# Truncated hexagonal tiling

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{{Short description|Semiregular tiling of a plane}}
{{Uniform tiles db|Uniform tiling stat table|Uth}}
In [geometry](/source/geometry), the '''truncated hexagonal tiling''' is a semiregular tiling of the [Euclidean plane](/source/Euclidean_plane). There are 2 [dodecagon](/source/dodecagon)s (12-sides) and one [triangle](/source/triangle) on each [vertex](/source/vertex_(geometry)).

As the name implies this tiling is constructed by a [truncation](/source/Truncation_(geometry)) operation applied to a [hexagonal tiling](/source/hexagonal_tiling), leaving dodecagons in place of the original [hexagon](/source/hexagon)s, and new triangles at the original vertex locations. It is given an extended [Schläfli symbol](/source/Schl%C3%A4fli_symbol) of ''t''{6,3}.

[Conway](/source/John_Horton_Conway) calls it a '''truncated hextille''', constructed as a [truncation](/source/truncation_(geometry)) operation applied to a [hexagonal tiling](/source/hexagonal_tiling) (hextille).

There are 3 [regular](/source/List_of_regular_polytopes) and 8 [semiregular tilings](/source/List_of_uniform_tilings) in the plane.

== Uniform colorings ==

There is only one [uniform coloring](/source/uniform_coloring) of a truncated hexagonal tiling. (Naming the colors by indices around a vertex: 122.)

100px

== Topologically identical tilings==
The [dodecagon](/source/dodecagon)al faces can be distorted into different geometries, such as:
{| class=wikitable
|150px
|150px
|-
|150px
|150px
|}

== Related polyhedra and tilings ==
[[File:Contracted truncated hexagonal tilings.png|thumb|A truncated hexagonal tiling can be contracted in one dimension, reducing dodecagons into decagons. Contracting in second direction reduces decagons into octagons. Contracting a third time make the [trihexagonal tiling](/source/trihexagonal_tiling).]]

=== Wythoff constructions from hexagonal and triangular tilings ===
Like the [uniform polyhedra](/source/Uniform_polyhedron) there are eight [uniform tiling](/source/uniform_tiling)s that can be based from the regular hexagonal tiling (or the dual [triangular tiling](/source/triangular_tiling)).

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms, 7 which are topologically distinct. (The ''truncated triangular tiling'' is topologically identical to the hexagonal tiling.)

{{Hexagonal tiling small table}}

=== Symmetry mutations===
This tiling is topologically related as a part of sequence of uniform [truncated](/source/Truncation_(geometry)) polyhedra with [vertex configuration](/source/vertex_configuration)s (3.2n.2n), and [n,3] [Coxeter group](/source/Coxeter_group) symmetry.

{{Truncated figure1 table}}

=== Related 2-uniform tilings===

Two [2-uniform tiling](/source/2-uniform_tiling)s are related by dissected the [dodecagon](/source/dodecagon)s into a central hexagonal and 6 surrounding triangles and 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=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
!1-uniform
!Dissection
!colspan=2|2-uniform dissections
|- align=center
|200px<br/>(3.12<sup>2</sup>)
|100px100px
|200px<br/>(3.4.6.4) & (3<sup>3</sup>.4<sup>2</sup>)
|200px<br/>(3.4.6.4) & (3<sup>2</sup>.4.3.4)
|-
!colspan=4|Dual Tilings
|- align = center
|alt=
O
|200x200px
|alt=
to DB
|alt=
to DC
|}

=== Circle packing ===
The truncated hexagonal tiling can be used as a [circle packing](/source/circle_packing), placing equal diameter circles at the center of every point.<ref name=Critchlow>Order in Space: A design source book, Keith Critchlow, p.74-75, pattern G</ref> Every circle is in contact with 3 other circles in the packing ([kissing number](/source/kissing_number)). <!--The packing density is ... % coverage.)--> This is the lowest density packing that can be created from a uniform tiling.
:200px

=== Triakis triangular tiling===
{{Infobox face-uniform tiling  |
  name=Triakis triangular tiling |
  Image_File      = Tiling truncated 6 dual simple.svg|
  Type = [Dual semiregular tiling](/source/List_of_uniform_tilings) |
  Cox={{CDD|node|3|node_f1|6|node_f1}} |
  Face_List       = [triangle](/source/triangle) |
  Symmetry_Group  = p6m, [6,3], (*632) |
  Rotation_Group  = p6, [6,3]<sup>+</sup>, (632) |
  Face_Type       = V3.12.1280px|right |
  Dual            = Truncated hexagonal tiling |
  Property_List   = [face-transitive](/source/face-transitive)|
}}
[[File:Wallpaper group-p6m-6.jpg|thumb|On painted [porcelain](/source/porcelain), [China](/source/China)]]
The '''triakis triangular tiling''' is a tiling of the Euclidean plane. It is an equilateral [triangular tiling](/source/triangular_tiling) with each triangle divided into three obtuse triangles (angles 30-30-120) from the center point. It is labeled by [face configuration](/source/face_configuration) V3.12.12 because each isosceles triangle face has two types of vertices: one with 3 triangles, and two with 12 triangles.

[Conway](/source/John_Horton_Conway) calls it a '''kisdeltille''',<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> constructed as a [kis](/source/Conway_kis_operator) operation applied to a [triangular tiling](/source/triangular_tiling) (deltille).

In Japan the pattern is called '''asanoha''' for ''hemp leaf'', although the name also applies to other triakis shapes like the [triakis icosahedron](/source/triakis_icosahedron) and [triakis octahedron](/source/triakis_octahedron).<ref>{{cite web|url=http://www.mikworks.com/originalwork/asanoha/|title=mikworks.com : Original Work : Asanoha|first=Mikio|last=Inose|website=www.mikworks.com|access-date=20 April 2018}}</ref>

It is the dual tessellation of the truncated hexagonal tiling which has one triangle and two dodecagons at each vertex.<ref>{{MathWorld | urlname=DualTessellation | title=Dual tessellation}}</ref>
:320px

It is one of eight [edge tessellation](/source/edge_tessellation)s, tessellations generated by reflections across each edge of a prototile.<ref>{{citation
 | last1 = Kirby | first1 = Matthew
 | last2 = Umble | first2 = Ronald
 | arxiv = 0908.3257
 | doi = 10.4169/math.mag.84.4.283
 | issue = 4
 | journal = Mathematics Magazine
 | mr = 2843659
 | pages = 283–289
 | title = Edge tessellations and stamp folding puzzles
 | volume = 84
 | year = 2011}}.</ref>

==== Related duals to uniform tilings====
It is one of 7 dual uniform tilings in hexagonal symmetry, including the regular duals.
{{Dual_hexagonal_tiling_table}}

== See also ==
{{Commons category|Uniform tiling 3-12-12 (truncated hexagonal tiling)}}
* [Tilings of regular polygons](/source/Tilings_of_regular_polygons)
* [List of uniform tilings](/source/List_of_uniform_tilings)

== 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 | author-link=Branko Grünbaum | 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=39}}
* Keith Critchlow, ''Order in Space: A design source book'', 1970, p.&nbsp;69-61, Pattern E, Dual p.&nbsp;77-76, pattern 1
* Dale Seymour and [Jill Britton](/source/Jill_Britton), ''Introduction to Tessellations'', 1989, {{isbn|978-0866514613}}, pp.&nbsp;50–56, dual p.&nbsp;117

==External links==
* {{MathWorld | urlname=SemiregularTessellation | title=Semiregular tessellation}}
* {{KlitzingPolytopes|flat.htm#2D|2D Euclidean tilings|o3x6x - toxat  - O7}}

{{Tessellation}}

Category:Euclidean tilings
Category:Hexagonal tilings
Category:Isogonal tilings
Category:Semiregular tilings
Category:Truncated tilings

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Adapted from the Wikipedia article [Truncated hexagonal tiling](https://en.wikipedia.org/wiki/Truncated_hexagonal_tiling) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Truncated_hexagonal_tiling?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.