View source for Square tiling

{{Short description|Regular tiling of the Euclidean plane}}

In [[Geometry|geometry]], the '''square tiling''', '''square tessellation''' or '''square grid''' is a regular tiling of the [[Euclidean plane]].  It has [[Schläfli symbol]] of {{math|{4,4},}} meaning it has 4 squares around every [[Vertex (geometry)|vertex]]. [[Biography:John Horton Conway|Conway]] called it a '''quadrille'''.

The internal angle of the square is 90 degrees so four squares at a point make a full 360 degrees. It is one of three regular tilings of the plane. The other two are the [[Triangular tiling|triangular tiling]] and the [[Hexagonal tiling|hexagonal tiling]].

== Uniform colorings ==
There are 9 distinct [[Uniform coloring|uniform coloring]]s of a square tiling. Naming the colors by indices on the 4 squares around a vertex: 1111, 1112(i), 1112(ii), 1122, 1123(i), 1123(ii), 1212, 1213, 1234. (i) cases have simple reflection symmetry, and (ii) glide reflection symmetry. Three can be seen in the same symmetry domain as reduced colorings: 1112<sub>i</sub> from 1213, 1123<sub>i</sub> from 1234, and 1112<sub>ii</sub> reduced from 1123<sub>ii</sub>.

{| class="wikitable collapsible collapsed"
!colspan=12|9 uniform colorings
|- align=center
!1111||1212||1213||1112<sub>i</sub>||1122
|- align=center
|[[File:Square tiling uniform coloring 1.png|75px]]
|[[File:Square tiling uniform coloring 7.png|75px]]
|[[File:Square tiling uniform coloring 8.png|75px]]
|[[File:Square tiling uniform coloring 2.png|75px]]
|[[File:Square tiling uniform coloring 4.png|75px]]
|- align=center
|colspan=2|p4m (*442)
|colspan=2|p4m (*442)
|pmm (*2222)
|- align=center
!1234||1123<sub>i</sub>||1123<sub>ii</sub>||1112<sub>ii</sub>
|- align=center
|[[File:Square tiling uniform coloring 9.png|75px]]
|[[File:Square tiling uniform coloring 5.png|75px]]
|[[File:Square tiling uniform coloring 6.png|75px]]
|[[File:Square tiling uniform coloring 3.png|75px]]
|- align=center
|colspan=2|pmm (*2222)
|colspan=2|cmm (2*22)
|}

== Related polyhedra and tilings ==
This tiling is topologically related as a part of sequence of regular polyhedra and tilings, extending into the hyperbolic plane: {4,p}, p=3,4,5...
{{Regular square tiling table}}

This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the [[Octahedron|octahedron]], with [[Schläfli symbol]] {n,4}, and Coxeter diagram {{CDD|node_1|n|node|4|node}}, with n progressing to infinity.



== Wythoff constructions from square tiling ==
Like the [[Uniform polyhedron|uniform polyhedra]] there are eight uniform tilings that can be based from the regular square tiling.

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, all 8 forms are distinct. However treating faces identically, there are only three topologically distinct forms: '''square tiling''', [[Truncated square tiling|truncated square tiling]], [[Snub square tiling|snub square tiling]].

{{Square tiling table}}

== Topologically equivalent tilings ==
[[File:Isogonal snub square tiling-8x8b.svg|150px|thumb|An isogonal variation with two types of faces, seen as a [[Snub square tiling|snub square tiling]] with trangle pairs combined into rhombi.]]
[[File:Plus tiling 5-colors2.svg|150px|thumb|Topological square tilings can be made with concave faces and more than one edge shared between two faces. This variation has 3 edges shared.]]
Other [[Quadrilateral|quadrilateral]] tilings can be made which are topologically equivalent to the square tiling (4 quads around every vertex). 
[[File:Isohedral tiling p4-51d.png|150px|thumb|A 2-isohedral variation with rhombic faces]]
Isohedral tilings have identical faces (face-transitivity) and vertex-transitivity, there are 18 variations, with 6 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.<ref>Tilings and Patterns, from list of 107 isohedral tilings, p.473-481</ref>

{| class=wikitable width=480
|+ Isohedral quadrilateral tilings 
|- align=center valign=top
|[[File:Isohedral tiling p4-56.png|90px]]
|[[File:Isohedral tiling p4-49.png|90px]]
|[[File:Isohedral tiling p4-54.png|90px]]
|[[File:Isohedral tiling p4-50.png|90px]]
|[[File:Isohedral tiling p4-51.png|90px]]
|[[File:Isohedral tiling p4-55.png|90px]]
|[[File:Isohedral tiling p4-51c.png|90px]]
|- 
![[Square]]<BR>p4m, (*442)
!Quadrilateral<BR>p4g, (4*2)
![[Rectangle]]<BR>pmm, (*2222)
![[Parallelogram]]<BR>p2, (2222)
!Parallelogram<BR>pmg, (22*)
![[Rhombus]]<BR>cmm, (2*22)
!Rhombus<BR>pmg, (22*)
|- align=center valign=top
|[[File:Isohedral tiling p4-52b.png|90px]]
|[[File:Isohedral tiling p4-52.png|90px]]
|[[File:Isohedral tiling p4-46.png|90px]]
|[[File:Isohedral tiling p4-53.png|90px]]
|[[File:Isohedral tiling p4-47.png|90px]]
|[[File:Isohedral tiling p4-43.png|90px]]
|- align=center valign=top
!colspan=2|[[Trapezoid]]<BR>cmm, (2*22)
!Quadrilateral<BR>pgg, (22×)
![[Kite (geometry)|Kite]]<BR>pmg, (22*)
!Quadrilateral<BR>pgg, (22×)
!Quadrilateral<BR>p2, (2222)

|}

{| class=wikitable width=480
|+ Degenerate quadrilaterals or non-edge-to-edge triangles
|- align=center valign=top
|[[File:Isohedral tiling p3-7.png|90px]]
|[[File:Isohedral tiling p3-4.png|90px]]
|[[File:Isohedral tiling p3-5.png|90px]]
|[[File:Isohedral tiling p3-3.png|90px]]
|[[File:Isohedral tiling p3-6.png|90px]]
|[[File:Isohedral tiling p3-2.png|90px]]
|- align=center valign=top
!Isosceles<BR>pmg, (22*)
!colspan=2|Isosceles<BR>pgg, (22×)
!colspan=2|Scalene<BR>pgg, (22×)
!Scalene<BR>p2, (2222)
|}

== Circle packing ==
The square tiling can be used as a [[Circle packing|circle packing]], placing equal diameter circles at the center of every point. Every circle is in contact with 4 other circles in the packing ([[Kissing number|kissing number]]).<ref name=Critchlow>Order in Space: A design source book, Keith Critchlow, p.74-75, circle pattern 3</ref> The packing density is π/4=78.54% coverage. There are 4 uniform colorings of the circle packings.
:[[File:1-uniform-5-circlepack.svg|200px]]

== 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.<ref>Coxeter, Regular Complex Polytopes, pp. 111-112, p. 136.</ref>
{| class=wikitable
!Self-dual||colspan=2|Duals
|-
|[[File:Complex apeirogon 4-4-4.png|160px]]
|[[File:Complex apeirogon 2-8-4.png|160px]]
|[[File:Complex apeirogon 4-8-2.png|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==
* Checkerboard
* List of regular polytopes
* List of uniform tilings
* [[Square lattice]]
* Tilings of regular polygons

== References ==
{{Reflist}}
* Coxeter, H.S.M. ''Regular Polytopes'', (3rd edition, 1973), Dover edition, {{isbn|0-486-61480-8}} p.&nbsp;296, Table II: Regular honeycombs
* {{KlitzingPolytopes|flat.htm#2D|2D Euclidean tilings|o4o4x - squat - O1}}
* {{The Geometrical Foundation of Natural Structure (book)}} p36
* {{cite book | author1=Grünbaum, Branko |  author2=Shephard, G. C. | title=Tilings and Patterns | location=New York | publisher=W. H. Freeman | year=1987 | isbn=0-7167-1193-1 | url-access=registration | url=https://archive.org/details/isbn_0716711931 }} (Chapter 2.1: ''Regular and uniform tilings'', p.&nbsp;58-65)
* John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, {{isbn|978-1-56881-220-5}} [https://web.archive.org/web/20100919143320/https://akpeters.com/product.asp?ProdCode=2205]

== 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]]

{{Sourceattribution|Square tiling}}