View source for Euclidean tilings by convex regular polygons

{{Short description|Subdivision of the plane into polygons that are all regular}}
{| class="wikitable floatright" style="width:300px;"
|+ Example periodic tilings
|- valign=top
|[[File:1-uniform n1.svg|150px]]<br/>A [[#Regular_tilings|regular tiling]] has one type of regular face.
|[[File:1-uniform n2.svg|150px]]<br/>A [[#Archimedean.2C_uniform_or_semiregular_tilings|semiregular or uniform tiling]] has one [[vertex configuration|type of vertex]], but two or more types of faces.
|- valign=top
|[[File:2-uniform n1.svg|150px]]<br/>A [[#k-uniform_tilings|''k''-uniform tiling]] has ''k'' types of vertices, and two or more types of regular faces.
|[[File:Distorted truncated square tiling.svg|150px]]<br/>A [[#Tilings_that_are_not_edge-to-edge|non-edge-to-edge tiling]] can have different-sized regular faces.
|}
Euclidean [[Plane (mathematics)|plane]] '''[[Tessellation|tilings]] by convex [[regular polygon]]s''' have been widely used since antiquity.  The first systematic mathematical treatment was that of [[Johannes Kepler|Kepler]] in his {{lang|la|[[Harmonice Mundi]]}} ([[Latin language|Latin]]: ''The Harmony of the World'', 1619).

== Notation of Euclidean tilings ==

Euclidean tilings are usually named after Cundy & Rollett’s notation.<ref>{{cite book |last1=Cundy |first1=H.M.|last2=Rollett |first2=A.P. |title=Mathematical Models; |date=1981 |publisher=Tarquin Publications |location=Stradbroke (UK)}}</ref> This notation represents (i) the number of vertices, (ii) the number of polygons around each vertex (arranged clockwise) and (iii) the number of sides to each of those polygons. For example: 3<sup>6</sup>; 3<sup>6</sup>; 3<sup>4</sup>.6, tells us there are 3 vertices with 2 different vertex types, so this tiling would be classed as a ‘3-uniform (2-vertex types)’ tiling. Broken down, 3<sup>6</sup>; 3<sup>6</sup> (both of different transitivity class), or (3<sup>6</sup>)<sup>2</sup>, tells us that there are 2 vertices (denoted by the superscript 2), each with 6 equilateral 3-sided polygons (triangles). With a final vertex 3<sup>4</sup>.6, 4 more contiguous equilateral triangles and a single regular hexagon.

However, this notation has two main problems related to ambiguous conformation and uniqueness <ref name="Gomez-Jauregui 2012">{{cite journal |last1=Gomez-Jauregui |first1=Valentin al.|last2=Otero |first2=Cesar |display-authors=etal |title=Generation and Nomenclature of Tessellations and Double-Layer Grids |journal=Journal of Structural Engineering |date=2012 |volume=138 |issue=7 |pages=843–852 |doi=10.1061/(ASCE)ST.1943-541X.0000532 |hdl=10902/5869 |url=https://ascelibrary.org/doi/10.1061/%28ASCE%29ST.1943-541X.0000532|hdl-access=free }}</ref> First, when it comes to k-uniform tilings, the notation does not explain the relationships between the vertices. This makes it impossible to generate a covered plane given the notation alone. And second, some tessellations have the same nomenclature, they are very similar but it can be noticed that the relative positions of the hexagons are different. Therefore, the second problem is that this nomenclature is not unique for each tessellation.

In order to solve those problems, GomJau-Hogg’s notation <ref>{{cite journal |last1=Gomez-Jauregui |first1=Valentin |last2=Hogg |first2=Harrison|display-authors=etal |title=GomJau-Hogg's Notation for Automatic Generation of k-Uniform Tessellations with ANTWERP v3.0 |journal=Symmetry |date=2021 |volume=13 |issue=12 |page=2376 |doi=10.3390/sym13122376 |bibcode=2021Symm...13.2376G |doi-access=free |hdl=10902/23907 |hdl-access=free }}</ref> is a slightly modified version of the research and notation presented in 2012,<ref name="Gomez-Jauregui 2012" /> about the generation and nomenclature of tessellations and double-layer grids. Antwerp v3.0,<ref>{{cite web |last1=Hogg |first1=Harrison |last2=Gomez-Jauregui |first2=Valentin |title=Antwerp 3.0 |url=https://antwerp.hogg.io/<}}</ref> a free online application, allows for the infinite generation of regular polygon tilings through a set of shape placement stages and iterative rotation and reflection operations, obtained directly from the GomJau-Hogg’s notation.

== Regular tilings ==

Following [[Branko Grünbaum|Grünbaum]] and Shephard (section 1.3), a tiling is said to be ''regular'' if the [[symmetry group]] of the tiling [[Group action (mathematics)#Types of actions|acts transitively]] on the ''flags'' of the tiling, where a flag is a triple consisting of a mutually incident [[Vertex (geometry)#Of a plane tiling|vertex]], edge and tile of the tiling. This means that, for every pair of flags, there is a symmetry operation mapping the first flag to the second. This is equivalent to the tiling being an [[edge-to-edge tiling]] by [[congruence (geometry)|congruent]] regular polygons.  There must be six [[equilateral triangle]]s, four [[square (geometry)|square]]s or three regular [[hexagon]]s at a vertex, yielding the [[List of regular polytopes and compounds#Euclidean tilings|three regular tessellations]].

{| class="wikitable"
|+ Regular tilings (3)
!colspan=2|p6m, *632
!p4m, *442
|- align=center
|[[File:1-uniform n11.svg|200px]]
|[[File:1-uniform n1.svg|200px]]
|[[File:1-uniform n5.svg|200px]]
|- align=center
|[[File:Vertex type 3-3-3-3-3-3.svg|100px]]<br/>C&R: [[Triangular tiling|3<sup>6</sup>]]<br/>GJ-H: [https://antwerp.hogg.io/?configuration=3%2Fm30%2Fr%28h2%29 3/m30/r(h2)]<br/>(''t'' = 1, ''e'' = 1)
|[[File:Vertex type 6-6-6.svg|100px]]<br/>C&R: [[Hexagonal tiling|6<sup>3</sup>]]<br/>GJ-H: [https://antwerp.hogg.io/?configuration=6%2Fm30%2Fr%28h1%29 6/m30/r(h1)]<br/>(''t'' = 1, ''e'' = 1)
|[[File:Vertex type 4-4-4-4.svg|100px]]<br/>C&R: [[Square tiling|4<sup>4</sup>]]<br/>GJ-H: [https://antwerp.hogg.io/?configuration=4%2Fm45%2Fr%28h1%29 4/m45/r(h1)]<br/>(''t'' = 1, ''e'' = 1)
|}

''C&R: Cundy & Rollet's notation'' <br/> 
''GJ-H: Notation of GomJau-Hogg''

== Archimedean, uniform or semiregular tilings ==<!-- This section is linked from [[Archimedean tiling]] -->
{{Further|List of Euclidean uniform tilings}}
[[Vertex-transitive|Vertex-transitivity]] means that for every pair of vertices there is a [[symmetry operation]] mapping the first vertex to the second.<ref name="Critchlow 1969">{{cite book |last1=Critchlow |first1=K. |title=Order in Space: A Design Source Book |date=1969 |publisher=Thames and Hudson |location=London |pages=60–61}}</ref>

If the requirement of flag-transitivity is relaxed to one of vertex-transitivity, while the condition that the tiling is edge-to-edge is kept, there are eight additional tilings possible, known as ''Archimedean'', ''[[Uniform tiling|uniform]]'' or ''semiregular'' tilings.  Note that there are two [[mirror image]] (enantiomorphic or [[Chirality (mathematics)|chiral]]) forms of 3<sup>4</sup>.6 (snub hexagonal) tiling, only one of which is shown in the following table.  All other regular and semiregular tilings are achiral.

{| class="wikitable"
|+ Uniform tilings (8)
|- align=center
!colspan=6|p6m, *632
|- align=center valign=top
|[[File:1-uniform n4.svg|160px]]<br/><br/>[[File:Vertex type 3-12-12.svg|150x150px|alt=]]<br/>C&R: [[Truncated hexagonal tiling|3.12<sup>2</sup>]]<br/>GJ-H: [https://antwerp.hogg.io/?configuration=12-3%2Fm30%2Fr%28h3%29 12-3/m30/r(h3)]<br/>(''t'' = 2, ''e'' = 2)<br/>''t''{6,3}
|[[File:1-uniform n6.svg|160px]]<br/><br/>[[File:Vertex type 3-4-6-4.svg|100x100px|alt=]]<br/>C&R: [[Rhombitrihexagonal tiling|3.4.6.4]]<br/>GJ-H: [https://antwerp.hogg.io/?configuration=6-4-3%2Fm30%2Fr%28c2%29 6-4-3/m30/r(c2)]<br/>(''t'' = 3, ''e'' = 2)<br/>''rr''{3,6}
|[[File:1-uniform n3.svg|160px]]<br/><br/>[[File:Vertex type 4-6-12.svg|150x150px|alt=]]<br/>C&R: [[Truncated trihexagonal tiling|4.6.12]]<br/>GJ-H: [https://antwerp.hogg.io/?configuration=12-6%2C4%2Fm30%2Fr%28c2%29 12-6,4/m30/r(c2)]<br/>(''t'' = 3, ''e'' = 3)<br/>''tr''{3,6}
|[[File:1-uniform n7.svg|160px]]<br/><br/>[[File:Vertex type 3-6-3-6.svg|150x150px|alt=]]<br/>C&R: [[Trihexagonal tiling|(3.6)<sup>2</sup>]]<br/>GJ-H: [https://antwerp.hogg.io/?configuration=6-3-6%2Fm30%2Fr%28v4%29 6-3-6/m30/r(v4)]<br/>(''t'' = 2, ''e'' = 1)<br/>''r''{6,3}
|- align=center valign=top
|[[File:1-uniform n2.svg|160px]]<br/><br/>[[File:Vertex type 4-8-8.svg|150x150px|alt=]]<br/>C&R: [[Truncated square tiling|4.8<sup>2</sup>]]<br/>GJ-H: [https://antwerp.hogg.io/?configuration=8-4%2Fm90%2Fr%28h4%29 8-4/m90/r(h4)]<br/>(''t'' = 2, ''e'' = 2)<br/>''t''{4,4}
|[[File:1-uniform n9.svg|160px]]<br/><br/>[[File:Vertex type 3-3-4-3-4.svg|150x150px|alt=]]<br/>C&R: [[Snub square tiling|3<sup>2</sup>.4.3.4]]<br/>GJ-H: [https://antwerp.hogg.io/?configuration=4-3-3%2C4%2Fr90%2Fr%28h2%29 4-3-3,4/r90/r(h2)]<br/>(''t'' = 2, ''e'' = 2)<br/>''s''{4,4}
|[[File:1-uniform n8.svg|160px]]<br/><br/>[[File:Vertex type 3-3-3-4-4.svg|150x150px|alt=]]<br/>C&R: [[Elongated triangular tiling|3<sup>3</sup>.4<sup>2</sup>]]<br/>GJ-H: [https://antwerp.hogg.io/?configuration=4-3%2Fm90%2Fr%28h2%29 4-3/m90/r(h2)]<br/>(''t'' = 2, ''e'' = 3)<br/>{3,6}:''e''
|[[File:1-uniform n10.svg|160px]]<br/><br/>[[File:Vertex type 3-3-3-3-6.svg|150x150px|alt=]]<br/>C&R: [[Snub hexagonal tiling|3<sup>4</sup>.6]]<br/>GJ-H: [https://antwerp.hogg.io/?configuration=6-3-3%2Fr60%2Fr%28h5%29 6-3-3/r60/r(h5)]<br/>(''t'' = 3, ''e'' = 3)<br/>''sr''{3,6}
|}
''C&R: Cundy & Rollet's notation''<br/>
''GJ-H: Notation of GomJau-Hogg''<br/>
<br/>
Grünbaum and Shephard distinguish the description of these tilings as ''Archimedean'' as referring only to the local property of the arrangement of tiles around each vertex being the same, and that as ''uniform'' as referring to the global property of vertex-transitivity.  Though these yield the same set of tilings in the plane, in other spaces there are Archimedean tilings which are not uniform.

==Plane-vertex tilings==
There are 17 combinations of regular convex polygons that form 21 types of [[Vertex (geometry)#Of a plane tiling|plane-vertex tilings]].<ref>{{citation|title=The Elements of Plane Practical Geometry, Etc|first=Elmslie William|last=Dallas|publisher=John W. Parker & Son|year=1855|page=134|url=https://books.google.com/books?id=y4BaAAAAcAAJ&pg=PA134}}</ref><ref>[[Tilings and patterns]], Figure 2.1.1, p.60</ref> Polygons in these meet at a point with no gap or overlap. Listing by their [[vertex figure]]s, one has 6 polygons, three have 5 polygons, seven have 4 polygons, and ten have 3 polygons.<ref>[[Tilings and patterns]], p.58-69</ref>

Three of them can make [[#Regular_tilings|regular tilings]] (6<sup>3</sup>, 4<sup>4</sup>, 3<sup>6</sup>), and eight more can make [[#Archimedean,_uniform_or_semiregular_tilings|semiregular or archimedean tilings]], (3.12.12, 4.6.12, 4.8.8, (3.6)<sup>2</sup>, 3.4.6.4, 3.3.4.3.4, 3.3.3.4.4, 3.3.3.3.6). Four of them can exist in higher [[#k-uniform_tilings|''k''-uniform tilings]] (3.3.4.12, 3.4.3.12, 3.3.6.6, 3.4.4.6), while six can not be used to completely tile the plane by regular polygons with no gaps or overlaps - they only tessellate space entirely when irregular polygons are included (3.7.42, 3.8.24, 3.9.18, 3.10.15, 4.5.20, 5.5.10).<ref>{{Cite web|url=https://blogs.ams.org/visualinsight/2015/02/01/pentagon-decagon-packing/|title=Pentagon-Decagon Packing|website=American Mathematical Society|publisher=AMS|access-date=2022-03-07}}</ref> 
 
{| class=wikitable
|+ Plane-vertex tilings 
|- align=center 
!6
|valign=bottom|[[File:Regular polygons meeting at vertex 6 3 3 3 3 3 3.svg|40px]]<BR>3<sup>6</sup>
|- align=center
!5
|valign=bottom|[[File:Regular polygons meeting at vertex 5 3 3 4 3 4.svg|40px]]<BR>3.3.4.3.4
|valign=bottom|[[File:Regular polygons meeting at vertex 5 3 3 3 4 4.svg|40px]]<BR>3.3.3.4.4
|valign=bottom|[[File:Regular polygons meeting at vertex 5 3 3 3 3 6.svg|40px]]<BR>3.3.3.3.6
|- align=center
!4
|valign=bottom|[[File:Regular polygons meeting at vertex 4 3 3 4 12.svg|40px]]<BR>3.3.4.12
|valign=bottom|[[File:Regular polygons meeting at vertex 4 3 4 3 12.svg|40px]]<BR>3.4.3.12
|valign=bottom|[[File:Regular polygons meeting at vertex 4 3 3 6 6.svg|40px]]<BR>3.3.6.6
|valign=bottom|[[File:Regular polygons meeting at vertex 4 3 6 3 6.svg|40px]]<BR>(3.6)<sup>2</sup>
|valign=bottom|[[File:Regular polygons meeting at vertex 4 3 4 4 6.svg|40px]]<BR>3.4.4.6
|valign=bottom|[[File:Regular polygons meeting at vertex 4 3 4 6 4.svg|40px]]<BR>3.4.6.4
|valign=bottom|[[File:Regular polygons meeting at vertex 4 4 4 4 4.svg|40px]]<BR>4<sup>4</sup>
|- align=center
!3
|valign=bottom|[[File:Regular polygons meeting at vertex 3 3 7 42.svg|50px]]<BR>3.7.42
|valign=bottom|[[File:Regular polygons meeting at vertex 3 3 8 24.svg|40px]]<BR>3.8.24
|valign=bottom|[[File:Regular polygons meeting at vertex 3 3 9 18.svg|40px]]<BR>3.9.18
|valign=bottom|[[File:Regular polygons meeting at vertex 3 3 10 15.svg|40px]]<BR>3.10.15
|valign=bottom|[[File:Regular polygons meeting at vertex 3 3 12 12.svg|40px]]<BR>3.12.12
|valign=bottom|[[File:Regular polygons meeting at vertex 3 4 5 20.svg|40px]]<BR>4.5.20
|valign=bottom|[[File:Regular polygons meeting at vertex 3 4 6 12.svg|40px]]<BR>4.6.12
|valign=bottom|[[File:Regular polygons meeting at vertex 3 4 8 8.svg|50px]]<BR>4.8.8
|valign=bottom|[[File:Regular polygons meeting at vertex 3 5 5 10.svg|40px]]<BR>5.5.10
|valign=bottom|[[File:Regular polygons meeting at vertex 3 6 6 6.svg|50px]]<BR>6<sup>3</sup>
|}

== ''k''-uniform tilings ==

Such periodic tilings may be classified by the number of [[Group action (mathematics)#Orbits and stabilizers|orbits]] of vertices, edges and tiles.  If there are {{mvar|k}} orbits of vertices, a tiling is known as {{mvar|k}}-uniform or {{mvar|k}}-isogonal; if there are {{mvar|t}} orbits of tiles, as {{mvar|t}}-isohedral; if there are {{mvar|e}} orbits of edges, as {{mvar|e}}-isotoxal.

''k''-uniform tilings with the same vertex figures can be further identified by their [[wallpaper group]] symmetry.

1-uniform tilings include 3 regular tilings, and 8 semiregular ones, with 2 or more types of regular polygon faces. There are 20 2-uniform tilings, 61 3-uniform tilings, 151 4-uniform tilings, 332 5-uniform tilings and 673 6-uniform tilings. Each can be grouped by the number ''m'' of distinct vertex figures, which are also called ''m''-Archimedean tilings.<ref>[http://www2.math.uu.se/research/pub/Lenngren1.pdf ''k''-uniform tilings by regular polygons] {{webarchive|url=https://web.archive.org/web/20150630004622/http://www2.math.uu.se/research/pub/Lenngren1.pdf |date=2015-06-30 }} Nils Lenngren, 2009</ref>

Finally, if the number of types of vertices is the same as the uniformity (''m'' = ''k'' below), then the tiling is said to be ''[[oeis:A068600|Krotenheerdt]]''. In general, the uniformity is greater than or equal to the number of types of vertices (''m'' ≥ ''k''), as different types of vertices necessarily have different orbits, but not vice versa. Setting ''m'' = ''n'' = ''k'', there are 11 such tilings for ''n'' = 1; 20 such tilings for ''n'' = 2; 39 such tilings for ''n'' = 3; 33 such tilings for ''n'' = 4; 15 such tilings for ''n'' = 5; 10 such tilings for ''n'' = 6; and 7 such tilings for ''n'' = 7.

Below is an example of a 3-unifom tiling:

{| class=wikitable align=left width=300
|+ Colored 3-uniform tiling #57 of 61 
|- align=center valign=top
|[[File:3-uniform 57.svg|150px]]<br/>by sides, yellow triangles, red squares (by polygons)
|[[File:3-uniform n57.svg|150px]]<br/>by 4-isohedral positions, 3 shaded colors of triangles (by orbits)
|}

{{Clr}}

{| class="wikitable" align=left style="margin: auto; text-align:center;"
|+ ''k''-uniform, ''m''-Archimedean tiling counts <ref>{{Cite web|url=http://probabilitysports.com/tilings.html|title=n-Uniform Tilings|website=probabilitysports.com|access-date=2019-06-21}}</ref><ref>{{Cite OEIS |A068599 |Number of n-uniform tilings. |access-date=2023-01-07 }}</ref><ref>{{Cite web|url=https://zenorogue.github.io/tes-catalog/?c=k-uniform%2F|title=Enumeration of n-uniform k-Archimedean tilings|website=zenorogue.github.io/tes-catalog/?c=|access-date=2024-08-24}}</ref>
!colspan=2 rowspan=2 | !! colspan="16" |''m''-Archimedean
|-
!|1||2||3||4||5||6||7||8
!9
!10
!11
!12
!13
!14
!≥ 15
!Total
|-
! rowspan="16" | ''k''-uniform
! scope="row"| 1
||'''11'''||0||0||0||0||0||0||0
|0
|0
|0
|0
|0
|0
|0
|11
|-
! scope="row"| [[#2-uniform tilings|2]]
||0||'''20'''||0||0||0||0||0||0
|0
|0
|0
|0
|0
|0
|0
|20
|-
! scope="row"| [[#3-uniform tilings|3]]
| 0||'''22'''||'''39'''||0||0||0||0||0
|0
|0
|0
|0
|0
|0
|0
|61
|-
! scope="row"| [[#4-uniform tilings|4]]
| 0||'''33'''||'''85'''||'''33'''||0||0||0||0
|0
|0
|0
|0
|0
|0
|0
|151
|-
! scope="row"| [[#5-uniform tilings|5]]
| 0||'''74'''||'''149'''||'''94'''||'''15'''||0||0||0
|0
|0
|0
|0
|0
|0
|0
|332
|-
! scope="row"| [[#6-uniform tilings|6]]
| 0||'''100'''||'''284'''||'''187'''||'''92'''||'''10'''||0||0
|0
|0
|0
|0
|0
|0
|0
|673
|-
! scope="row"| 7
| 0 || '''175''' || '''572''' || '''426''' || '''218''' || '''74''' ||'''7'''||0
|0
|0
|0
|0
|0
|0
|0
|1472
|-
! scope="row"| 8
||0 || '''298''' || '''1037''' || '''795''' || '''537''' || '''203''' || '''20'''||0|0
|0
|0
|0
|0
|0
|0
|0
|2850
|-
! scope="row"| 9
| 0 || '''424''' || '''1992''' || '''1608''' || '''1278'''  || '''570''' || '''80''' ||'''8'''||0|0
|0
|0
|0
|0
|0
|0
|5960
|-
! scope="row"| 10
| 0 || '''663''' || '''3772''' || '''2979''' || '''2745''' || '''1468''' || '''212''' ||'''27'''||0|0
|0
|0
|0
|0
|0
|0
|11866
|-
! scope="row"| 11
|0|| '''1086''' || '''7171''' || '''5798''' || '''5993''' || '''3711''' || '''647''' || '''52''' ||'''1'''||0
|0
|0
|0
|0
|0
|24459
|-
! scope="row"| 12
|0|| '''1607''' || '''13762''' || '''11006''' || '''12309''' || '''9230''' || '''1736''' || '''129''' || '''15''' ||0||0|0
|0
|0
|0
|0
|49794
|-
! scope="row"| 13
|0||{{Dunno}}||{{Dunno}}||{{Dunno}}||{{Dunno}}||{{Dunno}}||{{Dunno}}||{{Dunno}}||{{Dunno}}||{{Dunno}}||'''0'''||0||0||0|0
|0
|103082
|-
! scope="row"| 14
|0||{{Dunno}}||{{Dunno}}||{{Dunno}}||{{Dunno}}||{{Dunno}}||{{Dunno}}||{{Dunno}}||{{Dunno}}||{{Dunno}}||'''0'''||0||0||0||0
|{{Dunno}}
|-
! scope="row"| ≥ 15
|0||{{Dunno}}||{{Dunno}}||{{Dunno}}||{{Dunno}}||{{Dunno}}||{{Dunno}}||{{Dunno}}||{{Dunno}}||{{Dunno}}||{{Dunno}}||{{Dunno}}||{{Dunno}}||{{Dunno}}||'''0'''
|{{Dunno}}
|-
! scope="row" | Total
|'''11'''||0|'''∞'''
|0|'''∞'''
|'''∞'''
|'''∞'''
|'''∞'''
|'''∞'''
|'''∞'''
|'''∞'''
|'''∞'''
|'''∞'''
|'''∞'''
|'''∞'''
|'''∞'''
|'''0'''
|'''∞'''
|-
|}

{{Clr}}

=== 2-uniform tilings ===
There are twenty (20) '''2-uniform tilings''' of the Euclidean plane. (also called '''2-[[isogonal figure|isogonal]] tilings''' or '''[[demiregular tiling]]s''') {{r|"Critchlow 1969"|p=62-67}} <ref>[[Tilings and patterns]], Grünbaum and Shephard 1986, pp. 65-67</ref><ref>{{Cite web |url=http://www.math.nus.edu.sg/aslaksen/papers/Demiregular.pdf |title=In Search of Demiregular Tilings |access-date=2015-06-04 |archive-url=https://web.archive.org/web/20160507010400/http://www.math.nus.edu.sg/aslaksen/papers/Demiregular.pdf |archive-date=2016-05-07 |url-status=dead }}</ref> Vertex types are listed for each. If two tilings share the same two vertex types, they are given subscripts 1,2.

{| class="wikitable" style="text-align:center;"
|+ 2-uniform tilings (20)
|- valign=top
! colspan=6|p6m, *632 !! p4m, *442
|- valign=top
| [[File:2-uniform n18.svg|120px]]<br/>[[Rhombitrihexagonal_tiling#Related_tilings|[3<sup>6</sup>; 3<sup>2</sup>.4.3.4]]]<br/>3-4-3/m30/r(c3) <br/>(''t'' = 3, ''e'' = 3)
| [[File:2-uniform n9.svg|120px]]<br/>[[Truncated_hexagonal_tiling#Related_2-uniform_tilings|[3.4.6.4; 3<sup>2</sup>.4.3.4]]]<br/> 6-4-3,3/m30/r(h1)<br/>(''t'' = 4, ''e'' = 4)
| [[File:2-uniform n8.svg|120px]]<br/>[[Truncated_hexagonal_tiling#Related_2-uniform_tilings|[3.4.6.4; 3<sup>3</sup>.4<sup>2</sup>]]]<br/>6-4-3-3/m30/r(h5) <br/>(''t'' = 4, ''e'' = 4)
| [[File:2-uniform n5.svg|120px]]<br/>[3.4.6.4; 3.4<sup>2</sup>.6]<br/>6-4-3,4-6/m30/r(c4) <br/>(''t'' = 5, ''e'' = 5)
| [[File:2-uniform n1.svg|120px]]<br/>[[3-4-6-12 tiling|[4.6.12; 3.4.6.4]]]<br/>12-4,6-3/m30/r(c3) <br/>(''t'' = 4, ''e'' = 4)
| [[File:2-uniform n13.svg|120px]]<br/>[[Truncated_trihexagonal_tiling#Related_2-uniform_tilings|[3<sup>6</sup>; 3<sup>2</sup>.4.12]]]<br/>12-3,4-3/m30/r(c3) <br/>(''t'' = 4, ''e'' = 4)
| [[File:2-uniform n2.svg|120px]]<br/>[[3-4-3-12 tiling|[3.12.12; 3.4.3.12]]]<br/>12-0,3,3-0,4/m45/m(h1) <br/>(''t'' = 3, ''e'' = 3)
|- valign=top
! p6m, *632||p6, 632||p6, 632||cmm, 2*22||pmm, *2222||cmm, 2*22||pmm, *2222
|- valign=top
| [[File:2-uniform n10.svg|120px]]<br/>[[Hexagonal_tiling#Related_tilings|[3<sup>6</sup>; 3<sup>2</sup>.6<sup>2</sup>]]]<br/>3-6/m30/r(c2) <br/>(''t'' = 2, ''e'' = 3)
| [[File:2-uniform n19.svg|120px]]<br/>[[Hexagonal_tiling#Related_tilings|[3<sup>6</sup>; 3<sup>4</sup>.6]<sub>1</sub>]]<br/>6-3,3-3/m30/r(h1) <br/>(''t'' = 3, ''e'' = 3)
| [[File:2-uniform n20.svg|120px]]<br/>[3<sup>6</sup>; 3<sup>4</sup>.6]<sub>2</sub><br/>6-3-3,3-3/r60/r(h8) <br/>(''t'' = 5, ''e'' = 7)
| [[File:2-uniform n12.svg|120px]]<br/>[3<sup>2</sup>.6<sup>2</sup>; 3<sup>4</sup>.6]<br/>6-3/m90/r(h1) <br/>(''t'' = 2, ''e'' = 4)
| [[File:2-uniform n11.svg|120px]]<br/>[3.6.3.6; 3<sup>2</sup>.6<sup>2</sup>]<br/>6-3,6/m90/r(h3) <br/>(''t'' = 2, ''e'' = 3)
| [[File:2-uniform n6.svg|120px]]<br/>[[Rhombitrihexagonal_tiling#Related_polyhedra_and_tilings|[3.4<sup>2</sup>.6; 3.6.3.6]]]<sub>2</sub><br/>6-3,4-6-3,4-6,4/m90/r(c6) <br/>(''t'' = 3, ''e'' = 4)
| [[File:2-uniform n7.svg|120px]]<br/>[3.4<sup>2</sup>.6; 3.6.3.6]<sub>1</sub><br/>6-3,4/m90/r(h4) <br/>(''t'' = 4, ''e'' = 4)
|- valign=top
!p4g, 4*2||pgg, 22×||cmm, 2*22||cmm, 2*22||pmm, *2222||cmm, 2*22
|- valign=top
| [[File:2-uniform n16.svg|120px]]<br/>[[33344-33434 tiling|[3<sup>3</sup>.4<sup>2</sup>; 3<sup>2</sup>.4.3.4]<sub>1</sub>]]<br/>4-3,3-4,3/r90/m(h3) <br/>(''t'' = 4, ''e'' = 5)
| [[File:2-uniform n17-1.svg|120px]]<br/>[[33344-33434 tiling|[3<sup>3</sup>.4<sup>2</sup>; 3<sup>2</sup>.4.3.4]<sub>2</sub>]]<br/>4-3,3,3-4,3/r(c2)/r(h13)/r(h45)<br/>(''t'' = 3, ''e'' = 6)
| [[File:2-uniform n4.svg|120px]]<br/>[4<sup>4</sup>; 3<sup>3</sup>.4<sup>2</sup>]<sub>1</sub><br/>4-3/m(h4)/m(h3)/r(h2)<br/>(''t'' = 2, ''e'' = 4)
| [[File:2-uniform n3.svg|120px]]<br/>[4<sup>4</sup>; 3<sup>3</sup>.4<sup>2</sup>]<sub>2</sub><br/>4-4-3-3/m90/r(h3)<br/>(''t'' = 3, ''e'' = 5)
| [[File:2-uniform n14.svg|120px]]<br/>[3<sup>6</sup>; 3<sup>3</sup>.4<sup>2</sup>]<sub>1</sub><br/>4-3,4-3,3/m90/r(h3)<br/>(''t'' = 3, ''e'' = 4)
| [[File:2-uniform n15.svg|120px]]<br/>[3<sup>6</sup>; 3<sup>3</sup>.4<sup>2</sup>]<sub>2</sub><br/>4-3-3-3/m90/r(h7)/r(h5)<br/>(''t'' = 4, ''e'' = 5)
|}

== Higher ''k''-uniform tilings==
''k''-uniform tilings have been enumerated up to 6. There are 673 6-uniform tilings of the Euclidean plane. Brian Galebach's search reproduced Krotenheerdt's list of 10 6-uniform tilings with 6 distinct vertex types, as well as finding 92 of them with 5 vertex types, 187 of them with 4 vertex types, 284 of them with 3 vertex types, and 100 with 2 vertex types.

=== Fractalizing ''k''-uniform tilings ===
There are many ways of generating new ''k''-uniform tilings from old ''k''-uniform tilings. For example, notice that the 2-uniform [[3-4-3-12 tiling|[3.12.12; 3.4.3.12]]] tiling has a square lattice, the 4(3-1)-uniform [343.12; (3.12<sup>2</sup>)3] tiling has a snub square lattice, and the 5(3-1-1)-uniform [334.12; 343.12; (3.12.12)3] tiling has an elongated triangular lattice. These higher-order uniform tilings use the same lattice but possess greater complexity. The fractalizing basis for theses tilings is as follows:<ref>{{Cite journal|last=Chavey|first=Darrah|s2cid=33928615|date=2014|title=TILINGS BY REGULAR POLYGONS III: DODECAGON-DENSE TILINGS|journal=Symmetry-Culture and Science|volume=25|issue=3|pages=193–210}}</ref>
{| class="wikitable"
!
!Triangle
!Square
!Hexagon
!Dissected<BR>Dodecagon
|- valign="center"
! valign="center" |Shape
|[[File:The Triangle.svg|alt=|center|frameless|80px]]
|[[File:A Square Tile.svg|alt=|center|frameless|80px]]
|[[File:A Hexagon Tile.svg|center|frameless|120px]]
|[[File:A Dissected Dodecagon.svg|center|frameless|120px]]
|-
!Fractalizing
|[[File:Truncated Hexagonal Fractal Triangle.svg|center|frameless|80px]]
|[[File:Truncated Hexagonal Fractal Square.svg|center|frameless|80px]]
|[[File:Truncated Hexagonal Fractal Hexagon.svg|center|frameless|90px]]
|[[File:Truncated Hexagonal Fractal Dissected Dodecagon.svg|center|frameless|100px]]
|}
The side lengths are dilated by a factor of <math>2+\sqrt{3}</math>.

This can similarly be done with the truncated trihexagonal tiling as a basis, with corresponding dilation of <math>3+\sqrt{3}</math>.
<br />
{| class="wikitable"
!
!Triangle
!Square
!Hexagon
!Dissected<BR>Dodecagon
|- valign="center"
! valign="center" |Shape
|[[File:The Triangle.svg|alt=|center|frameless|80px]]
|[[File:A Square Tile.svg|alt=|center|frameless|80px]]
|[[File:A Hexagon Tile.svg|center|frameless|80px]]
|[[File:A Dissected Dodecagon.svg|center|frameless|100px]]
|-
!Fractalizing
|[[File:Truncated Trihexagonal Fractal Triangle.svg|center|frameless|143px]]
|[[File:Truncated Trihexagonal Fractal Square.svg|center|frameless|143px]]
|[[File:Truncated Trihexagonal Fractal Hexagon.svg|center|frameless|120px]]
|[[File:Truncated Trihexagonal Fractal Dissected Dodecagon.svg|center|frameless|120px]]
|}

==== Fractalizing examples ====
{| class="wikitable"
!
!Truncated Hexagonal Tiling
!Truncated Trihexagonal Tiling
|-
!Fractalizing
|[[File:Planar Fractalizing Truncated Hexagonal Tiling II.png|alt=|center|frameless|320px]]
|[[File:Planar Tiling Fractalizing the Truncated Trihexagonal Tiling.png|center|frameless|320px]]
|}

== Tilings that are not edge-to-edge ==

Convex regular polygons can also form plane tilings that are not edge-to-edge. Such tilings can be considered edge-to-edge as nonregular polygons with adjacent colinear edges.

There are seven families of [[isogonal figure]]s, each family having a real-valued parameter determining the overlap between sides of adjacent tiles or the ratio between the edge lengths of different tiles. Two of the families are generated from shifted square, either progressive or zig-zagging positions. Grünbaum and Shephard call these tilings ''uniform'' although it contradicts Coxeter's definition for uniformity which requires edge-to-edge regular polygons.<ref>{{Cite web |url=http://www.maa.org/sites/default/files/images/upload_library/22/Allendoerfer/1978/0025570x.di021102.02p0230f.pdf |title=Tilings by regular polygons |p=236 |archive-url=https://web.archive.org/web/20160303235526/http://www.maa.org/sites/default/files/images/upload_library/22/Allendoerfer/1978/0025570x.di021102.02p0230f.pdf |archive-date=2016-03-03 |url-status=dead}}</ref> Such isogonal tilings are actually topologically identical to the uniform tilings, with different geometric proportions.

{| class=wikitable width=600
|-
|+ Periodic [[isogonal figure|isogonal]] tilings by non-edge-to-edge convex regular polygons
!1
!2
!3
!4
!5
!6
!7
|- valign=top
|colspan=2|[[File:Square brick pattern.png|120px]]<br/>Rows of squares with horizontal offsets
|[[File:Half-offset triangular tiling.png|120px]]<br/>Rows of triangles with horizontal offsets
|[[File:Distorted truncated square tiling.svg|120px]]<br/>[[Pythagorean tiling|A tiling by squares]]
|[[File:Gyrated truncated hexagonal tiling.png|120px]]<br/>Three hexagons surround each triangle
|[[File:Gyrated hexagonal tiling2.svg|120px]]<br/>Six triangles surround every hexagon.
|[[File:Trihexagonal tiling unequal2.svg|120px]]<br/>Three size triangles
|-
!cmm (2*22)
!p2 (2222)
!cmm (2*22)
!p4m (*442)
!colspan=2|p6 (632)
!p3 (333)
|-
!colspan=2|[[Hexagonal tiling#Topologically equivalent tilings|Hexagonal tiling]]
![[Square tiling#Topologically equivalent tilings|Square tiling]]
![[Truncated square tiling#Variations|Truncated square tiling]]
![[Truncated hexagonal tiling#Topologically identical tilings|Truncated hexagonal tiling]]
![[Hexagonal tiling#Topologically equivalent tilings|Hexagonal tiling]]
![[Trihexagonal tiling#Topologically equivalent tilings|Trihexagonal tiling]]
|}

== See also ==
{{colbegin|colwidth=30em}}
* [[Grid (spatial index)]]
* [[Uniform tilings in hyperbolic plane]]
* [[List of uniform tilings]]
* [[Wythoff symbol]]
* [[Tessellation]]
* [[Wallpaper group]]
* [[Regular polyhedron]] (the [[Platonic solid]]s)
* [[Semiregular polyhedron]] (including the [[Archimedean solid]]s)
* [[Hyperbolic geometry]]
* [[Penrose tiling]]
* [[Tiling with rectangles]]
* [[Lattice (group)]]
{{colend}}

== References ==
{{Reflist}}
* {{Cite journal | first1=Branko | last1=Grünbaum | first2=Geoffrey C. | last2=Shephard
  |title=Tilings by regular polygons | journal = Math. Mag. | year= 1977 | volume=50
  |number=5 | pages=227–247 | doi=10.2307/2689529| jstor=2689529 }}
* {{Cite journal| first1=Branko | last1=Grünbaum | first2=G. C. | last2=Shephard
  |title=The ninety-one types of isogonal tilings in the plane
  |journal=Trans. Am. Math. Soc. | volume=252 | pages=335–353 | year=1978 | doi=10.1090/S0002-9947-1978-0496813-3 | mr = 496813| doi-access=free }}
* {{Cite journal|first1=I. | last1=Debroey | first2=F. | last2=Landuyt
  | s2cid=122636363 |journal= Geometriae Dedicata | year=1981 | volume=11 | number=1 | title =Equitransitive edge-to-edge tilings
   |pages=47–60 | doi=10.1007/BF00183189 }}
* {{Cite book | author1=Grünbaum, Branko | author-link=Branko Grünbaum | author2=Shephard, G. C. | author2-link=G.C. Shephard | 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 }}
* {{Cite journal| first1=Ding | last1=Ren | first2=John R. | last2= Reay
  | title=The boundary characteristic and Pick's theorem in the Archimedean planar tilings
  | journal=J. Comb. Theory A | year=1987 | volume=44 | number=1 | pages=110–119
  |doi=10.1016/0097-3165(87)90063-X | doi-access= }}
* {{Cite journal | first=D. |last=Chavey | title=Tilings by Regular Polygons&mdash;II: A Catalog of Tilings | url=https://www.beloit.edu/computerscience/faculty/chavey/catalog/ | journal=Computers & Mathematics with Applications | year=1989 | volume=17 | pages=147&ndash;165 | doi=10.1016/0898-1221(89)90156-9| doi-access= | url-access=subscription }}
* Order in Space: A design source book, Keith Critchlow, 1970 {{ISBN|978-0-670-52830-1}}
*{{Cite book| first1= Duncan MacLaren Young |last1=Sommerville| title=An Introduction to the Geometry of '''n''' Dimensions | publisher=Dover Publications| year=1958}} Chapter X: The Regular Polytopes
* {{Cite journal|first1=P. | last1=Préa | title=Distance sequences and percolation thresholds in Archimedean Tilings | journal=Mathl. Comput. Modelling | volume=26
   |number=8–10 | year=1997|pages=317–320 |doi=10.1016/S0895-7177(97)00216-1| doi-access= }}
* {{Cite journal| first1=Jurij | last1=Kovic | title=Symmetry-type graphs of Platonic and Archimedean solids
  |year=2011 | journal=Math. Commun. | volume=16 | number=2 |pages=491–507 | url=http://hrcak.srce.hr/74895
  }}
* {{Cite journal| first1=Daniel | last1=Pellicer | first2=Gordon | last2= Williams
  |title = Minimal Covers of the Archimedean Tilings, Part 1
  | journal= The Electronic Journal of Combinatorics| year=2012 | volume=19 | number=3 | page= #P6
  | doi=10.37236/2512 | url=http://www.combinatorics.org/ojs/index.php/eljc/article/view/v19i3p6
  | doi-access=free }}
* Dale Seymour and [[Jill Britton]], ''Introduction to Tessellations'', 1989, {{ISBN|978-0866514613}}, pp.&nbsp;50–57

== External links ==

Euclidean and general tiling links:
* [http://probabilitysports.com/tilings.html n-uniform tilings], Brian Galebach
* {{Cite web | author = Dutch, Steve | title = Uniform Tilings | url = http://www.uwgb.edu/dutchs/symmetry/uniftil.htm | access-date = 2006-09-09 | archive-url = https://web.archive.org/web/20060909053826/http://www.uwgb.edu/dutchs/SYMMETRY/uniftil.htm | archive-date = 2006-09-09 | url-status = dead }}
* {{Cite web | author = Mitchell, K | title = Semi-Regular Tilings | url = http://people.hws.edu/mitchell/tilings/Part1.html | access-date = 2006-09-09}}
* {{Mathworld |urlname=Tessellation |title=Tessellation}}
* {{MathWorld | urlname=SemiregularTessellation | title=Semiregular tessellation}}
* {{MathWorld | urlname=DemiregularTessellation | title=Demiregular tessellation}}

{{Tessellation}}

[[Category:Euclidean plane geometry]]
[[Category:Regular tilings]]
[[Category:Tessellation]]