Truncated hexagonal tiling
Semiregular tiling of a plane
title: "Truncated hexagonal tiling" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["euclidean-tilings", "hexagonal-tilings", "isogonal-tilings", "semiregular-tilings", "truncated-tilings"] description: "Semiregular tiling of a plane" topic_path: "general/euclidean-tilings" source: "https://en.wikipedia.org/wiki/Truncated_hexagonal_tiling" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
::summary Semiregular tiling of a plane ::
::data[format=table title="Infobox face-uniform tiling |"]
| Field | Value |
|---|---|
| name | Triakis triangular tiling |
| Image_File | Tiling truncated 6 dual simple.svg |
| Type | Dual semiregular tiling |
| Cox | |
| Face_List | triangle |
| Symmetry_Group | p6m, [6,3], (*632) |
| Rotation_Group | p6, [6,3]+, (632) |
| Face_Type | V3.12.12[[File:Tiling truncated 6 dual face.svg |
| Dual | Truncated hexagonal tiling |
| Property_List | face-transitive |
| :: |
In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. There are 2 dodecagons (12-sides) and one triangle on each vertex.
As the name implies this tiling is constructed by a truncation operation applied to a hexagonal tiling, leaving dodecagons in place of the original hexagons, and new triangles at the original vertex locations. It is given an extended Schläfli symbol of t{6,3}.
Conway calls it a truncated hextille, constructed as a truncation operation applied to a hexagonal tiling (hextille).
There are 3 regular and 8 semiregular tilings in the plane.
Uniform colorings
There is only one uniform coloring of a truncated hexagonal tiling. (Naming the colors by indices around a vertex: 122.)
::figure[src="https://upload.wikimedia.org/wikipedia/commons/6/61/Uniform_polyhedron-63-t01.png"] ::
Topologically identical tilings
The dodecagonal faces can be distorted into different geometries, such as: ::data[format=table] | [[File:Gyrated truncated hexagonal tiling3.svg|150px]] | [[File:Gyrated truncated hexagonal tiling2.svg|150px]] | |---|---| ::
Related polyhedra and tilings
::figure[src="https://upload.wikimedia.org/wikipedia/commons/c/c3/Contracted_truncated_hexagonal_tilings.png" caption="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]]."] ::
Wythoff constructions from hexagonal and triangular tilings
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.)
Symmetry mutations
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.
Related 2-uniform tilings
Two 2-uniform tilings are related by dissected the dodecagons into a central hexagonal and 6 surrounding triangles and squares. ::data[format=table]
| 1-uniform | Dissection | 2-uniform dissections | Dual Tilings |
|---|---|---|---|
| [[File:1-uniform n4.svg | 200px]] | ||
| (3.122) | [[File:Regular dodecagon.svg | 100px]][[File:Hexagonal cupola flat.svg | 100px]] |
| (3.4.6.4) & (33.42) | [[File:2-uniform n9.svg | 200px]] | |
| (3.4.6.4) & (32.4.3.4) | |||
| [[File:1-Uniform O.png | alt=]] | [[File:Inset_Variations_of_Dual_Uniform_Tiling.svg | 200x200px]] |
| :: |
Circle packing
The truncated hexagonal tiling can be used as a circle packing, placing equal diameter circles at the center of every point. 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. :[[File:1-uniform-4-circlepack.svg|200px]]
Triakis triangular tiling
name=Triakis triangular tiling | Image_File = Tiling truncated 6 dual simple.svg| Type = Dual semiregular tiling | Cox= | Face_List = triangle | Symmetry_Group = p6m, [6,3], (*632) | Rotation_Group = p6, [6,3]+, (632) | Face_Type = V3.12.12[[File:Tiling truncated 6 dual face.svg|80px|right]] | Dual = Truncated hexagonal tiling | Property_List = face-transitive| ::figure[src="https://upload.wikimedia.org/wikipedia/commons/3/31/Wallpaper_group-p6m-6.jpg" caption="On painted [[porcelain]], [[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, 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.
It is the dual tessellation of the truncated hexagonal tiling which has one triangle and two dodecagons at each vertex. :[[File:P4 dual.png|320px]]
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}}.
Related duals to uniform tilings
It is one of 7 dual uniform tilings in hexagonal symmetry, including the regular duals.
References
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, https://web.archive.org/web/20100919143320/https://akpeters.com/product.asp?ProdCode=2205
- (Chapter 2.1: Regular and uniform tilings, p. 58-65)
- 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, , pp. 50–56, dual p. 117
References
- Chavey, D.. (1989). "Tilings by Regular Polygons—II: A Catalog of Tilings". Computers & Mathematics with Applications.
- "Uniform Tilings".
- Order in Space: A design source book, Keith Critchlow, p.74-75, pattern G
- "A K Peters, LTD. - the Symmetries of Things".
- Inose, Mikio. "mikworks.com : Original Work : Asanoha".
- "Dual tessellation".
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