The truncated hexagonal tiling is a semiregular tiling of the Euclidean plane, featuring a vertex configuration of 3.12.12. It is created by a truncation applied to the hexagonal tiling, replacing hexagons with dodecagons and introducing triangles at original vertices. Its extended Schläfli symbol is t{6,3}, and it exhibits symmetry group p6m. Named the truncated hextille by Conway, this tiling is one of several uniform tilings, and its dual is the Triakis triangular tiling. It is also vertex-transitive, meaning all vertices are equivalent under its symmetries.
Uniform colorings
There is only one uniform coloring of a truncated hexagonal tiling. (Naming the colors by indices around a vertex: 122.)
Topologically identical tilings
The dodecagonal faces can be distorted into different geometries, such as:
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 | ||||||||
|---|---|---|---|---|---|---|---|---|
| Fundamentaldomains | 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} | |
| 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.
*n32 symmetry mutation of truncated tilings: t{n,3}
| |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Symmetry*n32[n,3] | Spherical | Euclid. | Compact hyperb. | Paraco. | Noncompact hyperbolic | ||||||
| *232[2,3] | *332[3,3] | *432[4,3] | *532[5,3] | *632[6,3] | *732[7,3] | *832[8,3]... | *∞32[∞,3] | [12i,3] | [9i,3] | [6i,3] | |
| Truncatedfigures | |||||||||||
| Symbol | t{2,3} | t{3,3} | t{4,3} | t{5,3} | t{6,3} | t{7,3} | t{8,3} | t{∞,3} | t{12i,3} | t{9i,3} | t{6i,3} |
| Triakisfigures | |||||||||||
| Config. | V3.4.4 | V3.6.6 | V3.8.8 | V3.10.10 | V3.12.12 | V3.14.14 | V3.16.16 | V3.∞.∞ | |||
Related 2-uniform tilings
Two 2-uniform tilings are related by dissected the dodecagons into a central hexagonal and 6 surrounding triangles and squares.12
| 1-uniform | Dissection | 2-uniform dissections | |
|---|---|---|---|
| (3.122) | (3.4.6.4) & (33.42) | (3.4.6.4) & (32.4.3.4) | |
| Dual Tilings | |||
O | to DB | 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
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.
Dual uniform hexagonal/triangular tilings| Symmetry: [6,3], (*632) | [6,3]+, (632) | |||||
|---|---|---|---|---|---|---|
| V63 | V3.122 | V(3.6)2 | V36 | V3.4.6.4 | V.4.6.12 | V34.6 |
See also
Wikimedia Commons has media related to Uniform tiling 3-12-12 (truncated hexagonal tiling).- John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 [1]
- 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)
- Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. p. 39. ISBN 0-486-23729-X.
- 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
External links
- Weisstein, Eric W. "Semiregular tessellation". MathWorld.
- Klitzing, Richard. "2D Euclidean tilings o3x6x - toxat - O7".
References
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. https://www.beloit.edu/computerscience/faculty/chavey/catalog/ ↩
"Uniform Tilings". Archived from the original on 2006-09-09. Retrieved 2006-09-09. https://web.archive.org/web/20060909053826/http://www.uwgb.edu/dutchs/SYMMETRY/uniftil.htm ↩
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, ISBN 978-1-56881-220-5 "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) /wiki/ISBN_(identifier) ↩
Inose, Mikio. "mikworks.com : Original Work : Asanoha". www.mikworks.com. Retrieved 20 April 2018. http://www.mikworks.com/originalwork/asanoha/ ↩
Weisstein, Eric W. "Dual tessellation". MathWorld. /wiki/Eric_W._Weisstein ↩
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. /wiki/ArXiv_(identifier) ↩