View source for Elongated triangular tiling

{{Uniform tiles db|Uniform tiling stat table|Uet}}
In [[geometry]], the '''elongated triangular tiling''' is a [[Tiling by regular polygons|semiregular tiling]] of the Euclidean plane. There are three triangles and two squares on each [[vertex (geometry)|vertex]]. It is named as a [[triangular tiling]] [[elongation (geometry)|elongated]] by rows of squares, and given [[Schläfli symbol]] {3,6}:e.

[[John Horton Conway|Conway]] calls it a '''isosnub quadrille'''.<ref name="Conway, 2008, p.288 table">Conway, 2008, p.288 table</ref>

There are 3 [[List of regular polytopes#Euclidean tilings|regular]] and 8 [[List of uniform tilings|semiregular tilings]] in the plane. This tiling is similar to the [[snub square tiling]] which also has 3 triangles and two squares on a vertex, but in a different order.

== Construction==
It is also the only convex [[uniform tiling]] that can not be created as a [[Wythoff construction]]. It can be constructed as alternate layers of [[apeirogonal prism]]s and [[apeirogonal antiprism]]s.

== Uniform colorings ==

There is one [[uniform coloring]]s of an elongated triangular tiling. Two 2-uniform colorings have a single vertex figure, 11123, with two colors of squares, but are not 1-uniform, repeated either by reflection or glide reflection, or in general each row of squares can be shifted around independently. The 2-uniform tilings are also called [[Archimedean coloring]]s. There are infinite variations of these Archimedean colorings by arbitrary shifts in the square row colorings.

{| class=wikitable
|-
!11122 (1-uniform)
!colspan=2|11123 (2-uniform or 1-Archimedean)
|-
|[[File:Elongated triangular tiling 1.png|150px]] 
|[[File:Elongated triangular tiling 3.png|150px]] 
|[[File:Elongated triangular tiling 2.png|150px]]
|- align=center
|cmm (2*22)
|pmg (22*)
|pgg (22×)
|}

== Circle packing ==
The elongated triangular 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 F</ref> <!--The packing density is (2-{{radic|3}})π ≈ 84.18% coverage.)-->
:[[File:1-uniform-8-circlepack.svg|200px]]

== Related tilings ==

Sections of stacked triangles and squares can be combined into radial forms. This mixes two vertex configurations, 3.3.3.4.4 and 3.3.4.3.4 on the transitions. Twelve copies are needed to fill the plane with different center arrangements. The duals will mix in [[cairo pentagonal tiling]] pentagons.<ref>[https://twitter.com/SumDumThum/status/1074929556203614209 aperiodic tilings by towers] Andrew Osborne 2018</ref>
{| class=wikitable
|+ Example radial forms
!Center||colspan=2|Triangle||colspan=2|Square||colspan=2|Hexagon
|- align=center
![[Dihedral group|Symmetry]]||[3]||[3]<sup>+</sup>||[2]||[4]<sup>+</sup>||[6]||[6]<sup>+</sup>
|- align=center
|align=center|[[File:Tower elonaged triangular tiling.svg|80px]]<BR>Tower
|[[File:Triangular_radial_elonaged_triangular_tiling.svg|160px]]
|[[File:Triangle2_elongated_triangular_tiling.svg|160px]]
|[[File:Square radial elongated triangular tiling.svg|160px]]
|[[File:Square2 radial elongated triangular tiling.svg|160px]]
|[[File:Point_radial_elongated_triangular_tiling.svg|160px]]
|[[File:Spiral_elongated_triangular_tiling.svg|160px]]
|-
|align=center|[[File:Dual tower elongated triangular tiling.svg|80px]]<BR>Dual
|[[File:Dual triangular_radial_elonaged_triangular_tiling.svg|160px]]
|[[File:Dual triangle2_elongated_triangular_tiling.svg|160px]]
|[[File:Dual square radial elongated triangular tiling.svg|160px]]
|[[File:Dual square2 radial elongated triangular tiling.svg|160px]]
|[[File:Dual point_radial_elongated_triangular_tiling.svg|160px]]
|[[File:Dual_spiral_elongated_triangular_tiling.svg|160px]]
|}

=== Symmetry mutations===
It is first in a series of symmetry mutations<ref>[http://www.google.com/search?q=Two-Dimensional+Symmetry+Mutation ''Two Dimensional symmetry Mutations'' by Daniel Huson]</ref> with [[Uniform tilings in hyperbolic plane#Quadrilateral domains|hyperbolic uniform tiling]]s with 2*''n''2 [[orbifold notation]] symmetry, [[vertex figure]] 4.''n''.4.3.3.3, and [[Coxeter diagram]] {{CDD|node|ultra|node_h|n|node_h|ultra|node_1}}. Their duals have hexagonal faces in the hyperbolic plane, with [[face configuration]] V4.''n''.4.3.3.3.
{| class=wikitable
|+ Symmetry mutation 2*n2 of uniform tilings: 4.''n''.4.3.3.3
|-
!4.2.4.3.3.3
!4.3.4.3.3.3
!4.4.4.3.3.3
|-
!2*22
!2*32
!2*42
|-
|[[File:Elongated triangular tiling 4.2.4.3.3.3.png|150px]]
|[[File:Uniform tiling 4.3.4.3.3.3.png|150px]]
|[[File:Hyper 4.4.4.3.3.3a.png|150px]]
|- align=center
|{{CDD|node|infin|node_h|2x|node_h|infin|node_1}}
|{{CDD|node|ultra|node_h|3|node_h|ultra|node_1}} or {{CDD|branch_hh|2a2b-cross|nodes_01}}
|{{CDD|node|ultra|node_h|4|node_h|ultra|node_1}} or {{CDD|label4|branch_hh|2a2b-cross|nodes_01}}
|}

There are four related [[2-uniform tiling]]s, mixing 2 or 3 rows of triangles or squares.<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=2015-06-03 |archive-url=https://web.archive.org/web/20060909053826/http://www.uwgb.edu/dutchs/SYMMETRY/uniftil.htm |archive-date=2006-09-09 |url-status=dead }}</ref>
{| class=wikitable
!Double elongated
!Triple elongated
!Half elongated
!One third elongated
|-
|[[File:2-uniform n4.svg|160px]]
|[[File:2-uniform n3.svg|160px]]
|[[File:2-uniform n14.svg|160px]]
|[[File:2-uniform n15.svg|160px]]
|}

=== Prismatic pentagonal tiling===
{{Infobox face-uniform tiling
|name=Prismatic pentagonal tiling
|image=File:1-uniform 8 dual.svg
|type=[[List of uniform planar tilings|Dual uniform tiling]]
|coxeter={{CDD|node|infin|node_fh|2x|node_fh|infin|node_f1}}<BR>{{CDD|node_fh|infin|node_fh|2x|node_fh|infin|node_f1}}
|symmetry=cmm, [∞,2<sup>+</sup>,∞], (2*22)
|faces=[[pentagon|irregular pentagons]] V3.3.3.4.4 [[File: V3.3.3.4.4.png|53px|right]]
|dual=Elongated triangular tiling
|properties=[[face-transitive]]|}}
The prismatic pentagonal tiling is a [[List of uniform planar tilings|dual uniform tiling]] in the Euclidean plane. It is one of 15 known [[isohedral figure|isohedral]] [[pentagon tiling]]s. It can be seen as a stretched [[hexagonal tiling]] with a set of parallel bisecting lines through the hexagons.

[[John Horton Conway|Conway]] calls it an {{not a typo|iso(4-)pentille}}.<ref name="Conway, 2008, p.288 table"/> Each of its pentagonal [[face (geometry)|faces]] has three 120° and two 90° angles.

It is related to the [[Cairo pentagonal tiling]] with [[face configuration]] V3.3.4.3.4.

==== Geometric variations====

Monohedral [[Pentagonal tiling#Monohedral convex pentagonal tilings|pentagonal tiling]] type 6 has the same topology, but two edge lengths and a lower p2 (2222) [[wallpaper group]] symmetry:
{| class=wikitable
|- align=center
|'''[[File:P5-type6.png|200px]]'''
|'''[[File:Prototile p5-type6.png|200px]]<BR>a=d=e, b=c<BR>B+D=180°, 2B=E'''
|}

====Related 2-uniform dual tilings====

There are four related 2-uniform dual tilings, mixing in rows of squares or hexagons (the prismatic pentagon is half-square half-hexagon).
{| class=wikitable
!Dual: Double Elongated
!Dual: Triple Elongated
!Dual: Half Elongated
!Dual: One-Third Elongated
|-
|[[File:2-uniform 4 dual.svg|160px]]
|[[File:2-uniform 3 dual.svg|160px]]
|[[File:2-uniform 14 dual.svg|160px]]
|[[File:2-uniform 15 dual.svg|160px]]
|-
|Dual: [[Euclidean tilings by convex regular polygons#/media/File:2-uniform n4.svg|[4<sup>4</sup>; 3<sup>3</sup>.4<sup>2</sup><nowiki>]</nowiki><sub>1</sub> (t=2,e=4)]]
|Dual: [[Euclidean tilings by convex regular polygons#/media/File:2-uniform n3.svg|[4<sup>4</sup>; 3<sup>3</sup>.4<sup>2</sup><nowiki>]</nowiki><sub>2</sub> (t=3,e=5)]]
|Dual: [[Euclidean tilings by convex regular polygons#/media/File:2-uniform n14.svg|[3<sup>6</sup>; 3<sup>3</sup>.4<sup>2</sup><nowiki>]</nowiki><sub>1</sub> (t=3,e=4)]]
|Dual: [[Euclidean tilings by convex regular polygons#/media/File:2-uniform n15.svg|[3<sup>6</sup>; 3<sup>3</sup>.4<sup>2</sup><nowiki>]</nowiki><sub>2</sub> (t=4,e=5)]]
|}

==See also==
* [[Tilings of regular polygons]]
* [[Elongated triangular prismatic honeycomb]]
* [[Gyroelongated triangular prismatic honeycomb]]

== Notes ==
{{Reflist}}

== References ==
{{Commons category|Uniform tiling 3-3-3-4-4 (elongated triangular tiling)}}
{{Commons category|Prismatic pentagonal tiling}}
* {{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)}} p37
* 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]
* Keith Critchlow, ''Order in Space: A design source book'', 1970, p.&nbsp;69-61, Pattern Q<sub>2</sub>, Dual p.&nbsp;77-76, pattern 6
* Dale Seymour and [[Jill Britton]], ''Introduction to Tessellations'', 1989, {{ISBN|978-0866514613}}, pp.&nbsp;50–56

==External links==
* {{MathWorld | urlname=UniformTessellation | title=Uniform tessellation}}
* {{MathWorld | urlname=SemiregularTessellation | title=Semiregular tessellation}}
* {{KlitzingPolytopes|flat.htm#2D|2D Euclidean tilings|elong( x3o6o ) - etrat - O4}}

{{Tessellation}}

[[Category:Euclidean tilings]]
[[Category:Isogonal tilings]]
[[Category:Semiregular tilings]]
[[Category:Triangular tilings]]