View source for Uniform tiling

{{Short description|Vertex-transitive tiling of the plane by regular polygons}}
In [[Geometry|geometry]], a '''uniform tiling''' is a [[Tessellation|tessellation]] of the plane by [[Regular polygon|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 polyhedron|uniform polyhedra]] which can be considered uniform tilings of the [[Sphere|sphere]].

Most uniform tilings can be made from a [[Wythoff construction]] starting with a [[Symmetry group|symmetry group]] and a singular generator point inside of the [[Fundamental domain|fundamental domain]]. A planar symmetry group has a polygonal [[Fundamental domain|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|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|vertex configuration]], the sequence of polygons around each vertex.

All uniform tilings can be constructed from various operations applied to regular tilings. These operations as named by [[Biography: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]] for more details.)

== Coxeter groups ==

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

For groups with whole number orders, including:

{| class=wikitable
|+ Euclidean plane
|-
![[Orbifold notation|Orbifold<br/>symmetry]]
!colspan=3|[[Coxeter group]]
!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
| (&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 ([[Frieze group|frieze]])
|- 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<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 | Paracompact
|- 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 ==
{{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 tilings, 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|apeirogonal prism]] and [[Apeirogonal antiprism|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|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]]
!
!{''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.2p.2p
!(p.q)<sup>2</sup>
!p.&nbsp;2q.2q
!q<sup>p</sup>
!p.&nbsp;4.q.4
!4.2p.2q
!3.3.p.&nbsp;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/>3.4.6.4
|[[Image:Uniform tiling 63-t012.svg|64px]]<br/>4.6.12
|[[Image:Uniform tiling 63-snub.svg|65px]]<br/>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 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.2p.q.2p
!(p.r)<sup>q</sup>
!q.2r.p.&nbsp;2r
!(q.r)<sup>p</sup>
!q.2r.p.&nbsp;2r
!r.2q.p.&nbsp;2q
!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 tiling 333-t01.png|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.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.svg|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|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|square tiling]].

== Uniform tilings of the hyperbolic plane ==
{{further|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 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.&nbsp;2q.2q)
!q<sup>p</sup>
!(p.&nbsp;4.q.4)
!(4.2p.2q)
!(3.3.p.&nbsp;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/>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/>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 &#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 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 figure]]
!
!(p.r)<sup>q</sup>
!(r.2p.q.2p)
!(p.q)<sup>r</sup>
!(q.2r.p.&nbsp;2r)
!(q.r)<sup>p</sup>
!(r.2q.p.&nbsp;2q)
!(2p.2q.2r)
!(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|vertex figure]]s for the six tilings with convex [[Regular polygon|regular polygon]]s and [[Apeirogon|apeirogon]] faces.(The [[Wythoff symbol|Wythoff symbol]] is given in red.)]]
[[Image:Twenty one uniform tiling vertex figures.png|320px|thumb|Vertex figures for 21 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]] tiles can be included.
# [[Apeirogon]]s, {&infin;}, 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 &infin;) (3/2 3 &infin;) (6 6/5 &infin;) (3 3/2 &infin;)

[[Biography:Branko Grünbaum|Branko Grünbaum]] and G. C. Shephard, 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 they call ''hollow tilings'' which included the first two expansions above, star polygon faces and vertex figures.<ref>Tiles and Patterns, Table 12.3.1 p.640</ref>

H.S.M. Coxeter, M. S. Longuet-Higgins, and [[Biography:J. C. P. Miller|J. C. P. Miller]], in the 1954 paper 'Uniform polyhedra', in ''Table 8: Uniform Tessellations'', use the first three expansions and enumerates 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'' listed 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]]|| &infin;.&infin; || ||p1m1 ||(Two half-plane tiles, [[Order-2 apeirogonal tiling|order-2 apeirogonal tiling]])
|-
|I2||[[File:Infinite prism alternating.svg|100px]]|| 4.4.&infin;  || '''&infin; 2 &#124; 2''' ||p1m1 ||[[Apeirogonal prism]]
|-
|I3||[[File:Infinite antiprism.svg|100px]]||3.3.3.&infin; || '''&#124; 2 2 &infin;''' ||p11g ||[[Apeirogonal antiprism]]
|}

For clarity, the tilings are not coloured from here onward (due to the overlaps). A set of polygons round 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 &#124; 2 4 || p4m
|- align=center
!I4||2.14
||[[File:Tiling 4,i,-4,i.png|150px]]||4.&infin;.4/3.&infin;<br/>4.&infin;.-4.&infin; || 4/3 4 &#124; &infin; ||p4m
|- align=center
!Convex||1.24
|| ||[[File:Tiling 6,6,6.png|150px]]||6.6.6||3 &#124; 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 &#124; 2 3 ||p6m
|- align=center
!I5||2.26
||[[File:Tiling 3,i,3,i,3,i.png|150px]]||(3.&infin;.3.&infin;.3.&infin;)/2 || 3/2 &#124; 3 &infin;||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 &#124; 3 6|| p6m
|- align=center
!I6||2.25
||[[File:Tiling 6,i,-6,i.png|150px]]|| 6.&infin;.6/5.&infin;<br/>6.&infin;.-6.&infin; || 6/5 6 &#124; &infin;||p6m
|- align=center
!I7||2.24
|| [[File:Tiling 3,i,-3,i.png|150px]]||&infin;.3.&infin;.3/2<br/>3.&infin;.-3.&infin; || 3/2 3 &#124; &infin;||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 &#124; 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 &#124; 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) &#124;||p6m
|- align=center
!Convex||1.5
| ||[[File:Tiling 4,8,8.png|150px]] || 4.8.8 || 2 4 &#124; 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.&infin;.8/3 || 4 &infin; &#124; 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 &infin;/2) &#124;||p4m
|- align=center
!2.6
|[[File:Tiling -4,8,i,8.png|150px]] || 8.4/3.8.&infin;<br/>-4.8.&infin;.8 || 4/3 &infin; &#124; 4||p4m
|- align=center
!Convex||1.20
| ||[[File:Tiling 3,12,12.png|150px]] || 3.12.12 || 2 3 &#124; 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.&infin;.12/5 ||6 &infin; &#124; 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 &infin;/2) &#124;||p6m
|- align=center
!2.16
|[[File:Tiling -6,12,i,12.png|150px]] || 12.6/5.12.&infin;<br/>-6.12.&infin;.12 || 6/5 &infin; &#124; 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 &#124; 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) &#124;||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 &#124; 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.&infin; || 4/3 4 &infin; &#124;||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.&infin; || 6/5 6 &infin; &#124;||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 &#124;||p4m
|- align=center
!Convex||1.11
| ||[[File:Tiling 4,6,12.png|150px]] || 4.6.12 || 2 3 6 &#124; || 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 &#124;||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 &#124;||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 &#124; 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 &#124; 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 || &#124; 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 || &#124; 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.&infin;<br/>3.4.3.-4.3.&infin; || &#124; 4/3 4 &infin; ||p4
|- align=center
!Convex||1.10
| ||[[File:Tiling 3,3,3,3,6.png|150px]] || 3.3.3.3.6 || &#124; 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.&infin;.3/2.&infin;.3/2.4/3.4/3<br>3.4.4.3.&infin;.3.&infin; || non-Wythoffian || cmm
|- align=center
!2.2
|| [[File:Tiling 3,-4,-4,3,i,3,i.png|150px]] || 3/2.&infin;.3/2.&infin;.3/2.4.4<br>3.-4.-4.3.&infin;.3.&infin; || non-Wythoffian || cmm
|- align=center
! ||2.3
| ||[[File:Tiling 3,4,4,3,-4,-4,3,i.png|150px]] || 3/2.&infin;.3/2.4.4.3/2.4/3.4/3<br>3.4.4.3.-4.-4.3.&infin; || non-Wythoffian || p3
|- align=center
!rowspan=2| ||2.8
|rowspan=2| || [[File:Tiling 4,8,83,-4,i.png|150px]] || 4.&infin;.4/3.8/3.8<br>4.8.8/3.-4.&infin; || non-Wythoffian || p4m
|- align=center
!2.9
|| [[File:Tiling -4,8,83,4,i.png|150px]] || 4.&infin;.4.8.8/3<br>-4.8.8/3.4.&infin; || non-Wythoffian || p4m
|- align=center
!rowspan=4| ||2.10
|rowspan=4| || [[File:Tiling 4,8,-4,8,-4,i.png|150px]] || 4.&infin;.4/3.8.4/3.8<br>4.8.-4.8.-4.&infin; || non-Wythoffian || p4m
|- align=center
!2.11
|| [[File:Tiling 4,8,-4,8,-4,i.png|150px]] || 4.&infin;.4/3.8.4/3.8<br>4.8.-4.8.-4.&infin; || non-Wythoffian || p4g
|- align=center
!2.12
|| [[File:Tiling 4,83,4,83,-4,i.png|150px]] || 4.&infin;.4/3.8/3.4.8/3<br>4.8/3.4.8/3.-4.&infin; || non-Wythoffian || p4m
|- align=center
!2.13
|| [[File:Tiling 4,83,4,83,-4,i.png|150px]] || 4.&infin;.4/3.8/3.4.8/3<br>4.8/3.4.8/3.-4.&infin; || 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.&infin;.3/2.4/3.4/3.3/2.4/3.4/3<br>3.4.4.3.4.4.3.&infin; || non-Wythoffian || p6m
|- align=center
!2.19
|| [[File:Tiling 3,-4,-4,3,-4,-4,3,i.png|150px]] || 3/2.&infin;.3/2.4.4.3/2.4.4<br>3.-4.-4.3.-4.-4.3.&infin; || non-Wythoffian || p6m
|- align=center
!2.20
|| [[File:Tiling 3,12,-6,12,3,i,3,i.png|150px]] || 3/2.&infin;.3/2.&infin;.3/2.12/11.6.12/11<br>3.12.-6.12.3.&infin;.3.&infin; || non-Wythoffian || p6m
|- align=center
!2.21
|| [[File:Tiling 3,-12,6,-12,3,i,3,i.png|150px]] || 3/2.&infin;.3/2.&infin;.3/2.12.6/5.12<br>3.-12.6.-12.3.&infin;.3.&infin; || non-Wythoffian || p6m
|- align=center
!2.22
|| [[File:Tiling 3,125,6,125,3,i,3,i.png|150px]] || 3/2.&infin;.3/2.&infin;.3/2.12/7.6/5.12/7<br>3.12/5.6.12/5.3.&infin;.3.&infin; || non-Wythoffian || p6m
|- align=center
!2.23
|| [[File:Tiling 3,-125,-6,-125,3,i,3,i.png|150px]] || 3/2.&infin;.3/2.&infin;.3/2.12/5.6.12/5<br>3.-12/5.-6.-12/5.3.&infin;.3.&infin; || non-Wythoffian || p6m
|}

There are two uniform tilings for the vertex figure 4.8.-4.8.-4.&infin; (Grünbaum et al. 2.10 and 2.11) and also two uniform tilings for the vertex figure 4.8/3.4.8/3.-4.&infin; (Grünbaum et al. 2.12 and 2.13), with different symmetries. There is also a third tiling for each vertex figure that is only pseudo-uniform (vertices come in two symmetry orbits). They use different sets of square faces. Hence, for star Euclidean tilings, the vertex figure 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. The notation {&infin;<sup>&alpha;</sup>} denotes a zigzag with angle 0 &lt; &alpha; &lt; &pi;. The apeirogon can be considered as the special case &alpha; = &pi;. The symmetries are given for the generic case: there are sometimes special values of &alpha; 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 &alpha; 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|square tiling]] (black) with its dual (red).]]

Tilings 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 star polygons==
[[File:Hexagon hexagram tiling2.png|thumb|This example, 4.8{{supsub|*|&pi;/8}}.4{{supsub|**|&pi;/4}}.8{{supsub|*|&pi;/4}} is considered not edge-to-edge due to the large square, although it can be interpreted as star polygon with pairs of colinear edges.]]
Seeing a [[Star polygon#Simple isotoxal star polygons|star polygon]] as a nonconvex polygon with twice as many sides allows star polygons, and counting these as regular polygons allows them to be used in a ''uniform tiling''. These polygons are labeled as {N<sub>&alpha;</sub>} for a [[Isotoxal figure|isotoxal]] nonconvex 2N-gon with external dihedral angle &alpha;. Its external vertices are labeled as N{{supsub|*|&alpha;}}, and internal N{{supsub|**|&alpha;}}. This expansion to the definition requires corners with only 2 polygons to not be considered vertices. The tiling is defined by its [[Vertex configuration|vertex configuration]] as a cyclic sequence of convex and nonconvex polygons around every vertex. There are 4 such uniform tilings with adjustable angles &alpha;, and 18 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 are topologically related to the ordinary uniform tilings with convex regular polygons, with 2-valence vertices ignored, and square faces as digons, reduced to a single edge.

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

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

=== Uniform tilings using alternating polygons===

Star polygons of the form {p<sub>&alpha;</sub>} can also represent convex 2''p''-gons alternating two angles, the simplest being a rhombus {2<sub>&alpha;</sub>}. Allowing these as regular polygons, creates more uniform tilings, with some example below.
{| class=wikitable
|+ Examples
|- align=center valign=bottom
|[[File:Hexatile-rhombic-snub-hex.svg|160px]]<BR>3.2*.6.2**<BR>Topological [[Rhombitrihexagonal tiling|3.4.6.4]]
|[[File:Octatile-rhombic0.svg|160px]]<BR>4.4.4.4<BR>Topological [[Square tiling|4.4.4.4]]
|[[File:Octatile-rhombic1.svg|320px]]<BR>(2{{supsub|*|&pi;/6}}.2{{supsub|**|&pi;/3}})<sup>2</sup><BR>Topological [[Square tiling|4.4.4.4]]
|[[File:Octatile-rhombic2.svg|160px]]<BR>2{{supsub|*|&pi;/6}}.2{{supsub|*|&pi;/6}}.2{{supsub|**|&pi;/3}}.2{{supsub|**|&pi;/3}}<BR>Topological [[Square tiling|4.4.4.4]]
|[[File:Octatile-rhombic3.svg|160px]]<BR>4.2{{supsub|*|&pi;/6}}.4.2{{supsub|**|&pi;/3}}<BR>Topological [[Square tiling|4.4.4.4]]
|}

== See also ==
* [[Wythoff symbol]]
* List of uniform tilings
* [[Uniform tilings in hyperbolic plane]]
* Uniform polytope

== References ==
{{Reflist}}
* [[Biography:Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
** [[Biography:Norman Johnson (mathematician)|N.W. Johnson]]: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966 
* {{cite book|last1=Grünbaum|first1=Branko|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, [[Biography:J. C. P. Miller|J. C. P. Miller]], ''Uniform polyhedra'', '''Phil. Trans.''' 1954, 246 A, 401&ndash;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| ]]

{{Sourceattribution|Uniform tiling}}