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In [[geometry]], a '''uniform tiling''' is a [[tessellation]] of the plane by [[regular polygon]] faces with the restriction of being [[vertex-uniform]].
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Uniform tilings can exist in both the [[Euclidean plane]] and [[Hyperbolic space|hyperbolic plane]]. Uniform tilings are related to the finite [[uniform polyhedron|uniform polyhedra]] which 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 the group name represented by the order of the mirrors in sequential vertices.
 
A fundamental domain triangle is (''p'' ''q'' ''r''), and a right triangle (''p'' ''q'' 2), where ''p'', ''q'', ''r'' are whole numbers greater than 1. 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 a number of symbolic schemes for naming these figures, from a modified [[Schläfli symbol]] for right triangle domains: (''p'' ''q'' 2) → {''p'', ''q''}. The [[Coxeter-Dynkin diagram]] is a triangular graph with ''p'', ''q'', ''r'' labeled on the edges. If ''r'' = 2, the graph is linear since order-2 domain nodes generate no reflections. The [[Wythoff symbol]] takes the 3 integers and separates them by a vertical bar (|). If the generator point is off the mirror opposite a domain node, it is given before the bar.
 
Finally tilings can be described by their [[vertex configuration]], the sequence of polygons around each 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). [[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 whole number orders, including:
 
{| class=wikitable
|+ Euclidean plane
|-
![[Orbifold notation|Orbifold<BR>symmetry]]
!colspan=3|[[Coxeter group]]
![[Coxeter-Dynkin diagram|Coxeter-Dynkin<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
| (&infin; 2 &infin; 2)
| <math>{\tilde{I}}_1</math> × <math>{\tilde{I}}_1</math>
| [&infin;,2,&infin;]
| {{CDD|node|infin|node|2|node|infin|node}}
| 3 reflective forms, 1 snub
|-
! colspan=6 | '''Noncompact'''
|- align=center
| *&infin;&infin;
| (&infin;)
| <math>{\tilde{I}}_1</math>
| [&infin;]
| {{CDD|node|infin|node}}
|
|- align=center
| *22&infin;
| (2 2 &infin;)
| <math>{\tilde{I}}_1</math> × <math>{\tilde{A}}_2</math>
| [&infin;,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-Dynkin<BR>diagram]]
!notes
|-
! colspan=5 | '''Compact'''
|- valign=top align=center
| *pq2
| (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
|-
! colspan=5 | '''Noncompact'''
|- align=center
| *&infin;p2
| (p &infin; 2)
| [p,&infin;]
| {{CDD|node|p|node|infin|node}}|| p>=3
|- align=center
| *&infin;pq
| (p q &infin;)
| [(p,q,&infin;)]
| {{CDD|3|node|p|node|q|node|infin}}|| p,q>=3, p+q>6
|- align=center
| *&infin;&infin;p
| (p &infin; &infin;)
| [(p,&infin;,&infin;)]
| {{CDD|3|node|p|node|infin|node|infin}}|| p>=3
|- align=center
| *&infin;&infin;&infin;
| (&infin; &infin; &infin;)
| [(&infin;,&infin;,&infin;)]
| {{CDD|3|node|infin|node|infin|node|infin}}||
|}
 
== Uniform tilings of the Euclidean plane ==
 
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 represented by (2 2 2 2) represents by two sets of parallel mirrors, which in general can have a rectangular fundamental domain. It generates no new tilings.
 
A further prismatic symmetry group represented by (&infin; 2 2) which 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'' &#124; ''p'' 2
! 2 ''q'' &#124; ''p''
! 2 &#124; ''p'' ''q''
! 2 ''p'' &#124; ''q''
! ''p'' &#124; ''q'' 2
! ''p'' ''q'' &#124; 2
! ''p'' ''q'' 2 &#124;
! &#124; ''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-Dynkin 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 figure]]
!
!p<sup>q</sup>
!(q.2p.2p)
!(p.q.p.q)
!(p.2q.2q)
!q<sup>p</sup>
!(p.4.q.4)
!(4.2p.2q)
!(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> [[List of uniform tilings|V4.8.8]]
|[[Image:Uniform tiling 44-t0.png|64px]]<BR>[[Square tiling|{4,4}]]
|[[Image:Uniform tiling 44-t01.png|64px]]<BR>[[Truncated square tiling|4.8.8]]
|[[Image:Uniform tiling 44-t1.png|64px]]<BR>[[Square tiling|4.4.4.4]]
|[[Image:Uniform tiling 44-t12.png|64px]]<BR>[[Truncated square tiling|4.8.8]]
|[[Image:Uniform tiling 44-t2.png|64px]]<BR>[[Square tiling|{4,4}]]
|[[Image:Uniform tiling 44-t02.png|64px]]<BR>[[Square tiling|4.4.4.4]]
|[[Image:Uniform tiling 44-t012.png|64px]]<BR>[[Truncated square tiling|4.8.8]]
|[[Image:Uniform tiling 44-snub.png|64px]]<BR>[[Snub square tiling|3.3.4.3.4]]
|-align=center
|[[Hexagonal tiling]]<BR>(6 3 2)
|[[Image:Tile V46b.svg|64px]] <br> [[List of uniform tilings|V4.6.12]]
|[[Image:Uniform tiling 63-t0.png|64px]]<BR>[[Hexagonal tiling|{6,3}]]
|[[Image:Uniform tiling 63-t01.png|64px]]<BR>[[Truncated hexagonal tiling|3.12.12]]
|[[Image:Uniform tiling 63-t1.png|64px]]<BR>[[Trihexagonal tiling|3.6.3.6]]
|[[Image:Uniform tiling 63-t12.png|64px]]<BR>[[Hexagonal tiling|6.6.6]]
|[[Image:Uniform tiling 63-t2.png|64px]]<BR>[[Triangular tiling|{3,6}]]
|[[Image:Uniform tiling 63-t02.png|64px]]<BR>[[Small rhombitrihexagonal tiling|3.4.6.4]]
|[[Image:Uniform tiling 63-t012.png|64px]]<BR>[[Great rhombitrihexagonal tiling|4.6.12]]
|[[Image:Uniform tiling 63-snub.png|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 &#124; p r
! r q &#124; p
! r &#124; p q
! r p &#124; q
! p &#124; q r
! p q &#124; r
! p q r &#124;
! &#124; p q r
|-
![[Coxeter-Dynkin diagram]]
!
!{{CDD|node_1|p|node|q|node|r}}
!{{CDD|node_1|p|node_1|q|node|r}}
!{{CDD|node|p|node_1|q|node|r}}
!{{CDD|node|p|node_1|q|node_1|r}}
!{{CDD|node|p|node|q|node_1|r}}
!{{CDD|node_1|p|node|q|node_1|r}}
!{{CDD|node_1|p|node_1|q|node_1|r}}
!{{CDD|node_h|p|node_h|q|node_h|r}}
|-
![[Vertex configuration|Vertex figure]]
!
!(p.q)<sup>r</sup>
!(r.2p.q.2p)
!(p.r)<sup>q</sup>
!(q.2r.p.2r)
!(q.r)<sup>p</sup>
!(q.2r.p.2r)
!(r.2q.p.2q)
!(3.r.3.q.3.p)
|-align=center
|Triangular<BR>(3 3 3)
|[[Image:Tiling_Regular_3-6_Triangular.svg|64px]] <br> [[List of uniform tilings|V6.6.6]]
|[[Image:Uniform tiling 333-t0.png|64px]]<BR>[[Triangular tiling|(3.3)<sup>3</sup>]]
|[[Image:Uniform tiling 333-t01.png|64px]]<BR>[[Trihexagonal tiling|3.6.3.6]]
|[[Image:Uniform tiling 333-t1.png|64px]]<BR>[[Triangular tiling|(3.3)<sup>3</sup>]]
|[[Image:Uniform tiling 333-t12.png|64px]]<BR>[[Trihexagonal tiling|3.6.3.6]]
|[[Image:Uniform tiling 333-t2.png|64px]]<BR>[[Triangular tiling|(3.3)<sup>3</sup>]]
|[[Image:Uniform tiling 333-t02.png|64px]]<BR>[[Trihexagonal tiling|3.6.3.6]]
|[[Image:Uniform tiling 333-t012.png|64px]]<BR>[[Hexagonal tiling|6.6.6]]
|[[Image:Uniform tiling 333-snub.png|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-Dynkin diagram]]: {{CDD|node|infin|node|2|node|infin|node}}. All forms generated from it become a [[square tiling]].
 
== Uniform tilings of the hyperbolic plane ==
{{details|Uniform tilings in hyperbolic plane}}
 
There are infinitely many uniform tilings of 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's fundamental domains that place vertices at infinity, such as (&infin; 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 &#124; p 2
! 2 q &#124; p
! 2 &#124; p q
! 2 p &#124; q
! p &#124; q 2
! p q &#124; 2
! p q 2 &#124;
! &#124; 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-Dynkin 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 figure]]
!
!p<sup>q</sup>
!(q.2p.2p)
!(p.q.p.q)
!(p.2q.2q)
!q<sup>p</sup>
!(p.4.q.4)
!(4.2p.2q)
!(3.3.p.3.q)
|-
|(Hyperbolic plane)<BR>(5 4 2)
|[[Image:Order-4 bisected pentagonal tiling.png|72px]]<br>V4.8.10
|[[Image:Uniform tiling 54-t0.png|64px]]<BR>{5,4}
|[[Image:Uniform tiling 54-t01.png|64px]]<BR>4.10.10
|[[Image:Uniform tiling 54-t1.png|64px]]<BR>4.5.4.5
|[[Image:Uniform tiling 54-t12.png|64px]]<BR>5.8.8
|[[Image:Uniform tiling 54-t2.png|64px]]<BR>{4,5}
|[[Image:Uniform tiling 54-t02.png|64px]]<BR>4.4.5.4
|[[Image:Uniform tiling 54-t012.png|64px]]<BR>4.8.10
|[[Image:Uniform tiling 54-snub.png|64px]]<BR>3.3.4.3.5
|-
|(Hyperbolic plane)<BR>(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
|-
|(Hyperbolic plane)<BR>(7 3 2)
|[[Image:Order-3 heptakis heptagonal tiling.png|72px]]<br>V4.6.14
|[[Image:Uniform tiling 73-t0.png|64px]]<BR>[[Order-3 heptagonal tiling|{7,3}]]
|[[Image:Uniform tiling 73-t01.png|64px]]<BR>3.14.14
|[[Image:Uniform tiling 73-t1.png|64px]]<BR>[[Triheptagonal tiling|3.7.3.7]]
|[[Image:Uniform tiling 73-t12.png|64px]]<BR>7.6.6
|[[Image:Uniform tiling 73-t2.png|64px]]<BR>[[Order-7 triangular tiling|{3,7}]]
|[[Image:Uniform tiling 73-t02.png|64px]]<BR>3.4.7.4
|[[Image:Uniform tiling 73-t012.png|64px]]<BR>[[Great rhombitriheptagonal tiling|4.6.14]]
|[[Image:Uniform tiling 73-snub.png|65px]]<BR>3.3.3.3.7
|-
|(Hyperbolic plane)<BR>(8 3 2)
|[[File:Order-3 octakis octagonal tiling.png|72px]]<br>V4.6.16
|[[Image:Uniform tiling 83-t0.png|64px]]<BR>[[Order-3 octagonal tiling|{8,3}]]
|[[Image:Uniform tiling 83-t01.png|64px]]<BR>3.16.16
|[[Image:Uniform tiling 83-t1.png|64px]]<BR>[[Trioctagonal tiling|3.8.3.8]]
|[[Image:Uniform tiling 83-t12.png|64px]]<BR>8.6.6
|[[Image:Uniform tiling 83-t2.png|64px]]<BR>[[Order-8 triangular tiling|{3,8}]]
|[[Image:Uniform tiling 83-t02.png|64px]]<BR>3.4.8.4
|[[Image:Uniform tiling 83-t012.png|64px]]<BR>[[Great rhombitrioctagonal tiling|4.6.16]]
|[[Image:Uniform tiling 83-snub.png|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 &#124; p r
! r q &#124; p
! r &#124; p q
! r p &#124; q
! p &#124; q r
! p q &#124; r
! p q r &#124;
! &#124; p q r
|-
![[Coxeter-Dynkin diagram]]
!
!{{CDD|node_1|p|node|q|node|r}}
!{{CDD|node_1|p|node_1|q|node|r}}
!{{CDD|node|p|node_1|q|node|r}}
!{{CDD|node|p|node_1|q|node_1|r}}
!{{CDD|node|p|node|q|node_1|r}}
!{{CDD|node_1|p|node|q|node_1|r}}
!{{CDD|node_1|p|node_1|q|node_1|r}}
!{{CDD|node_h|p|node_h|q|node_h|r}}
|-
![[Vertex configuration|Vertex figure]]
!
!(p.r)<sup>q</sup>
!(r.2p.q.2p)
!(p.q)<sup>r</sup>
!(q.2r.p.2r)
!(q.r)<sup>p</sup>
!(r.2q.p.2q)
!(2p.2q.2r)
!(3.r.3.q.3.p)
|-
|Hyperbolic<BR>(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
|-
|Hyperbolic<BR>(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
|-
|Hyperbolic<BR>(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 ==
There are a number 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]]s tiles can be included.
# [[Apeirogon]]s, {&infin;}, can be used as tiling faces.
# 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 &infin;) (3/2 3 &infin;) (6 6/5 &infin;) (3 3/2 &infin;)
 
[[Branko Grünbaum]], in the 1987 book ''Tilings and patterns'', in section 12.3 enumerates a list of 25 uniform tilings, including the 11 convex forms, and adds 14 more he calls ''hollow tilings'' which included the first two expansions above, star polygon faces and vertex figures.
 
[[H.S.M. Coxeter]] et al., in the 1954 paper 'Uniform polyhedra', in ''Table 8: Uniform Tessellations'', uses the first three expansions and enumerates a total of 38 uniform tilings.
 
Finally, if a tiling made of 2 apeirogons is also counted, the total can be considered 39 uniform tilings.
 
[[Image:Six uniform tiling vertex figures.png|320px|thumb|The [[vertex figure]]s for the six tilings with convex [[regular polygon]]s and [[apeirogon]] faces.(The [[Wythoff symbol]] is given in red.)]]
The 7 new tilings with {&infin;} tiles, given by [[vertex figure]] and [[Wythoff symbol]] are:
# &infin;.&infin; (Two half-plane tiles, infinite [[dihedron]])
# 4.4.&infin; - '''&infin; 2 | 2''' ([[Apeirogonal prism]])
# 3.3.3.&infin; - '''| 2 2 &infin;''' ([[Apeirogonal antiprism]])
# 4.&infin;.4/3.&infin; - '''4/3 4 | &infin;''' (alternate square tiling)
# 3.&infin;.3.&infin;.3.&infin; - '''3/2 | 3 &infin;''' (alternate triangular tiling)
# 6.&infin;.6/5.&infin; - '''6/5 6 | &infin;''' (alternate trihexagonal tiling with only hexagons)
# &infin;.3.&infin;.3/2 - '''3/2 3 | &infin;''' (alternate trihexagonal tiling with only triangles)
 
The remaining list includes 21 tilings, 7 with {&infin;} tiles (apeirogons). Drawn as edge-graphs there are only 14 unique tilings, and the first is identical to the ''3.4.6.4'' tiling.
[[Image:Twenty one uniform tiling vertex figures.png|320px|thumb|Vertex figures for 21 uniform tilings.]]
The 21 grouped by shared edge graphs, given by vertex figures and Wythoff symbol, are:
# '''Type 1'''
#* 3/2.12.6.12 - '''3/2 6 | 6'''
#* 4.12.4/3.12/11 - '''2 6 (3/2 3) |'''
# '''Type 2'''
#* 8/3.4.8/3.&infin; - '''4 &infin; | 4/3'''
#* 8/3.8.8/5.8/7 - '''4/3 4 (2 &infin;) |'''
#* 8.4/3.8.&infin; - '''4/3 &infin; | 4'''
# '''Type 3'''
#* 12/5.6.12/5.&infin; - '''6 &infin; | 6/5'''
#* 12/5.12.12/7.12/11 - '''6/5 6 (3 &infin;) |'''
#* 12.6/5.12.&infin; - '''6/5 &infin; | 6'''
# '''Type 4'''
#* 12/5.3.12/5.6/5 - '''3 6 | 6/5'''
#* 12/5.4.12/7.4/3 - '''2 6/5 (3/2 3) |'''
#* 4.3/2.4.6/5 - '''3/2 6 | 2'''
# '''Type 5'''
#* 8.8/3.&infin; - '''4/3 4 &infin; |'''
# '''Type 6'''
#* 12.12/5.&infin; - '''6/5 6 &infin; |'''
# '''Type 7'''
#* 8.4/3.8/5 - 2 '''4/3 4 |'''
# '''Type 8'''
#* 6.4/3.12/7 - '''2 3 6/5 |'''
# '''Type 9'''
#* 12.6/5.12/7 - '''3 6/5 6 |'''
# '''Type 10'''
#* 4.8/5.8/5 - '''2 4 | 4/3'''
# '''Type 11'''
#* 12/5.12/5.3/2 - '''2 3 | 6/5'''
# '''Type 12'''
#* 4.4.3/2.3/2.3/2 - [[non-Wythoffian]]
# '''Type 13'''
#* 4.3/2.4.3/2.3/2 - '''| 2 4/3 4/3''' (snub)
# '''Type 14'''
#* 3.4.3.4/3.3.&infin; - '''| 4/3 4 &infin;''' (snub)
 
== Self-dual tilings ==
 
Tilings can also be self-dual. The square tiling with [[Schlafli symbol]]s {4,4} is self-dual.
 
{| class=wikitable width=200
|[[Image:Self-dual square tiling.png]]<BR>The {4,4} [[square tiling]] (black) with its dual (red).
|}
 
== See also ==
{{Commonscat|Uniform tilings}}
* [[Uniform tessellation]]
* [[Wythoff symbol]]
* [[List of uniform tilings]]
* [[Uniform tilings in hyperbolic plane]]
* [[Uniform polytope]]
 
== References ==
* [[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 | author=[[Branko Grünbaum|Grünbaum, Branko]]; [[G.C. Shephard|Shephard, G. C.]] | title=Tilings and Patterns | publisher=W. H. Freeman and Company | year=1987 | id=ISBN 0-7167-1193-1}} (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&ndash;50 [[JSTOR]]: [http://links.jstor.org/sici?sici=0080-4614%2819540513%29246%3A916%3C401%3AUP%3E2.0.CO%3B2-4] (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]
* [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}}
 
[[Category:Tessellation]]

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