square tiling

{{Infobox face-uniform tiling

| image = Tiling 4b.svg

| name = Square tiling

| type = regular tiling

| tile = square

| vertex_config = 4.4.4.4

| schläfli = $\{4,4\}$

| symmetry = p4m

| properties = vertex-transitive, edge-transitive, face-transitive

| dual = self-dual

}}

In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane consisting of four squares around every vertex. 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, the 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 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.{{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, 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 on the flags of the tiling. In this case, the flag 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 (mapping the vertex of a tile to another), edge-transitive (mapping the edge to another), and face-transitive (mapping square tile to another). By meeting these three properties, the square tiling is categorized as one of three regular tilings; the remaining being triangular tiling and hexagonal tiling with its prototiles are equilateral triangles and regular hexagons, respectively.{{sfnp|Grünbaum|Shephard|1987|p=[https://books.google.com/books?id=0x0vDAAAQBAJ&pg=PA35 35]}} The symmetry group of a square tiling is p4m: there is an order-4 dihedral group of a tile and an order-2 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 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) and vertex-transitivity. 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.png

| image8 = Isohedral tiling p4-52.png

| 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.png

| image17 = Isohedral tiling p3-6.png

| image18 = Isohedral tiling p3-2.png

| 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 apeirogons, 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}}

class=wikitable

!Self-dual

colspan=2|Duals

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}}