Elongated triangular tiling

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

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

There are 3 regular and 8 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 prisms and apeirogonal antiprisms.

== Uniform colorings ==

There is one uniform colorings 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 colorings. 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) |- |150px |150px |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>  :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 !Symmetry||[3]||[3]+||[2]||[4]+||[6]||[6]+ |- align=center |align=center|80px Tower |160px |160px |160px |160px |160px |160px |- |align=center|80px Dual |160px |160px |160px |160px |160px |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 hyperbolic uniform tilings 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 |- |150px |150px |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 tilings, mixing 2 or 3 rows of triangles or squares.<ref>{{cite journal | first=D. |last=Chavey | title=Tilings by Regular Polygons—II: A Catalog of Tilings | url=https://www.beloit.edu/computerscience/faculty/chavey/catalog/ | journal=Computers & Mathematics with Applications | year=1989 | volume=17 | pages=147–165 | doi=10.1016/0898-1221(89)90156-9| doi-access= | url-access=subscription }}</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 |- |160px |160px |160px |160px |}

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

Conway calls it an {{not a typo|iso(4-)pentille}}.<ref name="Conway, 2008, p.288 table"/> Each of its pentagonal 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 type 6 has the same topology, but two edge lengths and a lower p2 (2222) wallpaper group symmetry: {| class=wikitable |- align=center |'''200px''' |'''200px a=d=e, b=c 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 |- |160px |160px |160px |160px |- |Dual: [[Euclidean tilings by convex regular polygons#/media/File:2-uniform n4.svg|[44; 33.42<nowiki>]</nowiki>1 (t=2,e=4)]] |Dual: [[Euclidean tilings by convex regular polygons#/media/File:2-uniform n3.svg|[44; 33.42<nowiki>]</nowiki>2 (t=3,e=5)]] |Dual: [[Euclidean tilings by convex regular polygons#/media/File:2-uniform n14.svg|[36; 33.42<nowiki>]</nowiki>1 (t=3,e=4)]] |Dual: [[Euclidean tilings by convex regular polygons#/media/File:2-uniform n15.svg|[36; 33.42<nowiki>]</nowiki>2 (t=4,e=5)]] |}

== 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. 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. 69-61, Pattern Q2, Dual p. 77-76, pattern 6 * Dale Seymour and Jill Britton, ''Introduction to Tessellations'', 1989, {{ISBN|978-0866514613}}, pp. 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}}

Category:Euclidean tilings Category:Isogonal tilings Category:Semiregular tilings Category:Triangular tilings