Truncated hexagonal tiling

Triakis triangular tiling
Triakis triangular tiling
Type	Dual semiregular tiling
Faces	triangle
Coxeter diagram
Symmetry group	p6m, [6,3], (*632)
Rotation group	p6, [6,3]+, (632)
Dual polyhedron	Truncated hexagonal tiling
Face configuration	V3.12.12
Properties	face-transitive

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 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.)

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
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} v t e
Symmetry *n32 [n,3]	Spherical				Euclid.	Compact hyperb.		Paraco.	Noncompact hyperbolic
Symmetry *n32 [n,3]	*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.∞.∞

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	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

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.
^ "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, 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)
^ Inose, Mikio. "mikworks.com : Original Work : Asanoha". www.mikworks.com. Retrieved 20 April 2018.
^ Weisstein, Eric W. "Dual tessellation". MathWorld.
^ 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, 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