Truncated hexagonal tiling

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Truncated hexagonal tiling
Truncated hexagonal tiling
Type
Semiregular tiling
Vertex configuration
3.12.12
Schläfli symbol t{6,3}
Wythoff symbol 2 3 | 6
Coxeter diagram
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

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.)

Topologically identical tilings

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

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.)

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}
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]
Truncated
figures
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}
Triakis
figures
Config.
V3.4.4
 V3.6.6 V3.8.8 V3.10.10
V3.12.12
 V3.14.14 V3.16.16 V3.∞.∞

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

(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

Triakis triangular tiling
face-transitive
On painted porcelain, China

The triakis triangular tiling is a tiling of the Euclidean plane. It is an equilateral

face configuration
V3.12.12 because each isosceles triangle face has two types of vertices: one with 3 triangles, and two with 12 triangles.

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]

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).
  • Tilings of regular polygons
  • List of uniform tilings

References

  1. doi:10.1016/0898-1221(89)90156-9
    .
  2. ^ "Uniform Tilings". Archived from the original on 2006-09-09. Retrieved 2006-09-09.
  3. ^ Order in Space: A design source book, Keith Critchlow, p.74-75, pattern G
  4. 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)
  5. ^ Inose, Mikio. "mikworks.com : Original Work : Asanoha". www.mikworks.com. Retrieved 20 April 2018.
  6. ^ Weisstein, Eric W. "Dual tessellation". MathWorld.
  7. MR 2843659
    .
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