Truncated hexagonal tiling
| Truncated hexagonal tiling | |
|---|---|
| Truncated hexagonal tiling | |
| Type | Semiregular tiling |
| Vertex configuration | File:Tiling truncated 6 vertfig.svg 3.12.12 |
| Schläfli symbol | t{6,3} |
| Wythoff symbol | 2 3 | 6 |
| Coxeter diagram | Template:CDD |
| Symmetry | p6m, [6,3], (*632) |
| Rotation symmetry | p6, [6,3]+, (632) |
| Bowers acronym | Toxat |
| Dual | Triakis triangular tiling |
| Properties | Vertex-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.)
File:Uniform polyhedron-63-t01.png
Topologically identical tilings
The dodecagonal faces can be distorted into different geometries, such as:
| File:Truncated hexagonal tiling0.svg | File:Gyrated truncated hexagonal tiling.svg |
| File:Gyrated truncated hexagonal tiling3.svg | File:Gyrated truncated hexagonal tiling2.svg |
Related polyhedra and tilings
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.)
| Uniform hexagonal/triangular tilings | ||||||||
|---|---|---|---|---|---|---|---|---|
| Fundamental domains |
Symmetry: [6,3], (*632) | [6,3]+, (632) | ||||||
| {6,3} | t{6,3} | r{6,3} | t{3,6} | {3,6} | rr{6,3} | tr{6,3} | sr{6,3} | |
| Template:CDD | Template:CDD | Template:CDD | Template:CDD | Template:CDD | Template:CDD | Template:CDD | Template:CDD | |
| File:Tiling Dual Semiregular V4-6-12 Bisected Hexagonal.svg | File:Uniform tiling 63-t0.svg | File:Uniform tiling 63-t01.svg | File:Uniform tiling 63-t1.svg | File:Uniform tiling 63-t12.svg | File:Uniform tiling 63-t2.svg | File:Uniform tiling 63-t02.svg | File:Uniform tiling 63-t012.svg | File:Uniform tiling 63-snub.svg |
| Config. | 63 | 3.12.12 | (6.3)2 | 6.6.6 | 36 | 3.4.6.4 | 4.6.12 | 3.3.3.3.6 |
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.[1][2]
| 1-uniform | Dissection | 2-uniform dissections | |
|---|---|---|---|
| File:1-uniform n4.svg (3.122) |
File:Regular dodecagon.svgFile:Hexagonal cupola flat.svg | File:2-uniform n8.svg (3.4.6.4) & (33.42) |
File:2-uniform n9.svg (3.4.6.4) & (32.4.3.4) |
| Dual Tilings | |||
| File:1-Uniform O.png
O |
File:Inset Variations of Dual Uniform Tiling.svg | File:O Inset to DB.gif
to DB |
File:O Inset to DC.gif
to DC |
Circle packing
The truncated hexagonal tiling can be used as a circle packing, placing equal diameter circles at the center of every point.[3] 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.
Triakis triangular tiling
Page Module:Infobox/styles.css has no content.
| Triakis triangular tiling | |
|---|---|
| Lua error in package.lua at line 80: module 'Module:InfoboxImage/data' not found. | |
| Type | Dual semiregular tiling |
| Coxeter diagram | Template:CDD |
| Wallpaper group | p6m, [6,3], (*632) |
| Rotation group | p6, [6,3]+, (632) |
| Dual | Truncated hexagonal tiling |
| Face configuration | V3.12.12 |
| Properties | face-transitive |
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,[4] 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.[5]
It is the dual tessellation of the truncated hexagonal tiling which has one triangle and two dodecagons at each vertex.[6]
It is one of eight edge tessellations, tessellations generated by reflections across each edge of a prototile.[7]
Related duals to uniform tilings
It is one of 7 dual uniform tilings in hexagonal symmetry, including the regular duals.
See also
Page Module:Side box/styles.css has no content.Page Template:Sister project/styles.css has no content.
References
Page Template:Reflist/styles.css has no content.
- ^ Page Module:Citation/CS1/styles.css has no content.Chavey, D. (1989). "Tilings by Regular Polygons—II: A Catalog of Tilings". Computers & Mathematics with Applications. 17: 147–165. doi:10.1016/0898-1221(89)90156-9.
- ^ Page Module:Citation/CS1/styles.css has no content."Uniform Tilings". Archived from the original on 2006-09-09. Retrieved 2006-09-09.
- ^ Order in Space: A design source book, Keith Critchlow, p.74-75, pattern G
- ^ John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, Template:Isbn Page Module:Citation/CS1/styles.css has no content."A K Peters, LTD. - the Symmetries of Things". Archived from the original on 2010-09-19. Retrieved 2012-01-20. (Chapter 21, Naming Archimedean and Catalan polyhedra and tilings, p288 table)
- ^ Page Module:Citation/CS1/styles.css has no content.Inose, Mikio. "mikworks.com : Original Work : Asanoha". www.mikworks.com. Retrieved 20 April 2018.
- ^ Script error: No such module "Template wrapper".
- ^ Page Module:Citation/CS1/styles.css has no content.Kirby, Matthew; Umble, Ronald (2011), "Edge tessellations and stamp folding puzzles", Mathematics Magazine, 84 (4): 283–289, arXiv:0908.3257, doi:10.4169/math.mag.84.4.283, MR 2843659.
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, Template:Isbn [1]
- Page Module:Citation/CS1/styles.css has no content.Grünbaum, Branko & Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1. (Chapter 2.1: Regular and uniform tilings, p. 58-65)
- Script error: No such module "SST".
- 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, Template:Isbn, pp. 50–56, dual p. 117
External links
- Script error: No such module "Template wrapper".
- Page Module:Citation/CS1/styles.css has no content.Klitzing, Richard. "2D Euclidean tilings o3x6x - toxat - O7".
Lua error in package.lua at line 80: module 'Module:Navbox/configuration' not found.
