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

Vertex figure: 3.12.12

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 applies 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.)

Related polyhedra and tilings

The dodecagonal faces can be distorted into different geometries, like:

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.

Dimensional family of truncated spherical polyhedra and tilings: 3.2n.2n
Symmetry
*n32
[n,3]
Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
*232
[2,3]
D3h
*332
[3,3]
Td
*432
[4,3]
Oh
*532
[5,3]
Ih
*632
[6,3]
P6m
*732
[7,3]

*832
[8,3]...

*∞32
[∞,3]

[12i,3] [9i,3] [6i,3] [3i,3]
Figures
Schläfli 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} t{3i,3}
Coxeter
Dual
figures
Triakis
figures

V3.4.4

V3.6.6

V3.8.8

V3.10.10

V3.12.12

V3.14.14

V3.16.16

V3.∞.∞
Coxeter

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
Symmetry: [6,3], (*632) [6,3]+
(632)
[1+,6,3]
(*333)
[6,3+]
(3*3)
{6,3} t{6,3} r{6,3}
r{3[3]}
t{3,6}
t{3[3]}
{3,6}
{3[3]}
rr{6,3}
s2{6,3}
tr{6,3} sr{6,3} h{6,3}
{3[3]}
h2{6,3}
r{3[3]}
s{3,6}
s{3[3]}

=

=

=
=
or
=
or

=

Uniform duals
V63 V3.122 V(3.6)2 V63 V36 V3.4.12.4 V.4.6.12 V34.6 V36 V(3.6)2 V36

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. The dodecagonal gaps can be filled perfectly with 7 circles, creating a denser 3-uniform packing.

Triakis triangular tiling

Triakis triangular tiling
Type Dual semiregular tiling
Coxeter diagram
Faces triangle
Face configuration V3.12.12
Symmetry group p6m, [6,3], (*632)
Rotation group p6, [6,3]+, (632)
Dual Truncated hexagonal tiling
Properties face-transitive
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 kisdeltile,[1] 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.[2]

It is the dual tessellation of the truncated hexagonal tiling which has one triangle and two dodecagons at each vertex.[3]