View source for Truncated square tiling

In [[Geometry|geometry]], the '''truncated square tiling''' is a semiregular tiling by regular polygons of the [[Euclidean plane]] with one square and two [[Octagon|octagon]]s on each [[Vertex (geometry)|vertex]]. This is the only edge-to-edge tiling by [[Regular polygon|regular]] [[Convex polygon|convex polygon]]s which contains an octagon. It has [[Schläfli symbol]] of ''t{4,4}''.

[[Biography:John Horton Conway|Conway]] calls it a '''truncated quadrille''', constructed as a [[Truncation (geometry)|truncation]] operation applied to a [[Square tiling|square tiling]] (quadrille).

Other names used for this pattern include '''Mediterranean tiling''' and '''octagonal tiling''', which is often represented by smaller squares, and nonregular octagons which alternate long and short edges.

There are 3 regular and 8 semiregular tilings in the plane.

== Uniform colorings ==

There are two distinct [[Uniform coloring|uniform coloring]]s of a truncated square tiling. (Naming the colors by indices around a vertex (4.8.8): 122, 123.)
{| class="wikitable"
|[[File:Uniform tiling 44-t12.svg|150px]]<BR/>2 colors: 122<BR/>{{CDD|node_1|4|node_1|4|node}}
|[[File:Uniform tiling 44-t012.png|150px]]<BR/>3 colors: 123<BR/>{{CDD|node_1|4|node_1|4|node_1}}
|}

== Circle packing ==
The truncated 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 3 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 H</ref> <!--The packing density is ...% coverage.-->

:[[File:1-uniform-2-circlepack.svg|200px]]

== Variations ==
[[File:Alternate-sized truncated square tiling.svg|thumb|The squares from the truncation can be alternately sized. In the limit, half of the vertices can remain untruncated, leading to a [[Chamfered square tiling|chamfered square tiling]].]]
[[File:Dodecatile-alt-skew-truncsquare.svg|thumb|A skew equilateral form with squares into rhombi, and flattened octagons.]]
One variations on this pattern, often called a ''Mediterranean pattern'', is shown in stone tiles with smaller squares and diagonally aligned with the borders. Other variations stretch the squares or octagons.

The [[Pythagorean tiling]] alternates large and small squares, and may be seen as topologically identical to the truncated square tiling. The squares are rotated 45 degrees and octagons are distorted into squares with mid-edge vertices.

A weaving pattern also has the same topology, with [[Octagon|octagon]]s flattened [[Rectangle|rectangle]]s.

{| class=wikitable width=600px
!colspan=2|p4m, (*442)
!colspan=2|p4, (442)
!colspan=2|p4g, (4*2)
!colspan=2|pmm (*2222)
|-
|[[File:Mediterranean tiling.png|100px]]
|[[File:Mediterranean tiling2.png|100px]]
|[[File:Distorted truncated square tiling.png|100px]]
|[[File:truncated_square_tiling_star_varation.svg|100px]]
|[[File:Weaved truncated square tiling0b.png|100px]]
|[[File:Distorted truncated square tiling 4.svg|100px]]
|[[File:Octagon rectangle tiling.png|100px]]
|[[File:Octagon rhombus tiling.png|100px]]
|-
!colspan=2|p4m, (*442)
!p4, (442)
!colspan=3|cmm, (2*22)
!colspan=2|pmm (*2222)
|-
|[[File:Truncated_square_tiling_isogonal2.png|100px]]
|[[File:Truncated_square_tiling_isogonal.png|100px]]
|[[File:Distorted truncated square tiling2.png|100px]]
|[[File:Flemish_bond_tiling.svg|100px]]
|[[File:Weaved truncated square tiling.png|100px]]
|[[File:Distorted truncated square tiling 3.svg|100px]]
|[[File:Truncated rhombic tiling.png|100px]]
|[[File:Truncated rectangular tiling.png|100px]]
|-
!colspan=2|Mediterranean
![[Pythagorean tiling|Pythagorean]]
!Flemish bond
!Weaving
!Twisted
!colspan=2|Rectangular/rhombic
|}

== Related polyhedra and tilings ==
[[File:Optical-illusion-checkerboard-twisted-cord2.svg|thumb|The truncated square tiling is used in an [[Physics:Optical illusion|optical illusion]] with truncated vertices divides and colored alternately, seeming to twist the grid.]]
The truncated square tiling is topologically related as a part of sequence of uniform polyhedra and tilings with vertex figures 4.2n.2n, extending into the hyperbolic plane:

The 3-dimensional [[Bitruncated cubic honeycomb|bitruncated cubic honeycomb]] projected into the plane shows two copies of a truncated tiling. In the plane it can be represented by a compound tiling, or combined can be seen as a [[Chamfered square tiling|chamfered square tiling]].
{| class="wikitable" width=450
|- align=center
|[[File:Uniform tiling 44-t01.png|200px]]<BR/>{{CDD|node_1|4|node_1|4|node}}
|[[File:Uniform tiling 44-t12.svg|200px]]<BR/>{{CDD|node|4|node_1|4|node_1}}
|[[File:Chamfered square tiling.png|200px]]<BR/>{{CDD|node_1|4|node_1|4|node}} + {{CDD|node|4|node_1|4|node_1}}
|}

=== Wythoff constructions from 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 unique topologically forms: [[Square tiling|square tiling]], truncated square tiling, [[Snub square tiling|snub square tiling]].

{{Square tiling table}}

=== Related tilings in other symmetries===

=== Tetrakis square tiling===
[[File:1-uniform 2 dual.svg|thumb|The [[Tetrakis square tiling|tetrakis square tiling]]]]
:{{main|Tetrakis square tiling}}
The '''tetrakis square tiling''' is the tiling of the Euclidean plane dual to the truncated square tiling. It can be constructed [[Square tiling|square tiling]] with each square divided into four [[Isosceles triangle|isosceles]] [[Right triangle|right triangle]]s from the center point, forming an infinite [[Arrangement of lines|arrangement of lines]]. It can also be formed by subdividing each square of a grid into two triangles by a diagonal, with the diagonals alternating in direction, or by overlaying two square grids, one rotated by 45 degrees from the other and scaled by a factor of [[Square root of 2|{{sqrt|2}}]].

[[Biography:John Horton Conway|Conway]] calls it a '''kisquadrille''',<ref>John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, {{isbn|978-1-56881-220-5}} {{cite web |url=http://www.akpeters.com/product.asp?ProdCode=2205 |title=A K Peters, LTD. - the Symmetries of Things |access-date=2012-01-20 |url-status=dead |archive-url=https://web.archive.org/web/20100919143320/https://akpeters.com/product.asp?ProdCode=2205 |archive-date=2010-09-19 }} (Chapter 21, Naming Archimedean and Catalan polyhedra and tilings, p288 table)</ref> represented by a kis operation that adds a center point and triangles to replace the faces of a [[Square tiling|square tiling]] (quadrille). It is also called the '''Union Jack lattice''' because of the resemblance to the UK flag of the triangles surrounding its degree-8 vertices.<ref>{{citation|title=Ising Model with Antiferromagnetic Next-Nearest-Neighbor Coupling: Spin Correlations and Disorder Points|journal=Phys. Rev. B |volume=1|issue=11|pages=4405–4409|doi=10.1103/PhysRevB.1.4405|first=John|last=Stephenson|year=1970 }}.</ref>

[[File:P1 dual.png|220x220px]]

== See also ==
* Tilings of regular polygons
* List of uniform tilings
* [[Percolation threshold]]

== References ==
{{Reflist}}
* 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]
* {{cite book | author=Grünbaum, Branko |  author2=Shephard, G. C. | name-list-style=amp | 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)
* {{The Geometrical Foundation of Natural Structure (book)|page=40}}
* Dale Seymour and Jill Britton, ''Introduction to Tessellations'', 1989, {{isbn|978-0866514613}}, pp.&nbsp;50–56

== External links ==
* {{MathWorld | urlname=SemiregularTessellation | title=Semiregular tessellation}}
* {{KlitzingPolytopes|flat.htm#2D|2D Euclidean tilings|o4x4x - tosquat - O6}}

{{Tessellation}}

[[Category:Euclidean tilings]]
[[Category:Isogonal tilings]]
[[Category:Semiregular tilings]]
[[Category:Square tilings]]
[[Category:Truncated tilings]]

{{Sourceattribution|Truncated square tiling}}