View source for Snub square tiling

{{Uniform tiles db|Uniform tiling stat table|Uns}}
In [[geometry]], the '''snub square tiling''' is a [[semiregular tiling]] of the [[Euclidean plane]]. There are three triangles and two squares on each [[vertex (geometry)|vertex]].  Its [[Schläfli symbol]] is ''s{4,4}''.

[[John Horton Conway|Conway]] calls it a '''snub quadrille''', constructed by a [[Snub (geometry)|snub]] operation applied to a [[square tiling]] (quadrille).

There are 3 [[List of regular polytopes#Euclidean tilings|regular]] and 8 [[List of uniform tilings|semiregular tilings]] in the plane.

== Uniform colorings ==

There are two distinct [[uniform coloring]]s of a snub square tiling. (Naming the colors by indices around a vertex (3.3.4.3.4): 11212, 11213.)
{| class=wikitable
|- align=center
!Coloring
|[[File:Uniform tiling 44-h01.svg|120px]]<BR>11212
|[[File:Uniform tiling 44-snub.svg|120px]]<BR>11213
|- align=center
![[List of planar symmetry groups#Wallpaper groups|Symmetry]]
|4*2, [4<sup>+</sup>,4], (p4g)
|442, [4,4]<sup>+</sup>, (p4)
|- align=center
![[Schläfli symbol]]
|s{4,4}
|sr{4,4}
|- align=center
![[Wythoff symbol]]
| 
| {{pipe}} 4 4 2
|- align=center
![[Coxeter diagram]]
|{{CDD|node_h|4|node_h|4|node}}
|{{CDD|node_h|4|node_h|4|node_h}}
|}

== Circle packing ==
The snub square tiling can be used as a [[circle packing]], placing equal diameter circles at the center of every point. Every circle is in contact with 5 other circles in the packing ([[kissing number]]).<ref name=Critchlow>Order in Space: A design source book, Keith Critchlow, p.74-75, circle pattern C</ref> <!--The packing density is ... % coverage.)-->
:[[File:1-uniform-9-circlepack.svg|alt=|200x200px]]

== Wythoff construction ==

The '''snub square tiling''' can be [[Wythoff construction|constructed]] as a [[Snub (geometry)|snub]] operation from the [[square tiling]], or as an [[Alternation (geometry)|alternate truncation]] from the [[truncated square tiling]].

An alternate truncation deletes every other vertex, creating a new triangular faces at the removed vertices, and reduces the original faces to half as many sides. In this case starting with a ''truncated square tiling'' with 2 [[octagon]]s and 1 [[Square (geometry)|square]] per vertex, the octagon faces into squares, and the square faces degenerate into edges and 2 new triangles appear at the truncated vertices around the original square.

If the original tiling is made of regular faces the new triangles will be isosceles. Starting with octagons which alternate long and short edge lengths, derived from a regular [[dodecagon]], will produce a snub tiling with perfect equilateral triangle faces.

Example:
{| class="wikitable"
|-
|align="center"|[[File:Uniform tiling 44-t012.png|150px]]<br />Regular octagons alternately truncated
|align="center"|<div style="font-size:24pt">→</div>(Alternate<br />truncation)
|align="center"|[[File:Nonuniform tiling 44-snub.png|150px]]<br />Isosceles triangles (Nonuniform tiling)
|-
|align="center"|[[File:Nonuniform tiling 44-t012-snub.svg|150px]]<br />Nonregular octagons alternately truncated
|align="center"|<div style="font-size:24pt">→</div>(Alternate<br />truncation)
|align="center"|[[File:Uniform tiling 44-snub.svg|150px]]<br />Equilateral triangles
|}

== Related tilings==
<gallery widths=240px heights=240px>
File:Snub snub square tiling.svg|A [[Conway snub operator|snub operator]] applied twice to the square tiling, while it doesn't have regular faces, is made of square with irregular triangles and pentagons.
File:Isogonal snub square tiling-8x8.svg|A related [[isogonal tiling]] that combines pairs of triangles into rhombi
File:Triangular heptagonal tiling.svg|A 2-isogonal tiling can be made by combining 2 squares and 3 triangles into heptagons.
File:P2_dual.png|The [[Cairo pentagonal tiling]] is dual to the snub square tiling.
</gallery>

=== Related k-uniform tilings===
This tiling is related to the [[elongated triangular tiling]] which also has 3 triangles and two squares on a vertex, but in a different order, 3.3.3.4.4. The two vertex figures can be mixed in many [[k-uniform tiling|''k''-uniform tilings]].<ref>{{cite journal | first=D. |last=Chavey | title=Tilings by Regular Polygons&mdash;II: A Catalog of Tilings | url=https://www.beloit.edu/computerscience/faculty/chavey/catalog/ | journal=Computers & Mathematics with Applications | year=1989 | volume=17 | pages=147&ndash;165 | doi=10.1016/0898-1221(89)90156-9| doi-access= }}</ref><ref>{{cite web |url=http://www.uwgb.edu/dutchs/symmetry/uniftil.htm |title=Uniform Tilings |access-date=2006-09-09 |url-status=dead |archive-url=https://web.archive.org/web/20060909053826/http://www.uwgb.edu/dutchs/SYMMETRY/uniftil.htm |archive-date=2006-09-09 }}</ref>
{{-}}
{| class="wikitable collapsible"
!colspan=6|Related tilings of triangles and squares
|-
!snub square
!elongated triangular
!colspan=2| [[Euclidean tilings by convex regular polygons#2-uniform tilings|2-uniform]]
!colspan=3| [[Euclidean tilings by convex regular polygons#3-uniform tilings|3-uniform]]
|- align=center
!p4g, (4*2)
!p2, (2222)
!p2, (2222)
!cmm, (2*22)
!colspan=2|p2, (2222)
|- align=center
|[[File:1-uniform n9.svg|100px]]<BR>[3<sup>2</sup>434]
|[[File:1-uniform n8.svg|100px]]<BR>[3<sup>3</sup>4<sup>2</sup>]
|[[File:2-uniform n17.svg|100px]]<BR>[3<sup>3</sup>4<sup>2</sup>; 3<sup>2</sup>434]
|[[File:2-uniform n16.svg|100px]]<BR>[3<sup>3</sup>4<sup>2</sup>; 3<sup>2</sup>434]
|[[File:3-uniform 53.svg|100px]]<BR>[2: 3<sup>3</sup>4<sup>2</sup>; 3<sup>2</sup>434]
|[[File:3-uniform 55.svg|100px]]<BR>[3<sup>3</sup>4<sup>2</sup>; 2: 3<sup>2</sup>434]
|- align=center
|[[File:Vertex type 3-3-4-3-4.svg|30px]]
|[[File:Vertex type 3-3-3-4-4.svg|30px]]
|[[File:Vertex type 3-3-3-4-4.svg|30px]] [[File:Vertex type 3-3-4-3-4.svg|30px]]
|[[File:Vertex type 3-3-3-4-4.svg|30px]] [[File:Vertex type 3-3-4-3-4.svg|30px]]
|[[File:Vertex type 3-3-3-4-4.svg|30px]] [[File:Vertex type 3-3-3-4-4.svg|30px]] [[File:Vertex type 3-3-4-3-4.svg|30px]]
|[[File:Vertex type 3-3-3-4-4.svg|30px]] [[File:Vertex type 3-3-4-3-4.svg|30px]] [[File:Vertex type 3-3-4-3-4.svg|30px]]
|}

=== Related topological series of polyhedra and tiling===
The ''snub square tiling'' is third in a series of snub polyhedra and tilings with [[vertex figure]] 3.3.4.3.''n''.
{{Snub4 table}}
The ''snub square tiling'' is third in a series of snub polyhedra and tilings with [[vertex figure]] 3.3.''n''.3.''n''.
{{Snub5 table}}

{{Square tiling table}}

==See also==
{{Commons category|Uniform tiling 3-3-4-3-4 (snub square tiling)}}
* [[List of uniform planar tilings]]
* [[Snub (geometry)]]
* [[Snub square prismatic honeycomb]]
* [[Tilings of regular polygons]]
* [[Elongated triangular tiling]]

== 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]
* {{KlitzingPolytopes|flat.htm#2D|2D Euclidean tilings|s4s4s - snasquat - O10}}
* {{cite book | author1=Grünbaum, Branko | author-link=Branko Grünbaum | 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)
*{{The Geometrical Foundation of Natural Structure (book)}} p38
* Dale Seymour and [[Jill Britton]], ''Introduction to Tessellations'', 1989, {{isbn|978-0866514613}}, pp.&nbsp;50–56, dual p.&nbsp;115

==External links==
* {{MathWorld | urlname=SemiregularTessellation | title=Semiregular tessellation}}

{{Tessellation}}

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