[File:Dodecadodecahedron vertfig.png|75px]]<br>''5.<sup>5</sup>/<sub>2</sub>.5.<sup>5</sup>/<sub>2</sub>''
|}

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

{| class=wikitable width=500
|- align=center
!Quasiregular (rectified)
|[[File:Rectified tetrahedron.png|75px]]<BR>Tetratetrahedron
|[[File:Cuboctahedron.png|75px]]<BR>Cuboctahedron
|[[File:Icosidodecahedron.png|75px]]<BR>Icosidodecahedron
|[[File:Great icosidodecahedron.png|75px]]<BR>Great icosidodecahedron
|[[File:Dodecadodecahedron.png|75px]]<BR>Dodecadodecahedron
|- align=center
!Quasiregular (hemipolyhedra)
|[[File:Tetrahemihexahedron.png|75px]]<BR>[[Tetrahemihexahedron]]<BR><sup>3</sup>/<sub>2</sub> 3 {{pipe}} 2
|[[File:Octahemioctahedron.png|75px]]<BR>[[Octahemioctahedron]]<BR><sup>3</sup>/<sub>2</sub> 3 {{pipe}} 3
|[[File:Small icosihemidodecahedron.png|75px]]<BR>[[Small icosihemidodecahedron|Small&nbsp;icosihemidodecahedron]]<BR><sup>3</sup>/<sub>2</sub> 3 {{pipe}} 5
|[[File:Great icosihemidodecahedron.png|75px]]<BR>[[Great icosihemidodecahedron|Great&nbsp;icosihemidodecahedron]]<BR><sup>3</sup>/<sub>2</sub> 3 {{pipe}} <sup>5</sup>/<sub>3</sub>
|[[File:Small dodecahemicosahedron.png|75px]]<BR>[[Small dodecahemicosahedron|Small&nbsp;dodecahemicosahedron]]<BR><sup>5</sup>/<sub>3</sub> <sup>5</sup>/<sub>2</sub> {{pipe}} 3
|- align=center
![[Vertex figure]]
|[[File:Tetrahemihexahedron vertfig.png|75px]]<br>''3.4.<sup>3</sup>/<sub>2</sub>.4''
|[[File:Octahemioctahedron vertfig.png|75px]]<br>''3.6.<sup>3</sup>/<sub>2</sub>.6''
|[[File:Small icosihemidodecahedron vertfig.png|75px]]''<br><BR>3.10.<sup>3</sup>/<sub>2</sub>.10''
|[[File:Great icosihemidodecahedron vertfig.png|75px]]''<br>3.<sup>10</sup>/<sub>3</sub>.<sup>3</sup>/<sub>2</sub>.<sup>10</sup>/<sub>3</sub>''
|[[File:Small dodecahemicosahedron vertfig.png|75px]]''<br><sup>5</sup>/<sub>2</sub>.6.<sup>5</sup>/<sub>3</sub>.6''
|- align=center
!Quasiregular (hemipolyhedra)
|&nbsp;
|[[File:Cubohemioctahedron.png|75px]]<BR>[[Cubohemioctahedron]]<BR><sup>4</sup>/<sub>3</sub> 4 {{pipe}} 3
|[[File:Small dodecahemidodecahedron.png|75px]]<BR>[[Small dodecahemidodecahedron|Small&nbsp;dodecahemidodecahedron]]<BR><sup>5</sup>/<sub>4</sub> 5 {{pipe}} 5
|[[File:Great dodecahemidodecahedron.png|75px]]<BR>[[Great dodecahemidodecahedron|Great&nbsp;dodecahemidodecahedron]]<BR><sup>5</sup>/<sub>3</sub> <sup>5</sup>/<sub>2</sub> {{pipe}} <sup>5</sup>/<sub>3</sub>
|[[File:Great dodecahemicosahedron.png|75px]]<BR>[[Great dodecahemicosahedron|Great&nbsp;dodecahemicosahedron]]<BR><sup>5</sup>/<sub>4</sub> 5 {{pipe}} 3
|- align=center
!Vertex figure
|&nbsp;
|[[File:Cubohemioctahedron vertfig.png|75px]]<br>''4.6.<sup>4</sup>/<sub>3</sub>.6''
|[[File:Small dodecahemidodecahedron vertfig.png|75px]]<br>''5.10.<sup>5</sup>/<sub>4</sub>.10''
|[[File:Great dodecahemidodecahedron vertfig.png|75px]]<br>''<sup>5</sup>/<sub>2</sub>.<sup>10</sup>/<sub>3</sub>.<sup>5</sup>/<sub>3</sub>.<sup>10</sup>/<sub>3</sub>''
|[[File:Great dodecahemicosahedron vertfig.png|75px]]<br>''5.6.<sup>5</sup>/<sub>4</sub>.6''
|}

Lastly there are three [[ditrigonal]] forms, all facetings of the regular dodecahedron, whose vertex figures contain three alternations of the two face types:
{| class="wikitable"
! Image
! Faceted form<br>[[Wythoff symbol]]<br>[[Coxeter diagram]]
! [[Vertex figure]]
|- align=center
| [[File:Ditrigonal dodecadodecahedron.png|125px]]
||[[Ditrigonal dodecadodecahedron]]<br>''3 | 5/3 5''<br>[[File:Ditrigonal dodecadodecahedron cd.png]] or {{CDD|node|5|node_h3|5-2|node}}
||[[File:Ditrigonal dodecadodecahedron vertfig.png|100px]]<br>(5.5/3)<sup>3</sup>
|- align=center
| [[File:Small ditrigonal icosidodecahedron.png|125px]]
||[[Small ditrigonal icosidodecahedron]]<br>''3 | 5/2 3''<br>[[File:Small ditrigonal icosidodecahedron cd.png]] or {{CDD|node_h3|5|node|3|node}}
||[[File:Small ditrigonal icosidodecahedron vertfig.png|100px]]<br>(3.5/2)<sup>3</sup>
|- align=center
| [[File:Great ditrigonal icosidodecahedron.png|125px]]
||[[Great ditrigonal icosidodecahedron]]<br>''3/2 | 3 5''<br>[[File:Great ditrigonal icosidodecahedron cd.png]] or {{CDD|node_h3|5-2|node|3|node}}
||[[File:Great ditrigonal icosidodecahedron vertfig.png|100px]]<br>((3.5)<sup>3</sup>)/2
|}

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

{| class="wikitable sortable"
|-
!Original<br/>rectified<BR/>tiling!!Edge<br/>diagram!!Solid!!Vertex<br/>Config!!Wythoff!!Symmetry group
|- align=center
|[[File:Uniform tiling 44-t1.svg|100px]]<BR>Square<BR>tiling||[[File:4.oo.4-3.oo tiling frame.png|100px]]||[[File:Star tiling sha.gif|150px]]||4.&infin;.4/3.&infin;<br/>4.&infin;.-4.&infin; || '''4/3 4 {{pipe}} &infin;''' ||p4m 
|- align=center
|[[File:Uniform tiling 333-t1.svg|100px]]<BR>Triangular<BR>tiling||[[File:3.oo.3.oo.3oo tiling-frame.png|100px]]||[[File:Star tiling ditatha.gif|150px]]||(3.&infin;.3.&infin;.3.&infin;)/2 || '''3/2 {{pipe}} 3 &infin;'''||rowspan=3|p6m 
|- align=center
|rowspan=2|[[File:Uniform tiling 63-t1.svg|100px]]<BR>Trihexagonal<BR>tiling
|rowspan=2|[[File:6.oo.6-5.oo tiling-frame.png|100px]]||[[File:Star tiling hoha.gif|150px]]|| 6.&infin;.6/5.&infin;<br/>6.&infin;.-6.&infin; || '''6/5 6 {{pipe}} &infin;''' 
|- align=center
|[[File:Star tiling tha.gif|150px]]||&infin;.3.&infin;.3/2<br/>&infin;.3.&infin;.-3 || '''3/2 3 {{pipe}} &infin;''' 
|}

==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 solid]]s. The convex ones are, in corresponding order as above:
#The [[rhombic dodecahedron]], with two ''types'' of alternating vertices, 8 with three rhombic faces, and 6 with four rhombic faces.
#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 polyhedron|regular]], can be made quasiregular if alternate vertices are given different colors.

Their [[face configuration]]s are of the form V3.n.3.n, and [[Coxeter-Dynkin diagram]] {{CDD|node|3|node_f1|n|node}}
{| class="wikitable"
|[[File:Hexahedron.svg|120px]]
|[[File:Rhombicdodecahedron.jpg|140px]]
|[[File:Rhombictriacontahedron.svg|120px]]
|[[File:Rhombic star tiling.svg|120px]]
|[[File:7-3 rhombille tiling.svg|120px]]
|[[File:H2-8-3-rhombic.svg|120px]]
|- align=center valign=bottom
|[[Cube]]<br>V(3.3)<sup>2</sup><br>{{CDD|node|3|node_f1|3|node}}
|[[Rhombic dodecahedron]]<br>V(3.4)<sup>2</sup><br>{{CDD|node|3|node_f1|4|node}}
|[[Rhombic triacontahedron]]<br>V(3.5)<sup>2</sup><br>{{CDD|node|3|node_f1|5|node}}
|[[Rhombille tiling]]<br>V(3.6)<sup>2</sup><br>{{CDD|node|3|node_f1|6|node}}
|V(3.7)<sup>2</sup><br>{{CDD|node|3|node_f1|7|node}}
|V(3.8)<sup>2</sup><br>{{CDD|node|3|node_f1|8|node}}
|}

These three quasiregular duals are also characterised by having [[rhombus|rhombic]] faces.

This rhombic-faced pattern continues as V(3.6)<sup>2</sup>, 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.<ref name=regpol/>

In Euclidean 4-space, the regular [[16-cell]] can also be seen as quasiregular as an alternated [[tesseract]], h{4,3,3}, [[Coxeter diagram]]s: {{CDD|node_h1|4|node|3|node|3|node}} = {{CDD|nodes_10ru|split2|node|3|node}}, composed of alternating [[tetrahedron]] and [[tetrahedron]] [[Cell (geometry)|cells]]. Its [[vertex figure]] is the quasiregular [[tetratetrahedron]] (an octahedron with tetrahedral symmetry), {{CDD|node|3|node_1|3|node}}.

The only quasiregular honeycomb in Euclidean 3-space is the [[alternated cubic honeycomb]], h{4,3,4}, Coxeter diagrams: {{CDD|node_h1|4|node|3|node|4|node}} = {{CDD|nodes_10ru|split2|node|4|node}}, composed of alternating tetrahedral and [[octahedron|octahedral]] [[Cell (geometry)|cells]]. Its vertex figure is the quasiregular [[cuboctahedron]], {{CDD|node|4|node_1|3|node}}.<ref name=regpol>Coxeter, Regular Polytopes, 4.7 Other honeycombs. p.69, p.88</ref>

In hyperbolic 3-space, one quasiregular honeycomb is the [[alternated order-5 cubic honeycomb]], h{4,3,5}, Coxeter diagrams: {{CDD|node_h1|4|node|3|node|5|node}} = {{CDD|nodes_10ru|split2|node|5|node}}, composed of alternating tetrahedral and [[icosahedron|icosahedral]] [[Cell (geometry)|cells]]. Its vertex figure is the quasiregular [[icosidodecahedron]], {{CDD|node|5|node_1|3|node}}.<ref>{{cite journal |last1=Coxeter |first1=H. S. M. |last2=Whitrow |first2=G. J. |date=26 April 1950 |title=World-Structure and Non-Euclidean Honeycombs |url= |journal=Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences |volume=201 |issue=1066 |pages=417–437 |doi=10.1098/rspa.1950.0070 |access-date=}}</ref> 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]], {{CDD|node|6|node_1|3|node}}.

{{Quasiregular polychora and honeycombs}}

Regular polychora or honeycombs of the form {p,3,4} or {{CDD|node_1|p|node|3|node|4|node}} can have their symmetry cut in half as {{CDD|node_1|p|node|3|node|4|node_h0}} into quasiregular form {{CDD|node_1|p|node|split1|nodes}}, creating alternately colored {p,3} cells. These cases include the Euclidean [[cubic honeycomb]] {4,3,4} with [[cube|cubic]] cells, and compact hyperbolic {5,3,4} with [[dodecahedron|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 figure]]s are quasiregular tetratetrahedra, {{CDD|node_1|3|node|4|node_h0}} = {{CDD|node_1|split1|nodes}}.
[[File:Uniform polyhedron-33-t1.svg|100px|thumb|Common vertex figure is the quasiregular tetratetrahedron, {{CDD|node_1|split1|nodes}}, same as regular [[octahedron]]]]
{{Regular and Quasiregular honeycombs}}

Similarly regular hyperbolic honeycombs of the form {p,3,6} or {{CDD|node_1|p|node|3|node|6|node}} can have their symmetry cut in half as {{CDD|node_1|p|node|3|node|6|node_h0}} into quasiregular form {{CDD|node_1|p|node|split1|branch}}, creating alternately colored {p,3} cells. They have six cells around each edge, alternating in 2 colors. Their [[vertex figure]]s are quasiregular [[triangular tiling]]s, {{CDD|node_1|split1|branch}}.
[[File:Uniform tiling 333-t1.svg|100px|thumb|The common [[vertex figure]] is a quasiregular [[triangular tiling]], {{CDD|node_1|3|node|6|node_h0}} = {{CDD|node_1|split1|branch}}]]

{{Triangular tiling vertex figure tessellations}}

== See also ==
* [[Chiral polytope]]
* [[Rectification (geometry)]]

==Notes ==
{{reflist}}

==References==
*Cromwell, P. ''Polyhedra'', Cambridge University Press (1977).
*[[Coxeter]], ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, {{isbn|0-486-61480-8}}, 2.3 ''Quasi-Regular Polyhedra.'' (p.&nbsp;17), Quasi-regular honeycombs p.69

==External links==
* {{Mathworld | urlname=QuasiregularPolyhedron | title=Quasiregular polyhedron }}
* {{Mathworld | urlname=UniformPolyhedron | title=Uniform polyhedron }} Quasi-regular polyhedra:  (p.q)<sup>r</sup>
* George Hart, [http://www.georgehart.com/virtual-polyhedra/quasi-regular-info.html Quasiregular polyhedra]

[[Category:Quasiregular polyhedra| ]]