List of Euclidean uniform tilings

{{Short description|None}} [[File:Semi-regular-floor-3464.JPG|thumb|An example of uniform tiling in the Archeological Museum of Seville, Sevilla, Spain: rhombitrihexagonal tiling]] [[File:Brückner Vielflache Fig. 96b - 98.jpg|thumb|Uniform tilings and their duals drawn by Max Brückner in ''Vielecke und Vielflache '' (1900)]] This table shows the 11 convex uniform tilings (regular and semiregular) of the Euclidean plane, and their dual tilings.

There are three regular and eight semiregular tilings in the plane. The semiregular tilings form new tilings from their duals, each made from one type of irregular face.

John Conway called these uniform duals '''''Catalan tilings''''', in parallel to the Catalan solid polyhedra.

Uniform tilings are listed by their vertex configuration, the sequence of faces that exist on each vertex. For example ''4.8.8'' means one square and two octagons on a vertex.

These 11 uniform tilings have 32 different ''uniform colorings''. A uniform coloring allows identical sided polygons at a vertex to be colored differently, while still maintaining vertex-uniformity and transformational congruence between vertices. (Note: Some of the tiling images shown below are ''not'' color-uniform.)

In addition to the 11 convex uniform tilings, there are also 14 known nonconvex tilings, using star polygons, and reverse orientation vertex configurations. A further 28 uniform tilings are known using apeirogons. If zigzags are also allowed, there are 23 more known uniform tilings and 10 more known families depending on a parameter: in 8 cases the parameter is continuous, and in the other 2 it is discrete. The set is not known to be complete.

== Laves tilings== In the 1987 book, ''Tilings and patterns'', Branko Grünbaum calls the vertex-uniform tilings '''''Archimedean''''', in parallel to the Archimedean solids. Their dual tilings are called '''''Laves tilings''''' in honor of crystallographer Fritz Laves.<ref>{{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 | pages=[https://archive.org/details/isbn_0716711931/page/59 59, 96] | url-access=registration | url=https://archive.org/details/isbn_0716711931/page/59 }}</ref><ref>{{cite book | first1 = John H. | last1 = Conway | author-link = John Horton Conway | first2 = Heidi | last2 = Burgiel | author-link2 = | first3 = Chaim | last3 = Goodman-Strauss | author-link3 = Chaim Goodman-Strauss | title = The Symmetries of Things | chapter = Chapter 21, ''Naming the Archimedean and Catalan polyhedra and tilings, Euclidean Plane Tessellations'' | date = April 18, 2008 | url = https://akpeters.com/product.asp?ProdCode=2205 | archive-url = https://web.archive.org/web/20100919143320/https://akpeters.com/product.asp?ProdCode=2205 | url-status = dead | archive-date = September 19, 2010 | location = | publisher = A K Peters / CRC Press | page = 288 | isbn = 978-1-56881-220-5 }}</ref> They're also called '''Shubnikov–Laves tilings''' after Aleksei Shubnikov.<ref>[https://books.google.com/books?id=5rPnCAAAQBAJ&dq=Shubnikov%E2%80%93Laves+tilings&pg=PA169 Encyclopaedia of Mathematics: Orbit - Rayleigh Equation], 1991</ref> John Conway called the uniform duals '''''Catalan tilings''''', in parallel to the Catalan solid polyhedra.

The Laves tilings have vertices at the centers of the regular polygons, and edges connecting centers of regular polygons that share an edge. The tiles of the Laves tilings are called ''planigons''. This includes the 3 regular tiles (triangle, square and hexagon) and 8 irregular ones.<ref>{{SpringerEOM|title=Planigon|id=Planigon&oldid=31578|first=A. B.|last=Ivanov}}</ref> Each vertex has edges evenly spaced around it. Three dimensional analogues of the ''planigons'' are called stereohedrons.

These dual tilings are listed by their face configuration, the number of faces at each vertex of a face. For example ''V4.8.8'' means isosceles triangle tiles with one corner with four triangles, and two corners containing eight triangles. The orientations of the vertex planigons (up to D12) are consistent with the vertex diagrams in the below sections.

{| class=wikitable |+ Eleven planigons !colspan=4|Triangles||colspan=3|Quadrilaterals||colspan=3|Pentagons||Hexagon |- valign=bottom align=center |alt=|x100px V63 {{CDD|node|split1|branch}} |alt=|x110px V4.82 {{CDD|node|4|node|4|node}} |alt=|x100px V4.6.12 {{CDD|node|6|node|3|node}} |alt=|x130px V3.122 {{CDD|2}} |alt=|x85px V44 {{CDD|labelinfin|branch|2|branch|labelinfin}} |alt=|x130px V(3.6)2 {{CDD|2}} |alt=|x100px V3.4.6.4 {{CDD|2}} |alt=|x100px V32.4.3.4 {{CDD|2}} |alt=|x80px V34.6 {{CDD|2}} |alt=|x90px V33.42 {{CDD|2}} |alt=|x100px V36 {{CDD|2}} |}

== Convex uniform tilings of the Euclidean plane ==

All reflectional forms can be made by Wythoff constructions, represented by Wythoff symbols, or Coxeter-Dynkin diagrams, each operating upon one of three Schwarz triangle (4,4,2), (6,3,2), or (3,3,3), with symmetry represented by Coxeter groups: [4,4], [6,3], or [3[3]]. Alternated forms such as the snub can also be represented by special markups within each system. Only one uniform tiling can't be constructed by a Wythoff process, but can be made by an elongation of the triangular tiling. An orthogonal mirror construction [∞,2,∞] also exists, seen as two sets of parallel mirrors making a rectangular fundamental domain. If the domain is square, this symmetry can be doubled by a diagonal mirror into the [4,4] family.

Families: * (4,4,2), <math>{\tilde{BC}}_2</math>, [4,4] – Symmetry of the regular square tiling ** <math>{\tilde{I}}_1^2</math>, [∞,2,∞] * (6,3,2), <math>{\tilde{G}}_2</math>, [6,3] – Symmetry of the regular hexagonal tiling and triangular tiling. ** (3,3,3), <math>{\tilde{A}}_2</math>, [3[3]]

=== {{anchor|The 4,4 group family}} The [4,4] group family ===

{| class="wikitable" !Uniform tilings (Platonic and Archimedean) !Vertex figure and dual face Wythoff symbol(s) Symmetry group Coxeter diagram(s) !Dual-uniform tilings (called Laves or Catalan tilings) |- {{Uniform tiles db|Uniform tiling list table|Us}} {{Uniform tiles db|Uniform tiling list table|Uts}} {{Uniform tiles db|Uniform tiling list table|Uns}} |}

=== {{anchor|The 6,3 group family}} The [6,3] group family ===

{| class="wikitable" !Platonic and Archimedean tilings !Vertex figure and dual face Wythoff symbol(s) Symmetry group Coxeter diagram(s) !Dual Laves tilings |- {{Uniform tiles db|Uniform tiling list table|Uh}} {{Uniform tiles db|Uniform tiling list table|Uht}} {{Uniform tiles db|Uniform tiling list table|Uth}} {{Uniform tiles db|Uniform tiling list table|Ut}} {{Uniform tiles db|Uniform tiling list table|Urth}} {{Uniform tiles db|Uniform tiling list table|Ugrth}} {{Uniform tiles db|Uniform tiling list table|Unh}} |}

=== Non-Wythoffian uniform tiling ===

{| class="wikitable" !Platonic and Archimedean tilings !Vertex figure and dual face Wythoff symbol(s) Symmetry group Coxeter diagram !Dual Laves tilings |- {{Uniform tiles db|Uniform tiling list table|Uet}} |}

== See also == * Convex uniform honeycomb – The 28 uniform 3-dimensional tessellations, a parallel construction to the convex uniform Euclidean plane tilings. * Euclidean tilings by convex regular polygons * List of tessellations * Percolation threshold * Uniform tilings in hyperbolic plane

== References == {{Reflist}}

==Further reading== * {{cite book | first1 = John H. | last1 = Conway | author-link = John Horton Conway | first2 = Heidi | last2 = Burgiel | author-link2 = | first3 = Chaim | last3 = Goodman-Strauss | author-link3 = Chaim Goodman-Strauss | title = The Symmetries of Things | chapter = Chapter 19, ''Archimedean tilings'', table 19.1 | date = April 18, 2008 | url = https://akpeters.com/product.asp?ProdCode=2205 | archive-url = https://web.archive.org/web/20100919143320/https://akpeters.com/product.asp?ProdCode=2205 | url-status = dead | archive-date = September 19, 2010 | location = | publisher = A K Peters / CRC Press | page = | isbn = 978-1-56881-220-5 }} * {{cite journal | first1 = H.S.M. | last1 = Coxeter | author-link = H.S.M. Coxeter | first2 = M.S. | last2 = Longuet-Higgins | author-link2 = Michael S. Longuet-Higgins | first3 = J.C.P. | last3 = Miller | author-link3 = J.C.P. Miller | title = Uniform polyhedra | date = 1954 | url = | journal = Phil. Trans. | volume = 246 A | pages = 401–450 }} * {{The Geometrical Foundation of Natural Structure (book)}} (Section 2–3 ''Circle packings, plane tessellations, and networks'', pp. 34–40). * {{cite web | first1 = Laura | last1 = Asaro | author-link = | first2 = John | last2 = Hyde | author-link2 = | first3 = Melanie | last3 = Jensen | author-link3 = | first4 = Casey | last4 = Mann | author-link4 = | first5 = Tyler | last5 = Schroeder | author-link5 = | title = Uniform edge-''c''-colorings of the Archimedean Tilings | date = | url = http://faculty.washington.edu/cemann/uniform-edge-coloring.pdf | location = | journal = | volume = | publisher = University of Washington | page = }} ([http://faculty.washington.edu/cemann/ Casey Mann at the University of Washington]) * {{cite web |first1 = Branko |last1 = Grünbaum |author-link = Branko Grünbaum |first2 = Geoffrey |last2 = Shepard |author-link2 = G.C. Shephard |title = Tilings by Regular polygons |date = November 1977 |url = http://www.maa.org/sites/default/files/images/upload_library/22/Allendoerfer/1978/0025570x.di021102.02p0230f.pdf |location = |journal = |volume = 50 |issue = 5 |publisher = |page = |access-date = 2014-07-05 |archive-date = 2016-03-03 |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 |url-status = dead }} * {{cite book |first1 = Dale |last1 = Seymour |author-link = |first2 = Jill |last2 = Britton |author-link2 = Jill Britton |title = Introduction to Tessellations |date = 1989 |url = https://archive.org/details/introductiontote0000seym/page/50 |location = |publisher = Dale Seymour Publications |pages = [https://archive.org/details/introductiontote0000seym/page/50 50–57, 71-74] |isbn = 978-0866514613 }}

== 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] * [https://web.archive.org/web/20080602030052/http://www.tess-elation.co.uk/index.htm 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]

{{DEFAULTSORT:Euclidean uniform tilings}} Uniform Category:Uniform tilings Uniform planar tilings