View source for Uniform tiling

{{Short description|Vertex-transitive tiling of the plane by regular polygons}}
In [[geometry]], a '''uniform tiling''' is a [[tessellation]] of the plane by [[regular polygon]] faces with the restriction of being [[vertex-transitive]].

Uniform tilings can exist in both the [[Euclidean plane]] and [[Hyperbolic space|hyperbolic plane]]. Uniform tilings are related to the finite [[uniform polyhedra]]; these can be considered uniform tilings of the [[sphere]].

Most uniform tilings can be made from a [[Wythoff construction]] starting with a [[symmetry group]] and a singular generator point inside of the [[fundamental domain]]. A planar symmetry group has a polygonal fundamental domain and can be represented by its group notation: the sequence of the reflection orders of the fundamental domain vertices.

A fundamental domain triangle is denoted (''p q r''), where ''p'', ''q'', ''r'' are whole numbers > 1, i.e. ≥ 2; a fundamental domain right triangle is denoted (''p q'' 2). The triangle may exist as a [[spherical triangle]], a Euclidean plane triangle, or a hyperbolic plane triangle, depending on the values of ''p'', ''q'', and ''r''.

There are several symbolic schemes for denoting these figures:
* The modified [[Schläfli symbol]] for a right triangle domain: (''p q'' 2) → {''p'', ''q''}.
* The [[Coxeter-Dynkin diagram]] is a triangular graph with ''p'', ''q'', ''r'' labeled on the edges. If ''r'' = 2, then the graph is linear, since diagram nodes with connectivity 2 are not connected to each other by a diagram branch (since domain mirrors meeting at 90 degrees generate no new mirrors).
* The [[Wythoff symbol]] takes the three integers and separates them by a vertical bar (|). If the generator point is off the mirror opposite to a domain vertex, then the reflection order of this domain vertex is given before the bar.
* Finally, a uniform tiling can be described by its [[vertex configuration]]: the (identical) sequence of polygons around each (equivalent) vertex.

All uniform tilings can be constructed from various operations applied to [[regular tiling]]s. These operations, as named by [[Norman Johnson (mathematician)|Norman Johnson]], are called [[Truncation (geometry)|truncation]] (cutting vertices), [[Rectification (geometry)|rectification]] (cutting vertices until edges disappear), and [[cantellation]] (cutting edges and vertices). [[Omnitruncation]] is an operation that combines truncation and cantellation. Snubbing is an operation of [[Alternation (geometry)|alternate truncation]] of the omnitruncated form. (See [[Uniform polyhedron#Wythoff construction operators]] for more details.)

== Coxeter groups ==
[[Coxeter group]]s for the plane define the Wythoff construction and can be represented by [[Coxeter-Dynkin diagram]]s:

For groups with integer reflection orders, including:

{| class=wikitable
|+ Euclidean plane
|-
![[Orbifold notation|Orbifold<br/>symmetry]]
!colspan=3|[[Coxeter group]]
![[Coxeter-Dynkin diagram|Coxeter<br/>diagram]]
!Notes
|-
! colspan=6 | Compact
|- valign=top align=center
| *333
| (3 3 3) 
| <math>{\tilde{A}}_2</math>
| [3<sup>[3]</sup>] 
| {{CDD|node|split1|branch}}
| 3 reflective forms, 1 snub
|- valign=top align=center
| *442
| (4 4 2) 
| <math>{\tilde{B}}_2</math>
| [4,4] 
| {{CDD|node|4|node|4|node}} 
| 5 reflective forms, 1 snub
|- valign=top align=center
| *632
| (6 3 2) 
| <math>{\tilde{G}}_2</math>
| [6,3] 
| {{CDD|node|6|node|3|node}} 
| 7 reflective forms, 1 snub
|- valign=top align=center
| *2222
| (∞ 2 ∞ 2) 
| <math>{\tilde{I}}_1</math> × <math>{\tilde{I}}_1</math>
| [∞,2,∞] 
| {{CDD|node|infin|node|2|node|infin|node}} 
| 3 reflective forms, 1 snub
|-
! colspan=6 | Noncompact ([[Frieze group|Frieze]])
|- align=center
| *∞∞
| (∞) 
| <math>{\tilde{I}}_1</math>
| [∞]
| {{CDD|node|infin|node}}
|
|- align=center
| *22∞
| (2 2 ∞) 
| <math>{\tilde{I}}_1</math> × <math>{\tilde{A}}_2</math>
| [∞,2]
| {{CDD|node|infin|node|2|node}}
| 2 reflective forms, 1 snub
|}

{| class=wikitable
|+ Hyperbolic plane
|-
![[Orbifold notation|Orbifold<br/>symmetry]]
!colspan=2|[[Coxeter group]]
![[Coxeter-Dynkin diagram|Coxeter<br/>diagram]]
!Notes
|-
! colspan=5 | Compact
|- valign=top align=center
| *''pq''2
| (''p q'' 2)
| [''p'',''q'']
| {{CDD|node|p|node|q|node}}
| 2(''p''+''q'') < ''pq''
|- valign=top align=center
| *''pqr''
| (''p q r'') 
| [(''p'',''q'',''r'')]
| {{CDD|3|node|p|node|q|node|r}}
| ''pq''+''pr''+''qr'' < ''pqr'', i.e. {{sfrac|1|''p''}} + {{sfrac|1|''q''}} + {{sfrac|1|''r''}} < 1
|-
! colspan=5 | Paracompact
|- align=center
| *∞''p''2
| (''p'' ∞ 2)
| [''p'',∞]
| {{CDD|node|p|node|infin|node}}|| ''p'' ≥ 3
|- align=center
| *∞''pq''
| (''p q'' ∞)
| [(''p'',''q'',∞)]
| {{CDD|3|node|p|node|q|node|infin}}|| ''p'',''q'' ≥ 3; ''p''+''q'' > 6
|- align=center
| *∞∞''p''
| (''p'' ∞ ∞)
| [(''p'',∞,∞)]
| {{CDD|3|node|p|node|infin|node|infin}}|| ''p'' ≥ 3
|- align=center
| *∞∞∞
| (∞ ∞ ∞)
| [(∞,∞,∞)]
| {{CDD|3|node|infin|node|infin|node|infin}}||
|}

== Uniform tilings of the Euclidean plane ==
{{further|List of k-uniform tilings}}
[[File:Elongated triangular tiling 1.png|thumb|right|The elongated triangular tiling, the only non-Wythoffian convex uniform tiling]]
There are symmetry groups on the Euclidean plane constructed from fundamental triangles: (4 4 2), (6 3 2), and (3 3 3). Each is represented by a set of lines of reflection that divide the plane into fundamental triangles.

These symmetry groups create 3 [[regular tiling]]s, and 7 semiregular ones. A number of the semiregular tilings are repeated from different symmetry constructors.

A prismatic symmetry group, (2 2 2 2), is represented by two sets of parallel mirrors, which in general can make a rectangular fundamental domain. It generates no new tilings.

A further prismatic symmetry group, (∞ 2 2), has an infinite fundamental domain. It constructs two uniform tilings: the [[apeirogonal prism]] and [[apeirogonal antiprism]].

The stacking of the finite faces of these two prismatic tilings constructs one [[non-Wythoffian]] uniform tiling of the plane. It is called the [[elongated triangular tiling]], composed of alternating layers of squares and triangles.

'''Right angle fundamental triangles: (''p q'' 2)'''
{| class="wikitable"
|-
!(''p q'' 2)
!Fund.<br/>triangles
!Parent
!Truncated
!Rectified
!Bitruncated
!Birectified<br/>(dual)
!Cantellated
!Omnitruncated<br/>(<small>Cantitruncated</small>)
!Snub
|-
![[Wythoff construction|Wythoff symbol]]
!
! ''q'' {{pipe}} ''p'' 2
! 2 ''q'' {{pipe}} ''p''
! 2 {{pipe}} ''p'' ''q''
! 2 ''p'' {{pipe}} ''q''
! ''p'' {{pipe}} ''q'' 2
! ''p'' ''q'' {{pipe}} 2
! ''p'' ''q'' 2 {{pipe}}
! {{pipe}} ''p'' ''q'' 2
|-
![[Schläfli symbol]]
!
!{''p'',''q''}
!t{''p'',''q''}
!r{''p'',''q''}
!2t{''p'',''q''}=t{''q'',''p''}
!2r{''p'',''q''}={''q'',''p''}
!rr{''p'',''q''}
!tr{''p'',''q''}
!sr{''p'',''q''}
|-
![[Coxeter diagram]]
!
!{{CDD|node_1|p|node|q|node}}
!{{CDD|node_1|p|node_1|q|node}}
!{{CDD|node|p|node_1|q|node}}
!{{CDD|node|p|node_1|q|node_1}}
!{{CDD|node|p|node|q|node_1}}
!{{CDD|node_1|p|node|q|node_1}}
!{{CDD|node_1|p|node_1|q|node_1}}
!{{CDD|node_h|p|node_h|q|node_h}}
|-
![[Vertex configuration|Vertex config.]]
!
!''p<sup>q</sup>''
!''q''.2''p''.2''p''
!(''p.q'')<sup>2</sup>
!''p''.2''q''.2''q''
!''q<sup>p</sup>''
!''p''.4.''q''.4
!4.2''p''.2''q''
!3.3.''p''.3.''q''
|- align=center
|[[Square tiling]]<br/>(4 4 2)
|[[Image:Tiling Dual Semiregular V4-8-8 Tetrakis Square-2-color-zoom.svg|64px]]<br/>{{0}}
|[[Image:Uniform tiling 44-t0.svg|64px]]<br/>[[Square tiling|{4,4}]]
|[[Image:Uniform tiling 44-t01.svg|64px]]<br/>[[Truncated square tiling|4.8.8]]
|[[Image:Uniform tiling 44-t1.svg|64px]]<br/>[[Square tiling|4.4.4.4]]
|[[Image:Uniform tiling 44-t12.svg|64px]]<br/>[[Truncated square tiling|4.8.8]]
|[[Image:Uniform tiling 44-t2.svg|64px]]<br/>[[Square tiling|{4,4}]]
|[[Image:Uniform tiling 44-t02.svg|64px]]<br/>[[Square tiling|4.4.4.4]]
|[[Image:Uniform tiling 44-t012.svg|64px]]<br/>[[Truncated square tiling|4.8.8]]
|[[Image:Uniform tiling 44-snub.svg|64px]]<br/>[[Snub square tiling|3.3.4.3.4]]
|-align=center
|[[Hexagonal tiling]]<br/>(6 3 2)
|[[Image:Tile V46b.svg|64px]]<br/>{{0}}
|[[Image:Uniform tiling 63-t0.svg|64px]]<br/>[[Hexagonal tiling|{6,3}]]
|[[Image:Uniform tiling 63-t01.svg|64px]]<br/>[[Truncated hexagonal tiling|3.12.12]]
|[[Image:Uniform tiling 63-t1.svg|64px]]<br/>[[Trihexagonal tiling|3.6.3.6]]
|[[Image:Uniform tiling 63-t12.svg|64px]]<br/>[[Hexagonal tiling|6.6.6]]
|[[Image:Uniform tiling 63-t2.svg|64px]]<br/>[[Triangular tiling|{3,6}]]
|[[Image:Uniform tiling 63-t02.svg|64px]]<br/>[[Small rhombitrihexagonal tiling|3.4.6.4]]
|[[Image:Uniform tiling 63-t012.svg|64px]]<br/>[[Great rhombitrihexagonal tiling|4.6.12]]
|[[Image:Uniform tiling 63-snub.svg|65px]]<br/>[[Snub hexagonal tiling|3.3.3.3.6]]
|}

'''General fundamental triangles: (''p q r'')'''
{| class="wikitable"
![[Wythoff construction|Wythoff symbol]]<br/>(''p q r'')
!Fund.<br/>triangles
! ''q'' {{pipe}} ''p r''
! ''r q'' {{pipe}} ''p''
! ''r'' {{pipe}} ''p q''
! ''r p'' {{pipe}} ''q''
! ''p'' {{pipe}} ''q r''
! ''p q'' {{pipe}} ''r''
! ''p q r'' {{pipe}}
! {{pipe}} ''p q r''
|-
![[Coxeter diagram]]
!
!{{CDD|3|node_1|p|node|q|node|r}}
!{{CDD|3|node_1|p|node_1|q|node|r}}
!{{CDD|3|node|p|node_1|q|node|r}}
!{{CDD|3|node|p|node_1|q|node_1|r}}
!{{CDD|3|node|p|node|q|node_1|r}}
!{{CDD|3|node_1|p|node|q|node_1|r}}
!{{CDD|3|node_1|p|node_1|q|node_1|r}}
!{{CDD|3|node_h|p|node_h|q|node_h|r}}
|-
![[Vertex configuration|Vertex config.]]
!
!(''p.q'')<sup>''r''</sup>
!''r''.2''p.q''.2''p''
!(''p.r'')<sup>''q''</sup>
!''q''.2''r.p''.2''r''
!(''q.r'')<sup>''p''</sup>
!''q''.2''r.p''.2''r''
!''r''.2''q.p''.2''q''
!3.''r''.3.''q''.3.''p''
|-align=center
|Triangular<br/>(3 3 3)
|[[Image:Tiling Regular 3-6 Triangular.svg|64px]]<br/>{{0}}
|[[Image:Uniform tiling 333-t0.svg|64px]]<br/>[[Triangular tiling|(3.3)<sup>3</sup>]]
|[[Image:Uniform polyhedron-63-t1-1.svg|64px]]<br/>[[Trihexagonal tiling|3.6.3.6]]
|[[Image:Uniform tiling 333-t1.svg|64px]]<br/>[[Triangular tiling|(3.3)<sup>3</sup>]]
|[[Image:Uniform tiling 333-t12.svg|64px]]<br/>[[Trihexagonal tiling|3.6.3.6]]
|[[Image:Uniform tiling 333-t2.svg|64px]]<br/>[[Triangular tiling|(3.3)<sup>3</sup>]]
|[[Image:Uniform tiling 333-t02.svg|64px]]<br/>[[Trihexagonal tiling|3.6.3.6]]
|[[Image:Uniform tiling 333-t012.svg|64px]]<br/>[[Hexagonal tiling|6.6.6]]
|[[Image:Uniform tiling 333-snub.svg|64px]]<br/>[[Triangular tiling|3.3.3.3.3.3]]
|}

'''Non-simplical fundamental domains'''

The only possible fundamental domain in Euclidean 2-space that is not a [[simplex]] is the rectangle (∞ 2 ∞ 2), with [[Coxeter diagram]]: {{CDD|node|infin|node|2|node|infin|node}}. All forms generated from it become a [[square tiling]].

== Uniform tilings of the hyperbolic plane ==
{{further|Uniform tilings in hyperbolic plane}}

There are infinitely many uniform tilings by convex regular polygons on the [[Hyperbolic space|hyperbolic plane]], each based on a different reflective symmetry group (''p q r'').

A sampling is shown here with a [[Poincaré disk model|Poincaré disk]] projection.

The [[Coxeter-Dynkin diagram]] is given in a linear form, although it is actually a triangle, with the trailing segment ''r'' connecting to the first node.

Further symmetry groups exist in the hyperbolic plane with quadrilateral fundamental domains — starting with (2 2 2 3), etc. — that can generate new forms. As well, there are fundamental domains that place vertices at infinity, such as (∞ 2 3), etc.

'''Right angle fundamental triangles: (''p q'' 2)'''
{| class="wikitable"
|-
!(''p q'' 2)
!Fund.<br/>triangles
!Parent
!Truncated
!Rectified
!Bitruncated
!Birectified<br/>(dual)
!Cantellated
!Omnitruncated<br/>(<small>Cantitruncated</small>)
!Snub
|-
![[Wythoff construction|Wythoff symbol]]
!
! ''q'' {{pipe}} ''p'' 2
! 2 ''q'' {{pipe}} ''p''
! 2 {{pipe}} ''p q''
! 2 ''p'' {{pipe}} ''q''
! ''p'' {{pipe}} ''q'' 2
! ''p q'' {{pipe}} 2
! ''p q'' 2 {{pipe}}
! {{pipe}} ''p q'' 2
|-
![[Schläfli symbol]]
!
!t{''p'',''q''}
!t{''p'',''q''}
!r{''p'',''q''}
!2t{''p'',''q''}=t{''q'',''p''}
!2r{''p'',''q''}={''q'',''p''}
!rr{''p'',''q''}
!tr{''p'',''q''}
!sr{''p'',''q''}
|-
![[Coxeter diagram]]
!
!{{CDD|node_1|p|node|q|node}}
!{{CDD|node_1|p|node_1|q|node}}
!{{CDD|node|p|node_1|q|node}}
!{{CDD|node|p|node_1|q|node_1}}
!{{CDD|node|p|node|q|node_1}}
!{{CDD|node_1|p|node|q|node_1}}
!{{CDD|node_1|p|node_1|q|node_1}}
!{{CDD|node_h|p|node_h|q|node_h}}
|-
![[Vertex configuration|Vertex config.]]
!
!''p<sup>q</sup>''
!''q''.2''p''.2''p''
!''p.q.p.q''
!''p''.2''q''.2''q''
!''q<sup>p</sup>''
!''p''.4.''q''.4
!4.2''p''.2''q''
!3.3.''p''.3.''q''
|- align=center
|(5 4 2)
|[[Image:H2-5-4-kisrhombille.svg|72px]]<br/>V4.8.10
|[[Image:H2-5-4-dual.svg|64px]]<br/>{5,4}
|[[Image:H2-5-4-trunc-dual.svg|64px]]<br/>4.10.10
|[[Image:H2-5-4-rectified.svg|64px]]<br/>4.5.4.5
|[[Image:H2-5-4-trunc-primal.svg|64px]]<br/>5.8.8
|[[Image:H2-5-4-primal.svg|64px]]<br/>{4,5}
|[[Image:H2-5-4-cantellated.svg|64px]]<br/>4.4.5.4
|[[Image:H2-5-4-omnitruncated.svg|64px]]<br/>4.8.10
|[[Image:H2-5-4-snub.svg|64px]]<br/>3.3.4.3.5
|- align=center
|(5 5 2)
|[[File:Order-5 bisected pentagonal tiling.png|72px]]<br/>V4.10.10
|[[Image:Uniform tiling 552-t0.png|64px]]<br/>{5,5}
|[[Image:Uniform tiling 552-t01.png|64px]]<br/>5.10.10
|[[Image:Uniform tiling 552-t1.png|64px]]<br/>5.5.5.5
|[[Image:Uniform tiling 552-t12.png|64px]]<br/>5.10.10
|[[Image:Uniform tiling 552-t2.png|64px]]<br/>{5,5}
|[[Image:Uniform tiling 552-t02.png|64px]]<br/>5.4.5.4
|[[Image:Uniform tiling 552-t012.png|64px]]<br/>4.10.10
|[[Image:Uniform tiling 552-snub.png|64px]]<br/>3.3.5.3.5
|- align=center
|(7 3 2)
|[[Image:3-7 kisrhombille.svg|72px]]<br/>V4.6.14
|[[Image:Heptagonal tiling.svg|64px]]<br/>[[Order-3 heptagonal tiling|{7,3}]]
|[[Image:Truncated heptagonal tiling.svg|64px]]<br/>3.14.14
|[[Image:Triheptagonal tiling.svg|64px]]<br/>[[Triheptagonal tiling|3.7.3.7]]
|[[Image:Truncated order-7 triangular tiling.svg|64px]]<br/>7.6.6
|[[Image:Order-7 triangular tiling.svg|64px]]<br/>[[Order-7 triangular tiling|{3,7}]]
|[[Image:Rhombitriheptagonal tiling.svg|64px]]<br/>3.4.7.4
|[[Image:Truncated triheptagonal tiling.svg|64px]]<br/>[[Great rhombitriheptagonal tiling|4.6.14]]
|[[Image:Snub triheptagonal tiling.svg|65px]]<br/>3.3.3.3.7
|- align=center
|(8 3 2)
|[[File:H2-8-3-kisrhombille.svg|72px]]<br/>V4.6.16
|[[Image:H2-8-3-dual.svg|64px]]<br/>[[Octagonal tiling|{8,3}]]
|[[Image:H2-8-3-trunc-dual.svg|64px]]<br/>3.16.16
|[[Image:H2-8-3-rectified.svg|64px]]<br/>[[Trioctagonal tiling|3.8.3.8]]
|[[Image:H2-8-3-trunc-primal.svg|64px]]<br/>8.6.6
|[[Image:H2-8-3-primal.svg|64px]]<br/>[[Order-8 triangular tiling|{3,8}]]
|[[Image:H2-8-3-cantellated.svg|64px]]<br/>3.4.8.4
|[[Image:H2-8-3-omnitruncated.svg|64px]]<br/>[[Great rhombitrioctagonal tiling|4.6.16]]
|[[Image:H2-8-3-snub.svg|65px]]<br/>3.3.3.3.8
|}

'''General fundamental triangles: (''p q r'')'''
{| class="wikitable"
![[Wythoff construction|Wythoff symbol]]<br/>(''p q r'')
!Fund.<br/>triangles
! ''q'' {{pipe}} ''p r''
! ''r q'' {{pipe}} ''p''
! ''r'' {{pipe}} ''p q''
! ''r p'' {{pipe}} ''q''
! ''p'' {{pipe}} ''q r''
! ''p q'' {{pipe}} ''r''
! ''p q r'' {{pipe}}
! {{pipe}} ''p q r''
|-
![[Coxeter diagram]]
!
!{{CDD|3|node_1|p|node|q|node|r}}
!{{CDD|3|node_1|p|node_1|q|node|r}}
!{{CDD|3|node|p|node_1|q|node|r}}
!{{CDD|3|node|p|node_1|q|node_1|r}}
!{{CDD|3|node|p|node|q|node_1|r}}
!{{CDD|3|node_1|p|node|q|node_1|r}}
!{{CDD|3|node_1|p|node_1|q|node_1|r}}
!{{CDD|3|node_h|p|node_h|q|node_h|r}}
|-
![[Vertex configuration|Vertex config.]]
!
!(''p.r'')<sup>''q''</sup>
!''r''.2''p.q''.2''p''
!(''p.q'')<sup>''r''</sup>
!''q''.2''r.p''.2''r''
!(''q.r'')<sup>''p''</sup>
!''r''.2''q.p''.2''q''
!2''p''.2''q''.2''r''
!3.''r''.3.''q''.3.''p''
|- align=center
|(4 3 3)
|[[Image:Uniform dual tiling 433-t012.png|72px]]<br/>V6.6.8
|[[Image:Uniform tiling 433-t0.png|64px]]<br/>(3.4)<sup>3</sup>
|[[Image:Uniform tiling 433-t01.png|64px]]<br/>3.8.3.8
|[[Image:Uniform tiling 433-t1.png|64px]]<br/>(3.4)<sup>3</sup>
|[[Image:Uniform tiling 433-t12.png|64px]]<br/>3.6.4.6
|[[Image:Uniform tiling 433-t2.png|64px]]<br/>(3.3)<sup>4</sup>
|[[Image:Uniform tiling 433-t02.png|64px]]<br/>3.6.4.6
|[[Image:Uniform tiling 433-t012.png|64px]]<br/>6.6.8
|[[Image:Uniform tiling 433-snub2.png|64px]]<br/>3.3.3.3.3.4
|- align=center
|(4 4 3)
|[[Image:Uniform dual tiling 443-t012.png|72px]]<br/>V6.8.8
|[[Image:Uniform tiling 443-t0.png|64px]]<br/>(3.4)<sup>4</sup>
|[[Image:Uniform tiling 443-t01.png|64px]]<br/>3.8.4.8
|[[Image:Uniform tiling 443-t1.png|64px]]<br/>(4.4)<sup>3</sup>
|[[Image:Uniform tiling 443-t12.png|64px]]<br/>3.6.4.6
|[[Image:Uniform tiling 443-t2.png|64px]]<br/>(3.4)<sup>4</sup>
|[[Image:Uniform tiling 443-t02.png|64px]]<br/>4.6.4.6
|[[Image:Uniform tiling 443-t012.png|64px]]<br/>6.8.8
|[[Image:Uniform tiling 443-snub1.png|64px]]<br/>3.3.3.4.3.4
|- align=center
|(4 4 4)
|[[Image:Uniform dual tiling 444-t012.png|72px]]<br/>V8.8.8
|[[Image:Uniform tiling 444-t0.png|64px]]<br/>(4.4)<sup>4</sup>
|[[Image:Uniform tiling 444-t01.png|64px]]<br/>4.8.4.8
|[[Image:Uniform tiling 444-t1.png|64px]]<br/>(4.4)<sup>4</sup>
|[[Image:Uniform tiling 444-t12.png|64px]]<br/>4.8.4.8
|[[Image:Uniform tiling 444-t2.png|64px]]<br/>(4.4)<sup>4</sup>
|[[Image:Uniform tiling 444-t02.png|64px]]<br/>4.8.4.8
|[[Image:Uniform tiling 444-t012.png|64px]]<br/>8.8.8
|[[Image:Uniform tiling 444-snub.png|64px]]<br/>3.4.3.4.3.4
|}

== Expanded lists of uniform tilings ==
[[Image:Six uniform tiling vertex figures.png|320px|thumb|The [[vertex figure]]s for the six tilings with convex [[regular polygon]] and [[apeirogon]] faces. (The [[Wythoff symbol]] is given in red.)]]
[[Image:Twenty one uniform tiling vertex figures.png|320px|thumb|Vertex figures for 21 uniform tilings.]]

There are several ways the list of uniform tilings can be expanded:
# Vertex figures can have retrograde faces and turn around the vertex more than once. 
# [[Star polygon]] tiles can be included.
# [[Apeirogon]]s, {∞}, can be used as tiling faces.
# [[Infinite skew polygon#Regular zig-zag skew apeirogons in two dimensions|Zigzags]] (apeirogons alternating between two angles) can also be used.
# The restriction that tiles meet edge-to-edge can be relaxed, allowing additional tilings such as the [[Pythagorean tiling]].

Symmetry group triangles with retrogrades include:
: (4/3 4/3 2), (6 3/2 2), (6/5 3 2), (6 6/5 3), (6 6 3/2).
Symmetry group triangles with infinity include:
: (4 4/3 ∞), (3/2 3 ∞), (6 6/5 ∞), (3 3/2 ∞).

[[Branko Grünbaum]] and [[Geoffrey Colin Shephard|G. C. Shephard]], in the 1987 book ''[[Tilings and patterns]]'', section 12.3, enumerate a list of 25 uniform tilings, including the 11 convex forms, and add 14 more they call ''hollow tilings'', using the first two expansions above: star polygon faces and generalized vertex figures.<ref>Tiles and Patterns, Table 12.3.1, p. 640</ref>

[[H. S. M. Coxeter]], [[M. S. Longuet-Higgins]], and [[J. C. P. Miller]], in the 1954 paper 'Uniform polyhedra', ''Table 8: Uniform Tessellations'', use the first three expansions and enumerate a total of 38 uniform tilings. If a tiling made of 2 apeirogons is also counted, the total can be considered 39 uniform tilings.

In 1981, Grünbaum, Miller, and Shephard, in their paper ''Uniform Tilings with Hollow Tiles'', list 25 tilings using the first two expansions and 28 more when the third is added (making 53 using Coxeter ''et al''.'s definition). When the fourth is added, they list an additional 23 uniform tilings and 10 families (8 depending on continuous parameters and 2 on discrete parameters).<ref name=Grunbaum1981>{{cite book |first1=Branko |last1=Grünbaum |first2=J. C. P. |last2=Miller |first3=G. C. |last3=Shephard |date=1981 |editor1-first=Chandler |editor1-last=Davis |editor2-first=Branko |editor2-last=Grünbaum |editor3-first=F. A. |editor3-last=Sherk |title=The Geometric Vein: The Coxeter Festschrift |publisher=Springer |pages=17–64 |chapter=Uniform Tilings with Hollow Tiles |isbn=978-1-4612-5650-2}}</ref>

Besides the 11 convex solutions, the 28 uniform star tilings listed by Coxeter ''et al.'', grouped by shared edge graphs, are shown below, followed by 15 more listed by Grünbaum ''et al.'' that meet Coxeter ''et al''.'s definition but were missed by them.

This set is not proved complete. By "2.25" is meant tiling 25 in Grünbaum ''et al''.'s table 2 from 1981.

The following three tilings are exceptional in that there is only finitely many of one face type: two apeirogons in each. Sometimes the order-2 apeirogonal tiling is not included, as its two faces meet at more than one edge.
{| class=wikitable
|+ [[Frieze group]] symmetry
!McNeill<ref name=mcneill/>||Diagram||Vertex<br/>Config.||Wythoff||Symmetry||Notes
|-
|I1||[[File:Apeirogonal tiling.svg|100px]]|| ∞.∞ || ||p1m1 ||(Two half-plane tiles, [[order-2 apeirogonal tiling]])
|-
|I2||[[File:Infinite prism alternating.svg|100px]]|| 4.4.∞ || '''∞ 2 {{pipe}} 2''' ||p1m1 ||[[Apeirogonal prism]]
|-
|I3||[[File:Infinite antiprism.svg|100px]]||3.3.3.∞ || '''{{pipe}} 2 2 ∞''' ||p11g ||[[Apeirogonal antiprism]]
|}

For clarity, the tilings are not colored from here onward (due to the overlaps). A set of polygons around one vertex is highlighted. McNeill only lists tilings given by Coxeter ''et al''. (1954). The eleven convex uniform tilings have been repeated for reference.

{| class="wikitable sortable"
!colspan=7| [[Wallpaper group]] symmetry
|-
!McNeill<ref name=mcneill>[http://www.orchidpalms.com/polyhedra/tessellations/tessel.htm Jim McNeill]</ref>||Grünbaum ''et al.'', 1981<ref name=Grunbaum1981/>||Edge<br/>diagram||Highlighted||Vertex<br/>Config.||Wythoff||Symmetry
|- align=center
!Convex||1.9
|rowspan=2|[[File:4.oo.4-3.oo tiling frame.png|100px]]||[[File:Tiling 4,4,4,4.png|150px]]||4.4.4.4|| 4 {{pipe}} 2 4 || p4m
|- align=center
!I4||2.14
||[[File:Tiling 4,i,-4,i.png|150px]]||4.∞.4/3.∞<br/>4.∞.-4.∞ || 4/3 4 {{pipe}} ∞ ||p4m
|- align=center
!Convex||1.24
|| ||[[File:Tiling 6,6,6.png|150px]]||6.6.6||3 {{pipe}} 2 6 ||p6m
|- align=center
!Convex||1.25
|rowspan=2|[[File:3.oo.3.oo.3oo tiling-frame.png|100px]]||[[File:Tiling 3,3,3,3,3,3.png|150px]]||3.3.3.3.3.3||6 {{pipe}} 2 3 ||p6m
|- align=center
!I5||2.26
||[[File:Tiling 3,i,3,i,3,i.png|150px]]||(3.∞.3.∞.3.∞)/2 || 3/2 {{pipe}} 3 ∞||p3m1
|- align=center
!Convex||1.23
|rowspan=3|[[File:6.oo.6-5.oo tiling-frame.png|100px]]||[[File:Tiling 3,6,3,6.png|150px]]||3.6.3.6||2 {{pipe}} 3 6|| p6m
|- align=center
!I6||2.25
||[[File:Tiling 6,i,-6,i.png|150px]]|| 6.∞.6/5.∞<br/>6.∞.-6.∞ || 6/5 6 {{pipe}} ∞||p6m
|- align=center
!I7||2.24
|| [[File:Tiling 3,i,-3,i.png|150px]]||∞.3.∞.3/2<br/>3.∞.-3.∞ || 3/2 3 {{pipe}} ∞||p6m
|-align=center
!Convex||1.14
|rowspan=3|[[File:12.3-2.12.6 tiling-frame.png|100px]]||[[File:Tiling 3,4,6,4.png|150px]] || 3.4.6.4 || 3 6 {{pipe}} 2 || p6m
|- align=center
!rowspan=2|1||1.15
||[[File:Tiling -3,12,6,12.png|150px]]
|3/2.12.6.12<br/>-3.12.6.12|| 3/2 6 {{pipe}} 6||p6m
|- align=center
!1.16
|[[File:Tiling 4,12,-4,-12.png|150px]]||4.12.4/3.12/11<br/>4.12.-4.-12 || 2 6 (3/2 6/2) {{pipe}}||p6m
|- align=center
!Convex||1.5
| ||[[File:Tiling 4,8,8.png|150px]] || 4.8.8 || 2 4 {{pipe}} 4 || p4m
|- align=center
!rowspan=3|2||2.7
|rowspan=3|[[File:8-3.4.8-3.oo tiling-frame.png|100px]]||[[File:Tiling 4,83,i,83.png|150px]]
| 4.8/3.∞.8/3 || 4 ∞ {{pipe}} 4/3||p4m
|- align=center
!1.7
|[[File:Tiling 8,83,-8,-83.png|150px]] || 8/3.8.8/5.8/7<br/>8.8/3.-8.-8/3 || 4/3 4 (4/2 ∞/2) {{pipe}}||p4m
|- align=center
!2.6
|[[File:Tiling -4,8,i,8.png|150px]] || 8.4/3.8.∞<br/>-4.8.∞.8 || 4/3 ∞ {{pipe}} 4||p4m
|- align=center
!Convex||1.20
| ||[[File:Tiling 3,12,12.png|150px]] || 3.12.12 || 2 3 {{pipe}} 6 || p6m
|- align=center
!rowspan=3|3||2.17
|rowspan=3|[[File:12-5.6.12-5.oo tiling frame.svg|100px]]||[[File:Tiling 6,125,i,125.png|150px]] || 6.12/5.∞.12/5 ||6 ∞ {{pipe}} 6/5||p6m
|- align=center
!1.21
|[[File:Tiling 12,125,-12,-125.png|150px]] || 12/5.12.12/7.12/11<br/>12.12/5.-12.-12/5 || 6/5 6 (6/2 ∞/2) {{pipe}}||p6m
|- align=center
!2.16
|[[File:Tiling -6,12,i,12.png|150px]] || 12.6/5.12.∞<br/>-6.12.∞.12 || 6/5 ∞ {{pipe}} 6||p6m
|- align=center
!rowspan=3|4||1.18
|rowspan=3|[[File:12-5.3.12-5.6-5 tiling-frame.png|100px]]||[[File:Tiling 3,125,-6,125.png|150px]] || 12/5.3.12/5.6/5<br>3.12/5.-6.12/5 || 3 6 {{pipe}} 6/5||p6m
|- align=center
!1.19
|[[File:Tiling 4,125,-4,-125.png|150px]] || 12/5.4.12/7.4/3<br/>4.12/5.-4.-12/5 || 2 6/5 (3/2 6/2) {{pipe}}||p6m
|- align=center
!1.17
|[[File:Tiling 3,-4,6,-4.png|150px]] || 4.3/2.4.6/5<br/>3.-4.6.-4 || 3/2 6 {{pipe}} 2||p6m
|- align=center
!5||2.5
|[[File:8.8-3.oo tiling-frame.png|100px]]||[[File:Tiling 8,83,i.png|150px]] || 8.8/3.∞ || 4/3 4 ∞ {{pipe}}||p4m
|- align=center
!6||2.15
||[[File:12.12-5.oo tiling-frame.png|100px]]||[[File:Tiling 12,125,i.png|150px]] || 12.12/5.∞ || 6/5 6 ∞ {{pipe}}||p6m
|- align=center
!7||1.6
|[[File:8.4-3.8-5 tiling-frame.png|100px]]||[[File:Tiling 4,-8,83.png|150px]] || 8.4/3.8/5<br/>4.-8.8/3 || 2 4/3 4 {{pipe}}||p4m
|- align=center
!Convex||1.11
| ||[[File:Tiling 4,6,12.png|150px]] || 4.6.12 || 2 3 6 {{pipe}} || p6m
|- align=center
!8||1.13
|[[File:6.4-3.12-7 tiling-frame.png|100px]]||[[File:Tiling 4,-6,125.png|150px]] || 6.4/3.12/7<br/>4.-6.12/5|| 2 3 6/5 {{pipe}}||p6m
|- align=center
!9||1.12
|[[File:12.6-5.12-7 tiling-frame.png|100px]]||[[File:Tiling 6,-12,125.png|150px]] || 12.6/5.12/7<br/>6.-12.12/5 || 3 6/5 6 {{pipe}}||p6m
|- align=center
!10||1.8
|[[File:4.8-5.8-5 tiling-frame.png|100px]]||[[File:Tiling -4,83,83.png|150px]] || 4.8/5.8/5<br/>-4.8/3.8/3 || 2 4 {{pipe}} 4/3||p4m
|- align=center
!11||1.22
|[[File:12-5.12-5.3-2 tiling-frame.png|100px]]||[[File:Tiling -3,125,125.png|150px]] || 12/5.12/5.3/2<br/>-3.12/5.12/5 || 2 3 {{pipe}} 6/5||p6m
|- align=center
!Convex||1.1
| ||[[File:Tiling 3,3,3,4,4.png|150px]] || 3.3.3.4.4 || [[non-Wythoffian]] || cmm
|- align=center
!12||1.2
|[[File:3-2.3-2.3-2.4.4 tiling-frame.png|100px]]||[[File:Tiling 3,3,3,-4,-4.png|150px]] || 4.4.3/2.3/2.3/2<br/>3.3.3.-4.-4 || non-Wythoffian||cmm
|- align=center
!Convex||1.3
| ||[[File:Tiling 3,3,4,3,4.png|150px]] || 3.3.4.3.4 || {{pipe}} 2 4 4 || p4g
|- align=center
!13||1.4
| || [[File:Tiling 3,3,-4,3,-4.png|150px]] || 4.3/2.4.3/2.3/2<br/>3.3.-4.3.-4 || {{pipe}} 2 4/3 4/3 ||p4g
|- align=center
!14||2.4
| || [[File:Tiling 3,4,3,-4,3,i.png|150px]] || 3.4.3.4/3.3.∞<br/>3.4.3.-4.3.∞ || {{pipe}} 4/3 4 ∞ ||p4
|- align=center
!Convex||1.10
| ||[[File:Tiling 3,3,3,3,6.png|150px]] || 3.3.3.3.6 || {{pipe}} 2 3 6 || p6
|- align=center
!rowspan=2| ||2.1
|rowspan=2| || [[File:Tiling 3,4,4,3,i,3,i.png|150px]] || 3/2.∞.3/2.∞.3/2.4/3.4/3<br>3.4.4.3.∞.3.∞ || non-Wythoffian || cmm
|- align=center
!2.2
|| [[File:Tiling 3,-4,-4,3,i,3,i.png|150px]] || 3/2.∞.3/2.∞.3/2.4.4<br>3.-4.-4.3.∞.3.∞ || non-Wythoffian || cmm
|- align=center
! ||2.3
| ||[[File:Tiling 3,4,4,3,-4,-4,3,i.png|150px]] || 3/2.∞.3/2.4.4.3/2.4/3.4/3<br>3.4.4.3.-4.-4.3.∞ || non-Wythoffian || p3
|- align=center
!rowspan=2| ||2.8
|rowspan=2| || [[File:Tiling 4,8,83,-4,i.png|150px]] || 4.∞.4/3.8/3.8<br>4.8.8/3.-4.∞ || non-Wythoffian || p4m
|- align=center
!2.9
|| [[File:Tiling -4,8,83,4,i.png|150px]] || 4.∞.4.8.8/3<br>-4.8.8/3.4.∞ || non-Wythoffian || p4m
|- align=center
!rowspan=4| ||2.10
|rowspan=4| || [[File:Tiling 4,8,-4,8,-4,i.png|150px]] || 4.∞.4/3.8.4/3.8<br>4.8.-4.8.-4.∞ || non-Wythoffian || p4m
|- align=center
!2.11
|| [[File:Tiling 4,8,-4,8,-4,i.png|150px]] || 4.∞.4/3.8.4/3.8<br>4.8.-4.8.-4.∞ || non-Wythoffian || p4g
|- align=center
!2.12
|| [[File:Tiling 4,83,4,83,-4,i.png|150px]] || 4.∞.4/3.8/3.4.8/3<br>4.8/3.4.8/3.-4.∞ || non-Wythoffian || p4m
|- align=center
!2.13
|| [[File:Tiling 4,83,4,83,-4,i.png|150px]] || 4.∞.4/3.8/3.4.8/3<br>4.8/3.4.8/3.-4.∞ || non-Wythoffian || p4g
|- align=center
!rowspan=6| ||2.18
|rowspan=6| || [[File:Tiling 3,4,4,3,4,4,3,i.png|150px]] || 3/2.∞.3/2.4/3.4/3.3/2.4/3.4/3<br>3.4.4.3.4.4.3.∞ || non-Wythoffian || p6m
|- align=center
!2.19
|| [[File:Tiling 3,-4,-4,3,-4,-4,3,i.png|150px]] || 3/2.∞.3/2.4.4.3/2.4.4<br>3.-4.-4.3.-4.-4.3.∞ || non-Wythoffian || p6m
|- align=center
!2.20
|| [[File:Tiling 3,12,-6,12,3,i,3,i.png|150px]] || 3/2.∞.3/2.∞.3/2.12/11.6.12/11<br>3.12.-6.12.3.∞.3.∞ || non-Wythoffian || p6m
|- align=center
!2.21
|| [[File:Tiling 3,-12,6,-12,3,i,3,i.png|150px]] || 3/2.∞.3/2.∞.3/2.12.6/5.12<br>3.-12.6.-12.3.∞.3.∞ || non-Wythoffian || p6m
|- align=center
!2.22
|| [[File:Tiling 3,125,6,125,3,i,3,i.png|150px]] || 3/2.∞.3/2.∞.3/2.12/7.6/5.12/7<br>3.12/5.6.12/5.3.∞.3.∞ || non-Wythoffian || p6m
|- align=center
!2.23
|| [[File:Tiling 3,-125,-6,-125,3,i,3,i.png|150px]] || 3/2.∞.3/2.∞.3/2.12/5.6.12/5<br>3.-12/5.-6.-12/5.3.∞.3.∞ || non-Wythoffian || p6m
|}

There are two uniform tilings for the vertex configuration 4.8.-4.8.-4.∞ (Grünbaum ''et al.'', 2.10 and 2.11) and also two uniform tilings for the vertex configuration 4.8/3.4.8/3.-4.∞ (Grünbaum ''et al.'', 2.12 and 2.13), with different symmetries. There is also a third tiling for each vertex configuration that is only [[Pseudo-uniform polyhedron|pseudo-uniform]] (vertices come in two symmetry orbits). They use different sets of square faces. Hence, for star Euclidean tilings, the vertex configuration does not necessarily determine the tiling.<ref name=Grunbaum1981/>

In the pictures below, the included squares with horizontal and vertical edges are marked with a central dot. A single square has edges highlighted.<ref name=Grunbaum1981/>
<gallery>
Grünbaum ambiguous tilings 1.png|2.10 and 2.12 (p4m)
Grünbaum ambiguous tilings 2.png|2.11 and 2.13 (p4g)
Grünbaum ambiguous tilings 3.png|Pseudo-uniform
</gallery>

The tilings with zigzags are listed below. {∞<sup>𝛼</sup>} denotes a zigzag with angle 0 < 𝛼 < π. The apeirogon can be considered the special case 𝛼 = π. The symmetries are given for the generic case, but there are sometimes special values of 𝛼 that increase the symmetry. Tilings 3.1 and 3.12 can even become regular; 3.32 already is (it has no free parameters). Sometimes, there are special values of 𝛼 that cause the tiling to degenerate.<ref name=Grunbaum1981/>
{| class="wikitable sortable"
!colspan=4| Tilings with zigzags
|-
!Grünbaum ''et al.'', 1981<ref name=Grunbaum1981/>||Diagram||Vertex<br/>Config.||Symmetry
|- align=center
!3.1
|[[File:Tiling ia,ib,ic.png|150px]]||∞<sup>𝛼</sup>.∞<sup>β</sup>.∞<sup>γ</sup><br>𝛼+β+γ=2π||p2
|- align=center
!3.2
|[[File:Tiling ia,ib,-ia+b.png|150px]]||∞<sup>𝛼</sup>.∞<sup>β</sup>.-∞<sup>𝛼+β</sup><br>0<𝛼+β≤π||p2
|- align=center
!3.3
|[[File:Tiling 3,3,ip-a,-3,i2p3+a.png|150px]]||3.3.∞<sup>π-𝛼</sup>.-3.∞<sup>𝛼+2π/3</sup><br>0≤𝛼≤π/6||pgg
|- align=center
!3.4
|[[File:Tiling 3,3,-ip-a,-3,i2p3-a.png|150px]]||3.3.-∞<sup>π-𝛼</sup>.-3.∞<sup>−𝛼+2π/3</sup><br>0≤𝛼<π/3||pgg
|- align=center
!3.5
|[[File:Tiling 4,4,if,4,4,-if.png|150px]]||4.4.∞<sup>φ</sup>.4.4.-∞<sup>φ</sup><br>φ=2 arctan(''n''/''k''), ''nk'' even, (''n'',''k'')=1<br>drawn for φ=2 arctan 2||pmg
|- align=center
!3.6
|[[File:Tiling 4,4,ips,-4,-4,ips.png|150px]]||4.4.∞<sup>φ</sup>.-4.-4.∞<sup>φ</sup><br>φ=2 arctan(''n''/''k''), ''nk'' even, (''n'',''k'')=1<br>drawn for φ=2 arctan 1/2||pmg
|- align=center
!3.7
|[[File:Tiling 3,4,4,3,-i2p3,-3,-i2pi3.png|150px]]||3.4.4.3.-∞<sup>2π/3</sup>.-3.-∞<sup>2π/3</sup>||cmm
|- align=center
!3.8
|[[File:Tiling 3,-4,-4,3,-i2p3,-3,-i2p3.png|150px]]||3.-4.-4.3.-∞<sup>2π/3</sup>.-3.-∞<sup>2π/3</sup>||cmm
|- align=center
!3.9
|[[File:Tiling 4,4,ip3,i,-ip3.png|150px]]||4.4.∞<sup>π/3</sup>.∞.-∞<sup>π/3</sup>||p2
|- align=center
!3.10
|[[File:Tiling 4,4,i2p3,i,-i2p3.png|150px]]||4.4.∞<sup>2π/3</sup>.∞.-∞<sup>2π/3</sup>||p2
|- align=center
!3.11
|[[File:Tiling i,ia,i,-ia.png|150px]]||∞.∞<sup>𝛼</sup>.∞.∞<sup>−𝛼</sup><br>0<𝛼<π||cmm
|- align=center
!3.12
|[[File:Tiling ia,ip-a,ia,ip-a.png|150px]]||∞<sup>𝛼</sup>.∞<sup>π-𝛼</sup>.∞<sup>𝛼</sup>.∞<sup>π-𝛼</sup><br>0<𝛼≤π/2||cmm
|- align=center
!3.13
|[[File:Tiling 3,ia,-3,-ia.png|150px]]||3.∞<sup>𝛼</sup>.-3.-∞<sup>𝛼</sup><br>π/3<𝛼<π||p31m
|- align=center
!3.14
|[[File:Tiling 4,4,i2p3,4,4,-i2p3.png|150px]]||4.4.∞<sup>2π/3</sup>.4.4.-∞<sup>2π/3</sup>||p31m
|- align=center
!3.15
|[[File:Tiling 4,4,ip3,-4,-4,-ip3.png|150px]]||4.4.∞<sup>π/3</sup>.-4.-4.-∞<sup>π/3</sup>||p31m
|- align=center
!3.16
|[[File:Tiling 4,ia,-4,-ia.png|150px]]||4.∞<sup>𝛼</sup>.-4.-∞<sup>𝛼</sup><br>0<𝛼<π, 𝛼≠π/2||p4g
|- align=center
!3.17
|[[File:Tiling 4,-8,ip2,i,-ip2,-8.png|150px]]||4.-8.∞<sup>π/2</sup>.∞.-∞<sup>π/2</sup>.-8||cmm
|- align=center
!3.18
|[[File:Tiling 4,-8,ip2,i,-ip2,-8.png|150px]]||4.-8.∞<sup>π/2</sup>.∞.-∞<sup>π/2</sup>.-8||p4<!--not a mistake, again two tilings with the same vertex configuration-->
|- align=center
!3.19
|[[File:Tiling 4,83,ip2,i,-ip2,83.png|150px]]||4.8/3.∞<sup>π/2</sup>.∞.-∞<sup>π/2</sup>.8/3||cmm
|- align=center
!3.20
|[[File:Tiling 4,83,ip2,i,-ip2,83.png|150px]]||4.8/3.∞<sup>π/2</sup>.∞.-∞<sup>π/2</sup>.8/3||p4<!--another pair-->
|- align=center
!3.21
|[[File:Tiling 6,-12,ip3,i,-ip3,-12.png|150px]]||6.-12.∞<sup>π/3</sup>.∞.-∞<sup>π/3</sup>.-12||p6
|- align=center
!3.22
|[[File:Tiling 6,-12,i2p3,i,-i2p3,-12.png|150px]]||6.-12.∞<sup>2π/3</sup>.∞.-∞<sup>2π/3</sup>.-12||p6
|- align=center
!3.23
|[[File:Tiling 6,125,ip3,i,-ip3,125.png|150px]]||6.12/5.∞<sup>π/3</sup>.∞.-∞<sup>π/3</sup>.12/5||p6
|- align=center
!3.24
|[[File:Tiling 6,125,i2p3,i,-i2p3,125.png|150px]]||6.12/5.∞<sup>2π/3</sup>.∞.-∞<sup>2π/3</sup>.12/5||p6
|- align=center
!3.25
|[[File:Tiling 3,3,3,i2p3,-3,i2p3.png|150px]]||3.3.3.∞<sup>2π/3</sup>.-3.∞<sup>2π/3</sup>||p31m
|- align=center
!3.26
|[[File:Tiling 3,i,3,-i2p3,-3,-i2p3.png|150px]]||3.∞.3.-∞<sup>2π/3</sup>.-3.-∞<sup>2π/3</sup>||cm
|- align=center
!3.27
|[[File:Tiling 3,i,-i2p3,i,-i2p3,i.png|150px]]||3.∞.-∞<sup>2π/3</sup>.∞.-∞<sup>2π/3</sup>.∞||p31m
|- align=center
!3.28
|[[File:Tiling 3,i2p3,i2p3,-3,-i2p3,-i2p3.png|150px]]||3.∞<sup>2π/3</sup>.∞<sup>2π/3</sup>.-3.-∞<sup>2π/3</sup>.-∞<sup>2π/3</sup>||p31m
|- align=center
!3.29
|[[File:Tiling i,ip3,ip3,i,-ip3,-ip3.png|150px]]||∞.∞<sup>π/3</sup>.∞<sup>π/3</sup>.∞.-∞<sup>π/3</sup>.-∞<sup>π/3</sup>||cmm
|- align=center
!3.30
|[[File:Tiling i,ip3,-i2p3,i,i2p3,-ip3.png|150px]]||∞.∞<sup>π/3</sup>.-∞<sup>2π/3</sup>.∞.∞<sup>2π/3</sup>.-∞<sup>π/3</sup>||p2
|- align=center
!3.31
|[[File:Tiling i,i2p3,i2p3,i,-i2p3,-i2p3.png|150px]]||∞.∞<sup>2π/3</sup>.∞<sup>2π/3</sup>.∞.-∞<sup>2π/3</sup>.-∞<sup>2π/3</sup>||cmm
|- align=center
!3.32
|[[File:Tiling ip3,ip3,ip3,ip3,ip3,ip3.png|150px]]||∞<sup>π/3</sup>.∞<sup>π/3</sup>.∞<sup>π/3</sup>.∞<sup>π/3</sup>.∞<sup>π/3</sup>.∞<sup>π/3</sup>||p6m
|- align=center
!3.33
|[[File:Tiling ip3,-i2p3,-i2p3,ip3,-i2p3,-i2p3.png|150px]]||∞<sup>π/3</sup>.-∞<sup>2π/3</sup>.-∞<sup>2π/3</sup>.∞<sup>π/3</sup>.-∞<sup>2π/3</sup>.-∞<sup>2π/3</sup>||cmm
|}

The tiling pairs 3.17 and 3.18, as well as 3.19 and 3.20, have identical vertex configurations but different symmetries.<ref name=Grunbaum1981/>

Tilings 3.7 through 3.10 have the same edge arrangement as 2.1 and 2.2; 3.17 through 3.20 have the same edge arrangement as 2.10 through 2.13; 3.21 through 3.24 have the same edge arrangement as 2.18 through 2.23; and 3.25 through 3.33 have the same edge arrangement as 1.25 (the regular triangular tiling).<ref name=Grunbaum1981/><!--3.3 only matches snub square tiling in a special case-->

== Self-dual tilings ==
<!--[[Self-dual tiling]] & [[Self-dual tilings]] redirect directly here.-->
[[Image:Self-dual square tiling.png|right|The {4,4} [[square tiling]] (black) with its dual (red).]]

A tiling can also be '''self-dual'''. The square tiling, with [[Schläfli symbol]] {4,4}, is self-dual; shown here are two square tilings (red and black), dual to each other.

{{clear}}

== Uniform tilings using regular or isotoxal polygrams as nonconvex isotoxal simple polygons ==
[[File:Hexagon hexagram tiling2.png|thumb|This example, 4.8{{supsub|*|π/4}}.4{{supsub|**|π/2}}.8{{supsub|*|π/4}}, is considered not edge-to-edge due to the large square, although the latter can be interpreted as a star polygon with four pairs of collinear edges.]]

Seeing a [[Star polygon#Regular star polygon|regular star polygon]] as a [[nonconvex]] isotoxal [[simple polygon]] with twice as many (shorter) sides but alternating the same outer and "inner" [[internal angle]]s allows regular star polygons to be used in a tiling, and seeing [[isotoxal]] simple polygons as "regular" allows regular star polygons to (but not all of them can) be used in a "uniform" tiling.

Also, the outlines of certain non-regular isotoxal star polygons are nonconvex isotoxal (simple) polygons with ''as many'' (shorter) sides and alternating the same outer and "inner" internal angles; seeing this kind of isotoxal star polygons as their outlines allows it to be used in a tiling, and seeing isotoxal simple polygons as "regular" allows this kind of isotoxal star polygons to (but not all of them can) be used in a "uniform" tiling.

An isotoxal simple '''2'''''n''-gon with outer internal angle 𝛼 is denoted by {''n''<sub>𝛼</sub>}; its outer vertices are labeled as ''n''{{supsub|*|𝛼}}, and inner ones as ''n''{{supsub|**|𝛼}}.

These expansions to the definition for a tiling require corners with only 2 polygons to not be considered vertices — since the [[vertex configuration]] for vertices with at least 3 polygons suffices to define such a "uniform" tiling, and so that the latter has one vertex configuration alright (otherwise it would have two) —. There are 4 such uniform tilings with adjustable angles 𝛼, and 18 such uniform tilings that only work with specific angles, yielding a total of 22 uniform tilings that use star polygons.<ref>''Tilings and Patterns'', Branko Gruenbaum, G. C. Shephard, 1987, 2.5 Tilings using star polygons, pp. 82–85.</ref>

All of these tilings, with possible order-2 vertices ignored, with possible double edges and triple edges reduced to single edges, are topologically related to the ordinary uniform tilings (using only convex regular polygons).

{| class=wikitable
|+ 4 "uniform" tilings using star polygons with adjustable angles 𝛼
|- align=center valign=bottom
|[[File:Uniform-star-tiling-36s6s-e.svg|160px]]<BR>3.6{{supsub|*|𝛼}}.6{{supsub|**|𝛼}}<BR>Topol. related to [[Truncated hexagonal tiling|3.12.12]]
|[[File:Uniform-star-tiling-44s4s-a.svg|160px]]<BR>4.4{{supsub|*|𝛼}}.4{{supsub|**|𝛼}}<BR>Topol. related to [[Truncated square tiling|4.8.8]]
|[[File:Uniform-star-tiling-63s3s-a.svg|160px]]<BR>6.3{{supsub|*|𝛼}}.3{{supsub|**|𝛼}}<BR>Topol. related to [[Hexagonal tiling|6.6.6]]
|[[File:Uniform-star-tiling-33s33s-a.svg|160px]]<BR>3.3{{supsub|*|𝛼}}.3.3{{supsub|**|𝛼}}<BR>Topol. related to [[Trihexagonal tiling|3.6.3.6]]
|}

{| class=wikitable
|+ 18 "uniform" tilings using star polygons with specific angles
|- align=center valign=bottom
|[[File:Uniform-star-tiling-g.svg|160px]]<BR>4.6.4{{supsub|*|π/6}}.6<BR>Topol. related to [[Square tiling|4.4.4.4]]
|[[File:Uniform-star-tiling-l.svg|160px]]<BR>(8.4{{supsub|*|π/4}})<sup>2</sup><BR>Topol. related to [[Square tiling|4.4.4.4]]
|[[File:Uniform-star-tiling-o.svg|160px]]<BR>12.12.4{{supsub|*|π/3}}<BR>Topol. related to [[Truncated square tiling|4.8.8]]
|[[File:Uniform-star-tiling-c.svg|160px]]<BR>{{nowrap|3.3.8{{supsub|*|π/12}}.4{{supsub|**|π/3}}.8{{supsub|*|π/12}}}}<BR>Topol. related to [[Truncated square tiling|4.8.8]]
|[[File:Uniform-star-tiling-b.svg|160px]]<BR>{{nowrap|3.3.8{{supsub|*|π/12}}.3.4.3.8{{supsub|*|π/12}}}}<BR>Topol. related to [[Truncated square tiling|4.8.8]]
|[[File:Uniform-star-tiling-e.svg|160px]]<BR>3.4.8.3.8{{supsub|*|π/12}}<BR>Topol. related to [[Truncated square tiling|4.8.8]]
|- align=center valign=bottom
|[[File:Uniform-star-tiling-q.svg|160px]]<BR>5.5.4{{supsub|*|π/10}}.5.4{{supsub|*|π/10}}<BR>Topol. related to [[Snub square tiling|3.3.4.3.4]]
|[[File:Uniform-star-tiling-i.svg|160px]]<BR>4.6{{supsub|*|π/6}}.6{{supsub|**|π/2}}.6{{supsub|*|π/6}}<BR>Topol. related to [[Truncated triangular tiling|6.6.6]]
|[[File:Uniform-star-tiling-h.svg|160px]]<BR>(4.6{{supsub|*|π/6}})<sup>3</sup><BR>Topol. related to [[Hexagonal tiling|6.6.6]]
|[[File:Uniform-star-tiling-m.svg|160px]]<BR>9.9.6{{supsub|*|4π/9}}<BR>Topol. related to [[Truncated triangular tiling|6.6.6]]
|[[File:Uniform-star-tiling-j.svg|160px]]<BR>(6.6{{supsub|*|π/3}})<sup>2</sup><BR>Topol. related to [[Trihexagonal tiling|3.6.3.6]]
|[[File:Uniform-star-tiling-n.svg|160px]]<BR>(12.3{{supsub|*|π/6}})<sup>2</sup><BR>Topol. related to [[Trihexagonal tiling|3.6.3.6]]
|- align=center valign=bottom
|[[File:Uniform-star-tiling-d.svg|160px]]<BR>3.4.6.3.12{{supsub|*|π/6}}<BR>Topol. related to [[Truncated trihexagonal tiling|4.6.12]]
|[[File:Uniform-star-tiling-a.svg|160px]]<BR>{{nowrap|3.3.3.12{{supsub|*|π/6}}.3.3.12{{supsub|*|π/6}}}}<BR>Topol. related to [[Truncated hexagonal tiling|3.12.12]]
|[[File:Uniform-star-tiling-p.svg|160px]]<BR>18.18.3{{supsub|*|2π/9}}<BR>Topol. related to [[Truncated hexagonal tiling|3.12.12]]
|[[File:Uniform-star-tiling-f.svg|160px]]<BR>3.6.6{{supsub|*|π/3}}.6<BR>Topol. related to [[Rhombitrihexagonal tiling|3.4.6.4]]
|[[File:Uniform-star-tiling-k.svg|160px]]<BR>8.3{{supsub|*|π/12}}.8.6{{supsub|*|5π/12}}<BR>Topol. related to [[Rhombitrihexagonal tiling|3.4.6.4]]
|[[File:Uniform-star-tiling-9393s.svg|160px]]<BR>9.3.9.3{{supsub|*|π/9}}<BR>Topol. related to [[Trihexagonal tiling|3.6.3.6]]
|- align=center valign=bottom
|}

== Uniform tilings using convex isotoxal simple polygons ==
Non-regular isotoxal either star or simple 2''n''-gons always alternate two angles. Isotoxal simple 2''n''-gons, {''n''<sub>𝛼</sub>}, can be [[Convex polygon|convex]]; the simplest ones are the rhombi (2×2-gons), {2<sub>𝛼</sub>}. Considering these convex {''n''<sub>𝛼</sub>} as "regular" polygons allows more tilings to be considered "uniform".
{| class=wikitable
|+ Examples of "uniform" tilings using convex isotoxal simple 2''n''-gons
|- align=center valign=bottom
|[[File:Hexatile-rhombic-snub-hex.svg|160px]]<BR>3.2{{supsub|*|π/3}}.6.2{{supsub|**|π/3}}<BR>Topol. related to [[Rhombitrihexagonal tiling|3.4.6.4]]
|[[File:Octatile-rhombic0.svg|160px]]<BR>4.4.4.4<BR>Topol. related to [[Square tiling|4.4.4.4]]
|[[File:Octatile-rhombic1.svg|320px]]<BR>(2{{supsub|*|π/4}}.2{{supsub|**|π/4}})<sup>2</sup><BR>Topol. related to [[Square tiling|4.4.4.4]]
|[[File:Octatile-rhombic2.svg|160px]]<BR>2{{supsub|*|π/4}}.2{{supsub|*|π/4}}.2{{supsub|**|π/4}}.2{{supsub|**|π/4}}<BR>Topol. related to [[Square tiling|4.4.4.4]]
|[[File:Octatile-rhombic3.svg|160px]]<BR>4.2{{supsub|*|π/4}}.4.2{{supsub|**|π/4}}<BR>Topol. related to [[Square tiling|4.4.4.4]]
|}

== See also ==
{{Commons category|Uniform tilings}}
* [[Wythoff symbol]]
* [[List of uniform tilings]]
* [[Uniform tilings in hyperbolic plane]]
* [[Uniform polytope]]

== References ==
{{Reflist}}
* [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
** [[Norman Johnson (mathematician)|N. W. Johnson]]: ''The Theory of Uniform Polytopes and Honeycombs'', Ph. D. Dissertation, University of Toronto, 1966 
* {{cite book|author1-link=Branko Grünbaum|last1=Grünbaum|first1=Branko|author2-link=G. C. Shephard|last2=Shephard|first2=G. C.|title=Tilings and Patterns|publisher=W. H. Freeman and Company|year=1987|isbn=0-7167-1193-1|url-access=registration|url=https://archive.org/details/isbn_0716711931}} (Star tilings section 12.3)
* [[H. S. M. Coxeter]], [[M. S. Longuet-Higgins]], [[J. C. P. Miller]], ''Uniform polyhedra'', '''Phil. Trans.''', 1954, 246 A, 401–50 {{JSTOR|91532}} (Table 8)

== External links ==
* {{MathWorld | urlname=UniformTessellation | title=Uniform tessellation}}
* [http://www2u.biglobe.ne.jp/~hsaka/mandara/ue2 Uniform Tessellations on the Euclid plane]
* [http://www.orchidpalms.com/polyhedra/tessellations/tessel.htm Tessellations of the Plane]
* [http://www.tess-elation.co.uk/ David Bailey's World of Tessellations]
* [https://web.archive.org/web/20060909053826/http://www.uwgb.edu/dutchs/SYMMETRY/uniftil.htm ''k''-uniform tilings]
* [http://probabilitysports.com/tilings.html ''n''-uniform tilings]
* {{KlitzingPolytopes|flat.htm|4D|Euclidean tilings}}

{{Honeycombs}}
{{Tessellation}}

[[Category:Uniform tilings| ]]