# Square tiling > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Square_tiling > Markdown URL: https://mediated.wiki/source/Square_tiling.md > Source: https://en.wikipedia.org/wiki/Square_tiling > Source revision: 1356784600 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) {{Short description|Regular tiling of the Euclidean plane}} {{Infobox face-uniform tiling | image = Tiling 4b.svg | name = Square tiling | type = [regular tiling](/source/regular_tiling) | tile = [square](/source/square) | vertex_config = 4.4.4.4 | schläfli = \{4,4\} | symmetry = p4m | properties = [vertex-transitive](/source/vertex-transitive), [edge-transitive](/source/edge-transitive), [face-transitive](/source/face-transitive) | dual = [self-dual](/source/self-dual) }} In [geometry](/source/geometry), the '''square tiling''', '''square tessellation''' or '''square grid''' is a [regular tiling](/source/regular_tiling) of the [Euclidean plane](/source/Euclidean_plane) consisting of four [squares](/source/Square_(geometry)) around every [vertex](/source/Vertex_(geometry)). [John Horton Conway](/source/John_Horton_Conway) called it a '''quadrille'''.{{cite book | first1 = John H. | last1 = Conway | author1-link = John Horton Conway | first2 = Heidi | last2 = Burgiel | first3 = Chaim | last3 = Goodman-Strauss | title = The Symmetries of Things | title-link = The Symmetries of Things | year = 2008 | publisher = AK Peters | isbn = 978-1-56881-220-5 | page = [https://books.google.com/books?id=Drj1CwAAQBAJ&pg=PA288 288] }} == Structure and properties == {{multiple image | image1 = Lustenau, Rheinstraße 4, Küche, Fliesenboden.jpg | image2 = Chess.board.fabric.png | footer = Flooring and game board | total_width = 260 }} The square tiling has a structure consisting of one type of congruent [prototile](/source/prototile), the [square](/source/square), sharing two vertices with other identical ones. This is an example of monohedral tiling.{{cite book | title = The Tiling Book: An Introduction to the Mathematical Theory of Tilings | first = Colin | last = Adams | publisher = American Mathematical Society | year = 2022 | isbn = 9781470468972 | pages = [https://books.google.com/books?id=LvGGEAAAQBAJ&pg=PA23 23] }} Each vertex at the tiling is surrounded by four squares, which denotes in a [vertex configuration](/source/vertex_configuration) as 4.4.4.4 or 4^4 .{{sfnp|Grünbaum|Shephard|1987|p=[https://books.google.com/books?id=0x0vDAAAQBAJ&pg=PA59 59]}} The vertices of a square can be considered as the lattice, so the square tiling can be formed through the [square lattice](/source/square_lattice).{{cite book | last1 = Grünbaum | first1 = Branko | author-link1 = Branko Grunbaum | last2 = Shephard | first2 = G. C. | year = 1987 | title = Tilings and Patterns | title-link = Tilings and patterns | publisher = W. H. Freeman | page = [https://books.google.com/books?id=0x0vDAAAQBAJ&pg=PA21 21], [https://books.google.com/books?id=0x0vDAAAQBAJ&pg=PA29 29] }} This tiling is commonly familiar with the flooring and game boards.{{cite book | first = Sadun | last = Lorenzo | title = Topology of Tiling Spaces | url = https://books.google.com/books?id=dL8FCAAAQBAJ&pg=PA1 | page = 1 | publisher = American Mathematical Society | year = 2008 | isbn = 978-0-8218-4727-5 }} It is [self-dual](/source/self-dual), meaning the center of each square connects to another of the adjacent tile, forming square tiling itself.{{cite journal | last1 = Nelson | first1 = Roice | last2 = Segerman | first2 = Henry | title = Visualizing hyperbolic honeycombs | journal = Journal of Mathematics and the Arts | year = 2017 | volume = 11 | issue = 1 | pages = 4–39 | doi = 10.1080/17513472.2016.1263789 | arxiv = 1511.02851 }} The square tiling [acts transitively](/source/Group_action_(mathematics)) on the ''flags'' of the tiling. In this case, the [flag](/source/Flag_(geometry)) consists of a mutually incident vertex, edge, and tile of the tiling. Simply put, every pair of flags has a symmetry operation mapping the first flag to the second: they are [vertex-transitive](/source/vertex-transitive) (mapping the vertex of a tile to another), [edge-transitive](/source/edge-transitive) (mapping the edge to another), and [face-transitive](/source/face-transitive) (mapping square tile to another). By meeting these three properties, the square tiling is categorized as one of three [regular tiling](/source/regular_tiling)s; the remaining being [triangular tiling](/source/triangular_tiling) and [hexagonal tiling](/source/hexagonal_tiling) with its prototiles are [equilateral triangle](/source/equilateral_triangle)s and [regular hexagon](/source/regular_hexagon)s, respectively.{{sfnp|Grünbaum|Shephard|1987|p=[https://books.google.com/books?id=0x0vDAAAQBAJ&pg=PA35 35]}} The [symmetry group](/source/symmetry_group) of a square tiling is p4m: there is an [order-4 dihedral group](/source/Dihedral_group_of_order_4) of a tile and an order-2 [dihedral group](/source/dihedral_group) around the vertex surrounded by four squares lying on the line of reflection.{{sfnp|Grünbaum|Shephard|1987|p=[https://books.google.com/books?id=0x0vDAAAQBAJ&pg=PA42 42]|loc=see p. [https://books.google.com/books?id=0x0vDAAAQBAJ&pg=PA38 38] for detail of symbols}} The square tiling is alternatively formed by the [assemblage of infinitely many circles](/source/Circle_packing) arranged vertically and horizontally, wherein their equal diameter at the center of every point contact with four other circles.{{cite book | last = Williams | first = Robert | year = 1979 | title = The Geometrical Foundation of Natural Structure: A Source Book of Design | publisher = Dover Publications | isbn = 0-486-23729-X | page = 36 | url = https://archive.org/details/geometricalfound00will/page/36/mode/1up?view=theater }} Its densest packing is \frac{\pi}{4} \approx 0.785 .{{cite journal | title = Plane nets in crystal chemistry | first1 = M. | last1 = O'Keeffe | first2 = B. G. | last2 = Hyde | journal = Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences | volume = 295 | issue = 1417 | year = 1980 | pages = 553–618 | jstor = 36648 | doi = 10.1098/rsta.1980.0150 | bibcode = 1980RSPTA.295..553O | s2cid = 121456259 }} == Topologically equivalent tilings == Isohedral tilings have identical faces ([face-transitivity](/source/Face-transitive)) and [vertex-transitivity](/source/vertex-transitive). There are eighteen variations, with six identified as triangles that do not connect edge-to-edge, or as quadrilateral with two collinear edges. Symmetry given assumes all faces are the same color.{{sfnp|Grünbaum|Shephard|1987|p=[https://books.google.com/books?id=0x0vDAAAQBAJ&pg=PA473 473–481]}} {{multiple image | perrow = 6 | align = center | image1 = Isohedral tiling p4-49.svg | image2 = Lattice of rectangles.svg | image3 = Lattice of rhomboids.svg | image4 = Isohedral tiling p4-51c.svg | image5 = Lattice of rhombuses.svg | image6 = Isohedral tiling p4-51c.svg | image7 = Isohedral tiling p4-52b.svg | image8 = Isohedral tiling p4-52.svg | image9 = Isohedral tiling p4-46.svg | image10 = Isohedral tiling p4-53.svg | image11 = Isohedral tiling p4-47.svg | image12 = Isohedral tiling p4-43.svg | image13 = Isohedral tiling p3-7.svg | image14 = Isohedral tiling p3-4.svg | image15 = Isohedral tiling p3-5.svg | image16 = Isohedral tiling p3-3.svg | image17 = Isohedral tiling p3-6.svg | image18 = Isohedral tiling p3-2.svg | total_width = 700 | footer = Twelve isohedral quadrilateral tilings, and six triangular tilings that do not tile edge-to-edge }} == Related regular complex apeirogons == There are 3 [regular complex apeirogon](/source/regular_complex_apeirogon)s, sharing the vertices of the square tiling. Regular complex apeirogons have vertices and edges, where edges can contain 2 or more vertices. Regular apeirogons p{q}r are constrained by: 1/''p'' + 2/''q'' + 1/''r'' = 1. Edges have ''p'' vertices, and vertex figures are ''r''-gonal.Coxeter, Regular Complex Polytopes, pp. 111-112, p. 136. {| class=wikitable !Self-dual||colspan=2|Duals |- |160px |160px |160px |- !4{4}4 or {{CDD|4node_1|4|4node}} !2{8}4 or {{CDD|node_1|8|4node}} !4{8}2 or {{CDD|4node_1|8|node}} |} ==See also== {{Commons category|Order-4 square tiling}} * [Tiling with rectangles](/source/Tiling_with_rectangles) * [Fenestrane](/source/Fenestrane) * [Langton's ant](/source/Langton's_ant) * [OpenStructures](/source/OpenStructures), design pattern consisting of four squares around every vertex * [Polyomino](/source/Polyomino) == References == {{reflist}} * [Coxeter, H.S.M.](/source/Coxeter) ''[Regular Polytopes](/source/Regular_Polytopes_(book))'', (3rd edition, 1973), Dover edition, {{isbn|0-486-61480-8}} p. 296, Table II: Regular honeycombs == External links == * {{MathWorld | urlname=SquareGrid | title=Square Grid}} * {{MathWorld | urlname=RegularTessellation | title=Regular tessellation}} * {{MathWorld | urlname=UniformTessellation | title=Uniform tessellation}} {{Honeycombs}} {{Tessellation}} Category:Euclidean tilings Category:Isohedral tilings Category:Isogonal tilings Category:Polyhedra Category:Regular tilings Category:Self-dual tilings Category:Square tilings Category:Regular tessellations --- Adapted from the Wikipedia article [Square tiling](https://en.wikipedia.org/wiki/Square_tiling) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Square_tiling?action=history)). 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