Truncated hexagonal tiling

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

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The dodecagonal faces can be distorted into different geometries, such as:

File:Contracted truncated hexagonal tilings.png.]]

Like the uniform polyhedra there are eight uniform tilings that can be based from the regular hexagonal tiling (or the dual 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}}

This tiling is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (3.2n.2n), and [n,3] Coxeter group symmetry.

{{Truncated figure1 table}}

Two 2-uniform tilings are related by dissected the dodecagons into a central hexagonal and 6 surrounding triangles and squares.{{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= }}{{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 }}

The truncated hexagonal tiling can be used as a circle packing, placing equal diameter circles at the center of every point.Order in Space: A design source book, Keith Critchlow, p.74-75, pattern G Every circle is in contact with 3 other circles in the packing (kissing number). This is the lowest density packing that can be created from a uniform tiling.

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{{Infobox face-uniform tiling |

name=Triakis triangular tiling |

Image_File = Tiling truncated 6 dual simple.svg|

Type = Dual semiregular tiling |

Cox={{CDD|node|3|node_f1|6|node_f1}} |

Face_List = triangle |

Symmetry_Group = p6m, [6,3], (*632) |

Rotation_Group = p6, [6,3]⁺, (632) |

Face_Type = V3.12.12File:Tiling truncated 6 dual face.svg |

Dual = Truncated hexagonal tiling |

Property_List = face-transitive|

}}

File:Wallpaper group-p6m-6.jpg, China]]

The triakis triangular tiling is a tiling of the Euclidean plane. It is an equilateral triangular tiling with each triangle divided into three obtuse triangles (angles 30-30-120) from the center point. It is labeled by 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 calls it a kisdeltille,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) constructed as a kis operation applied to a 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 and triakis octahedron.{{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}}

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

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It is one of eight edge tessellations, tessellations generated by reflections across each edge of a prototile.{{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}}.

It is one of 7 dual uniform tilings in hexagonal symmetry, including the regular duals.

{{Dual_hexagonal_tiling_table}}

{{Commons category|Uniform tiling 3-12-12 (truncated hexagonal tiling)}}

• Tilings of regular polygons
• List of uniform tilings

{{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. 58-65)
• {{The Geometrical Foundation of Natural Structure (book)|page=39}}
• Keith Critchlow, Order in Space: A design source book, 1970, p. 69-61, Pattern E, Dual p. 77-76, pattern 1
• Dale Seymour and Jill Britton, Introduction to Tessellations, 1989, {{isbn|978-0866514613}}, pp. 50–56, dual p. 117

• {{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